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 |
|---|---|---|---|---|
153,186 | <p>How can I prove that the double negation elimination is not provable in constructive logic?<br>
To clarify, double negation elimination is the following statement: </p>
<p>$$\neg\neg q \rightarrow q$$</p>
| Joseph Vidal-Rosset | 62,171 | <p>A very simple and efficient tree proof method (i.e. a tableau method without signed formulae) for intuitionistic logic has been published by Bell, DeVidi and Solomon in "Logical Options: An Introduction to Classical and Alternative Logics"
<a href="http://books.google.fr/books/about/Logical_Options.html?id=zUVYx-bT... |
130,235 | <p>I need to find </p>
<p>$$f(n) = \int^\infty_0 t^{n-1} e^{-t} dt$$</p>
<p>So I think I find the indefinate integral first? But what do I do with $n$, since I am integrating with respect to $t$?</p>
| Pedro | 23,350 | <p>This integral can be easily computed by means of integration by parts. Let's call it $$\Gamma(n) =\int\limits_0^\infty t^{n-1}e^{-t}dt$$</p>
<p>We want to show that $\Gamma(n)$ follows some law. </p>
<p>Let's integrate by parts, with $u=t^{n-1}$. Then </p>
<p>$$\int\limits_0^\infty t^{n-1}e^{-t}dt=\left.-e^{-t}t^... |
1,615,201 | <p>I just solved a long problem for my physics w/calculus homework that required a simplification using a quadratic formula. The "textbook" (flipItPhysics) came up with a different simplification than mine but it turns out they are equivalent. I can't, for the life of me, figure out how to simplify mine into theirs.</p... | mweiss | 124,095 | <p>Notice that both expressions have a factor of $d$ (you have it in the numerator, and the textbook has it as the first factor), and both expressions have a factor of $q-Q$ in the denominator. So let's ignore those and compare what's left. We need to show that
$$q \pm \sqrt{Qq}$$
and
$$q \cdot \left(1 \pm \sqrt{\fra... |
2,599,978 | <p>This is a homework problem I have.</p>
<blockquote>
<p>Let a random variable <span class="math-container">$x$</span> follow a Gaussian distribution with mean <span class="math-container">$\mu = 10$</span> and variance <span class="math-container">$1$</span>.</p>
<p>Which of the following samples of size 4 has the la... | Jack D'Aurizio | 44,121 | <p>$$f(a,b,c)=\int_{-\pi/2}^{\pi/2}\left[\sin(x)-ax^2-bx-c\right]^2\cos(x)\,dx $$
is a regular function. By equating to zero $\frac{\partial f}{\partial b}$ you get $b=\frac{\pi}{2(\pi^2-8)}$, then by defining $g(a,c)=f\left(a,\frac{\pi}{2(\pi^2-8)},c\right)$ and equating to zero $\frac{\partial g}{\partial a}$ and $\f... |
2,599,978 | <p>This is a homework problem I have.</p>
<blockquote>
<p>Let a random variable <span class="math-container">$x$</span> follow a Gaussian distribution with mean <span class="math-container">$\mu = 10$</span> and variance <span class="math-container">$1$</span>.</p>
<p>Which of the following samples of size 4 has the la... | Disintegrating By Parts | 112,478 | <p>The expression $\langle f,g\rangle = \int_{-\pi/2}^{\pi/2}f\overline{g}\cos x dx$ does define a complex inner product on $C[-\pi,\pi]$. The problem can then be restated as finding scalars $\alpha,\beta,\gamma$ such that $\|\sin x-\alpha -\beta x-\gamma x^2\|$ is minimized, which is achieved iff $(\sin x -\alpha-\bet... |
2,982,205 | <p>I need help with a question that asks this:</p>
<p>Let <span class="math-container">$f(x)= -1/x$</span> and <span class="math-container">$g(x)=e^x$</span> find the domain , range ,monotonicity intervals. Draw the graph of <span class="math-container">$g(f(x))$</span>. </p>
<p>Everything else except the graph is p... | Dr. Sonnhard Graubner | 175,066 | <p>I have got <span class="math-container">$$\overline{h}h^2+\overline{h}z^2+2h\overline{h}z+h^2\overline{z}+2hz\overline{z}$$</span></p>
|
100,433 | <p>I am trying to export some largish integer arrays to HDF5 and know that every entry in them would fit into an unsigned 8 bit integer array. As the default "DataFormat" for export that Mathematica is using is a 32bit integer array the resulting files are unnecessarily large. Does anyone know the correct syntax to exp... | ScotMartin | 42,448 | <p>You can see:</p>
<p><a href="https://github.com/scotmartin1234/HDF5Mathematica" rel="nofollow">https://github.com/scotmartin1234/HDF5Mathematica</a></p>
<p>The package has the functionality you need for writing at low level. The high level functions are designed more for reading. The package could be expanded as o... |
3,398,164 | <p>I am working through the math behind homographies, but my math skills are a bit rusty.</p>
<p>A homography can be calculated with 8 corresponding points (4-4) because the homography matrix has 8 degrees of freedom. This is because, <a href="https://docs.opencv.org/master/d9/dab/tutorial_homography.html#lecture_16" ... | dan_fulea | 550,003 | <p>The following discussion may be an explanation, but if it is not, please try to ask where there is something hard to catch.</p>
<p>My strategy is always to simplify things, but keep the problem related (and slightly simpler, but essentially the same). Here, i propose to understand homographies from a (projective) l... |
3,646,694 | <p>Let <span class="math-container">$M$</span> be an elliptic element of <span class="math-container">$SL_2(\mathbb R)$</span>. Then it is conjugate to a rotation <span class="math-container">$R(\theta)$</span>. Note that we can calculate <span class="math-container">$\theta$</span> in terms of the trace of <span class... | Quanto | 686,284 | <p>Note </p>
<p><span class="math-container">$$\alpha (\alpha^2+2\alpha -4)\beta =-1$$</span></p>
<p>Thus,</p>
<p><span class="math-container">$$\beta =- \frac1{\alpha (\alpha^2+2\alpha -4)}
=- \frac1{\alpha^3+2\alpha^2-4\alpha}
=- \frac1{5\alpha -1 -4\alpha} =\frac1{1-\alpha}
$$</span></p>
|
569,814 | <p>I was watching a video on YouTube on Quantum Mechanics Concepts and saw that if you wanted to convert a probability amplitude to a probability, you square it. In the video he said that this was equivalent to multiply by it's complex conjugate. So is this correct? Is squaring the same as multiplying by the complex co... | Clive Newstead | 19,542 | <p>A probability is a <em>real</em> number (between $0$ and $1$). The magnitude of a complex number $z$ is $|z|$, which is real. Now $z \bar z = |z|^2$, so multiplying by the complex conjugate gives the square of its <em>magnitude</em>, not the square of the complex number itself.</p>
<p>For example, $i^2 = -1$ but $|... |
354,512 | <p>Find the minimal polynomial of the matrix M:
\begin{pmatrix}
0 & 0 & 0 & \dots & 0 & a_{1}\\
1 & 0 & 0 & \dots & 0 & a_{2}\\
0 & 1 & 0 & \dots & 0 & a_{3} \\
\dots & \dots \\
0 & 0 & 0 & \dots & 1 & a_{n}
\end{pmatrix}</p>
<p>Let's ... | user1551 | 1,551 | <p>Consider a monic polynomial $q(x) = b_nx^n + b_{n-1}x^{n-1} + b_{n-2}x^{n-2} + ... + b_1x + b_0$ of $M$ of degree $m\le n$ (if $m<n$, we have $b_m=1$ and $b_{m+1}=b_{m+2}=\cdots=b_n=0$) and let $\{e_1,e_2,\ldots,e_n\}$ be the standard basis of $\mathbb{R}^n$. If $q(M)=0$, we have, in particular,
$$q(M)e_1=b_n\sum... |
468,510 | <p>Given $n^2$, how many fourth powers $(x^4)$ are between 0 and $n^2$? </p>
<p>$n,x\in \mathbb{Z}$</p>
<p>Does this just reduce down to how many squares are below $n$?</p>
| Brian M. Scott | 12,042 | <p>Yes, it does: assuming that $x$ is a non-negative integer, $0\le x^4\le n^2$ if and only if $0\le x^2\le n$. You can express the number of squares exactly using the floor (greatest integer) function; I’ve left the answer spoiler-protected below.</p>
<blockquote class="spoiler">
<p> $\left\lfloor\sqrt{n}\right\rfl... |
1,650,638 | <p>I have a sequence of reals $S = s_1,s_2,\dots,s_n$ such that $s_i-s_{i-1}$ is a Gaussian distribution. From histogram of sequence $S$ (10000 elements) it appears that it is uniform distribution. Is it true ? If yes, can we prove it ? If no, what can we say about $s_i$ ?</p>
| Laurent Duval | 257,503 | <p>My interpretation is that diagonalization, when possible, simplifies the interpretation (and subsequent computations) of a matrix. In the eigenbasis, operations can be performed "coordinate-wise", or basis vector by basis vector, almost independently. </p>
<p>Suppose you have a vector $x^a = (x_1,x_2,\dots,x_n)$, ... |
1,373,927 | <p>We know that a tree with n edges have n+1 nodes.So if $|B_{n+1}|$ is the number of all possible ordered trees with n+1 nodes then its true that $C_{n+1} = |B_{n+1}|$ where $C$ is the Catalan number.Let it be $|L_k|$ the number of all possible ordered trees with n+1 nodes and k leaves and Its true that
$|B_{n+1}| = ... | Marko Riedel | 44,883 | <p>Here is a contribution using basic complex variables.
<P></p>
<p>We will compute the number of trees on <span class="math-container">$n$</span> nodes and having <span class="math-container">$q$</span>
leaves.</p>
<p>The combinatorial class equation for ordered rooted trees with leaves marked is
<span cla... |
2,006,437 | <p>Is it possible to get an example of two matrices $A,B\in M_4(\mathbb{R})$ both having $rank<2$ but $\det(A-\lambda B)\ne 0$ i.e it is not identically a zero polynomial. where $\lambda$ is indeterminate, I mean a variable. I want to say $(A-\lambda B)$ is of full rank matrix, assuming entries of $A-\lambda B$ is a... | Daniel McLaury | 3,296 | <p>No. If $\operatorname{rank} A < 2$ then either $A = 0$ or $\operatorname{rank} A = 1$, and similarly for $B$. Obviously if $A = 0$ then
$$\det(A - \lambda B) = - \lambda \det B = 0$$
since $B$ is not full-rank, and similarly if $B = 0$.</p>
<p>So suppose $\operatorname{rank} A = \operatorname{rank} B = 1$. Th... |
3,143,532 | <blockquote>
<p>Find <span class="math-container">$\det A$</span> and <span class="math-container">$\text{Tr} A$</span> for the matrix <span class="math-container">$A\in M_n(\mathbb{Q})$</span> such that <span class="math-container">$\sqrt[n]{p}$</span> is an eigenvalue of <span class="math-container">$A$</span>, whe... | Zacky | 515,527 | <p>We start off by some <span class="math-container">$x\rightarrow \frac{1}{x}$</span> substitutions while derivating under the integral sign: <span class="math-container">$$I(\theta)=\int_0^\infty \log \left(1-2\frac{\cos 2\theta}{x^2}+\frac{1}{x^4} \right)dx\overset{x\rightarrow \frac{1}{x}}=\int_0^\infty \frac{\ln(1... |
3,143,532 | <blockquote>
<p>Find <span class="math-container">$\det A$</span> and <span class="math-container">$\text{Tr} A$</span> for the matrix <span class="math-container">$A\in M_n(\mathbb{Q})$</span> such that <span class="math-container">$\sqrt[n]{p}$</span> is an eigenvalue of <span class="math-container">$A$</span>, whe... | Random Variable | 16,033 | <p>Assume that <span class="math-container">$\theta \in (0, \pi).$</span></p>
<p>Then</p>
<p><span class="math-container">$$\begin{align}\int_{0}^{\infty} \log \left(1 -\frac{2 \cos (2\theta)}{x^{2}} +\frac{1}{x^{4}} \right) \, \mathrm dx &= \int_{0}^{\infty} \frac{ \log \left(1- 2 \cos(2 \theta) u^{2} +u^{4}\right... |
4,367,330 | <p>Let <span class="math-container">$X_1, X_2,...$</span> be a sequence of independent uniform random variables on <span class="math-container">$(0,1)$</span>. Define: <span class="math-container">$$N := \text{min} \{n\geq 2: X_n < X_{n-1}\}.$$</span> Calculate <span class="math-container">$E(N)$</span>.</p>
<p>I th... | drhab | 75,923 | <p>Observe that: <span class="math-container">$$N>n\iff X_1\leq X_2\leq\cdots\leq X_n\text{ for }n=2,3,\dots$$</span>so that by symmetry: <span class="math-container">$$\Pr(N>n)=\frac1{n!}\text{ for }n=2,3,\dots$$</span>
Now apply:<span class="math-container">$$\mathbb EN=\sum_{n=0}^{\infty}\Pr(N>n)$$</span><... |
412,051 | <p>Suppose
$X_i \sim N(0,1)$ (independent, identical normal distributions) </p>
<p>Then by Law of large number,
$$
\sqrt{1-\delta} \frac{1}{n}\sum_i^\infty e^{\frac{\delta}{2}X_i^2} \rightarrow \sqrt{1-\delta} \int e^{\frac{\delta}{2}x^2}\frac{1}{\sqrt{2\pi}}e^{\frac{1}{2}x^2}dx =1
$$</p>
<p>However, according to si... | Avitus | 80,800 | <p>Your general solution is not correct.
The differential equation you are dealing with is the Laplace equation.</p>
<p>For the general theory of Laplace equation, please have a look at the introductory exposition in</p>
<p><a href="http://en.wikipedia.org/wiki/Laplace" rel="nofollow">http://en.wikipedia.org/wiki/La... |
222,312 | <p>How to prove that if you have $x^*$ such that $x^*=\text{psuedoinverse}(A) b$, and $Ay=b$, then
$$\Vert x^* \Vert_2 \leq \Vert y \Vert_2$$</p>
| chaohuang | 27,973 | <p>Let $A: \mathbb{U} \mapsto \mathbb{V}$, then in terms of SVD, we can write $A$ as $$A=\sum_{n=1}^R\sigma_nv_nu_n^{\dagger},$$ where $\sigma_n$ is a nonzero singular value; $u_n$ and $v_n$ are the right and left singular vector, respectively; $R$ is the rank of $A$.</p>
<p>Since {$u_n$} form an orthonormal basis of ... |
577,490 | <p>I wonder if I have approached this in the right way. I'm not sure if I have interpreted the question correctly, or made correct use of the successor function. Thank you in advance</p>
<blockquote>
<p><strong>Question:</strong> Prove that $¬∃n∅ = n^+$</p>
</blockquote>
<p>My approach:</p>
<p>Prove that $¬∃n∅... | MarnixKlooster ReinstateMonica | 11,994 | <p>Here is how I would approach this, resulting in a proof which is in essence the same as the one from the first answer.</p>
<p>In a slightly different notation, you're asked to prove $\;\lnot \langle \exists n :: n^+ = \emptyset \rangle \;$.</p>
<p>Let's see what we can do with $\;n^+ = \emptyset\;$ :
\begin{align}... |
2,433,765 | <p><a href="https://i.stack.imgur.com/AolxA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AolxA.png" alt="enter image description here"></a></p>
<p>In the following diagram of a triangle, $\overline{AB} = \overline{BC} = \overline{CD}$ and $\overline{AD} = \overline{BD}$. Find the measure of angle... | Edward Porcella | 403,946 | <p>$\triangle ABD$ is isosceles with each base angle double the angle at the vertex. The angle at D is therefore $36^o$. Euclid constructs this triangle in <em>Elements</em>,IV,10, as necessary for construction of the regular pentagon. </p>
|
778,140 | <p>I want to draw a box plot, which requires that I know the median, the lower and upper quartiles, and the minimum and maximum values of my data.</p>
<p>I understand that the quartiles are simply the value on certainly "percentile" of the cumulative frequency of the data.</p>
<p>So lower quartile = the value of the... | Thomas Rasberry | 265,575 | <p>Intro textbooks tend to use one of two different methods for boxplot/five number summary construction, in my experience. </p>
<p>The first method is to apply your percentile to the total number of observations; for example, the first quartile of 14 data points ordered least to greatest is the data value located in ... |
17,103 | <p>I'm reading some very old papers (by Birch et al) on quadratic forms and I don't get the following point: </p>
<blockquote>
<p>If <span class="math-container">$f$</span> is a quadratic form in <span class="math-container">$X_1,X_2,\cdots,X_n$</span> over a
finite field, then one can change variables such that <... | Pete L. Clark | 1,149 | <p>Let me try a similar explanation with different words. (Note that my explanation does not cover characteristic <span class="math-container">$2$</span>.)</p>
<p>A quadratic form is <strong>nondegenerate</strong> if any of its associated symmetric matrices has nonzero determinant. (Alternately, if the associated bil... |
2,552,720 | <p>How would you prove algebraically that the assertion $\exists (x,y) \in (0,1)^2. x +y > 1 + xy$ <a href="http://www.wolframalpha.com/input/?i=x%20%2B%20y%20%3E%20%201%20%2B%20xy,%200%20%3C%20x%20%3C%201,%200%20%3C%20y%20%3C%201" rel="nofollow noreferrer">is false</a>?</p>
| nonuser | 463,553 | <p>$$L-R =x+y-1-xy = x(1-y)-(1-y)= (1-y)(x-1)<0$$</p>
|
2,552,720 | <p>How would you prove algebraically that the assertion $\exists (x,y) \in (0,1)^2. x +y > 1 + xy$ <a href="http://www.wolframalpha.com/input/?i=x%20%2B%20y%20%3E%20%201%20%2B%20xy,%200%20%3C%20x%20%3C%201,%200%20%3C%20y%20%3C%201" rel="nofollow noreferrer">is false</a>?</p>
| PunkZebra | 120,252 | <p>I'll just work assuming that $x$ and $y$ can't equal $1$, otherwise the inequality isn't true for $x=0$ and $y=1$.</p>
<p>Squaring them both gets rid of the $xy$ term
$$x+y>1+xy$$
$$x^2+2xy+y^2>1+2xy+x^2y^2$$
$$x^2+y^2>1+x^2y^2$$
Now $x^2+y^2$ is a positive term less or equal to $1$ since it's inside the u... |
2,895,129 | <blockquote>
<p>Let $f(x) \leq g(x)$ for all $x$ in an open set $D$, and let $a$ be an interior point of $D$ such that $f(a) = g(a)$ and both $f, g$ are differentiable at $a$.
Does it follows that $f'(a)=g'(a)$?</p>
</blockquote>
<p>Is that right, and if so, why? If not then please give examples.</p>
| Jack M | 30,481 | <p>Fundamentally, the intuition here is that if two curves touch each other ($f(a)=g(a)$) and have different derivatives at that point, then one has to actually <em>cross</em> the other. If we are to have $f\leq g$ on the entire interval, then the only way those two curves can touch is if they just "kiss", tangent to e... |
3,621,836 | <p>Let's say I have a large determinant with scalar elements:</p>
<p><span class="math-container">$$\begin{vmatrix} x\cdot a & x\cdot b & x\cdot c & x\cdot d \\ x\cdot e & x\cdot f & x\cdot g & x\cdot h \\ x\cdot i & x\cdot g & x\cdot k & x\cdot l \\ x\cdot m & x\cdot n & x\... | lhf | 589 | <p>Since <span class="math-container">$\omega^3=1$</span> and <span class="math-container">$\omega^2+\omega+1=0$</span>, we have
<span class="math-container">$$
\begin{align}
(2-\omega)(2-\omega^2)(2-\omega^{19})(2-\omega^{23})
&=
(2-\omega)(2-\omega^2)(2-\omega^{1})(2-\omega^{2})
\\&=
((2-\omega)(2-\omega^2))^... |
77,290 | <p>How would you determine all integers $m$ such that the following is true? </p>
<p>$$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$</p>
<p>Note that $\lfloor \cdot \rfloor$ means the greatest integer function. Also, $x$ must be a positive real number.</p>
| Robert Israel | 8,508 | <p>If $k + j/10 \le x < k + (j+1)/10$ where $k$ and $j$ are nonnegative integers and $j \le 9$, $\lfloor 2x \rfloor = \begin{cases} 2k & 0 \le j \le 4 \\ 2k+1 & 5 \le j \le 9 \end{cases}$ while $\lfloor 5x \rfloor = \begin{cases} 5k + j/2 & j \ \text{even} \\ 5k + (j-1)/2 & j \ \text{odd} \end{cases... |
2,082,062 | <p>In Avner Friendman's Mondern Analysis book, he makes a statement that has stumped me when proving that for $\{f_n\}$ a sequence of measurable functions, $\sup_n f_n$ And $\inf f_n$ Are measurable.</p>
<blockquote>
<p>The assertion for $\inf_n f_n $ follows from $\inf (f_n )=- \sup (-f_n)$ .</p>
</blockquote>
<p>... | copper.hat | 27,978 | <p>It is straightforward to check that if $L_1, L_2$ are linear operators then
so is $L_1 \circ L_2$.</p>
<p>It is straightforward to check that $n(x) = -x$ is a linear functional.</p>
<p>Hence given some linear $L$, the map $n \circ L$ is linear.</p>
|
254,926 | <p>I was working on a problem involving perturbation methods and it asked me to sketch the graph of $\ln(x) = \epsilon x$ and explain why it must have 2 solutions. Clearly there is a solution near $x=1$ which depends on the value of $\epsilon$, but I fail to see why there must be a solution near $x \rightarrow \infty$... | Spenser | 39,285 | <p>Right after the first solution, you have that $\ln x>\epsilon x \Rightarrow \ln x-\epsilon x>0$. However,
$$\lim_{x\to\infty}(\ln x-\epsilon x)=\lim_{x\to\infty}\ln\frac{x}{e^{\epsilon x}}\to-\infty
$$
since $\frac{x}{e^{\epsilon x}}\to 0$. (Assuming $\epsilon>0$)</p>
<p>Hence, $\ln x-\epsilon x$, must cr... |
4,517,597 | <p>Suppose <span class="math-container">$\{f_n\}_{n \in \mathbb{N}}$</span> is a family of bounded, differentiable, monotone increasing functions on <span class="math-container">$[0,1]$</span>, which converge uniformly to a limit <span class="math-container">$f$</span>. Also, suppose we know that <span class="math-con... | Jacob Manaker | 330,413 | <p>Yes. I will prove this via functional analysis; I do not know if there is a simpler method.</p>
<ol>
<li>Note that <span class="math-container">$f_n$</span> are uniformly bounded, since they converge uniformly.</li>
<li>Suppose <span class="math-container">$m\leq f_n\leq M$</span> for any <span class="math-containe... |
1,136,109 | <p>Suppose that $a,b$ are reals such that the roots of $ax^3-x^2+bx-1=0$ are all positive real numbers. Prove that: </p>
<p>$(i)~~0\le 3ab\le 1$<br>
$(ii)~~b\ge \sqrt3$. </p>
<p>My attempt: </p>
<p>I could solve the first part by Vieta's theorem. But, I am stuck on the second part. Please help. Thank you.</p>
| Macavity | 58,320 | <p>Let $x, y, z > 0$ be the three roots. Then, $x+y+z = xyz = \dfrac1a$ and $xy+yz+zx = \dfrac{b}a$</p>
<p>$(i),\quad $ Clearly, $a, b > 0$. Also $(x+y+z)^2 \ge 3(xy+yz+zx) \implies \dfrac1{a^2} \ge 3\dfrac{b}a \implies 1 \ge 3ab$. </p>
<p>For $(ii),\quad (xy+yz+zx)^2 \ge 3xyz(x+y+z) \implies \dfrac{b^2}{a^2}... |
478,912 | <p>Ok so I get the basics of this, I just can't put it all together. $U$ is contained in the cartesian product of $\mathbb R^n \times \mathbb R$. What is $\mathbb R^n \times \mathbb R$ though? I know what a cartesian product is, but this seems weird to me. Someone please help explain this part, thanks!</p>
| azimut | 61,691 | <p>$\mathbb R^n\times \mathbb R$ is the set of all pairs
whose first component comes from the set $\mathbb R^n$ and the second component comes from the set $\mathbb R$.
So
$$U = (\mathbf v, x)$$
with $\mathbf v\in\mathbb R^n$ and $x\in \mathbb R$.</p>
<p>Since $\mathbb R^n$ is the set of all $n$-tuples over $\mathbb R... |
28,348 | <p>Thomson et al. provide a proof that <span class="math-container">$\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$</span> in <a href="http://classicalrealanalysis.info/documents/TBB-AllChapters-Landscape.pdf#page=95" rel="noreferrer">this book (page 73)</a>. It has to do with using an inequality that relies on the binomial... | robjohn | 13,854 | <p>Let $n$ be an integer $n>2$ and real $x>0$, the binomial theorem says
$$
(1+x)^n>1+nx+\frac{n(n-1)}{2}x^2
$$
Let $N(x)=\max(2,1+\frac{2}{x^2})$. For $n>N(x)$, we get that $\frac{n(n-1)}{2}x^2>n$. Thus, for any $x>0$, we get that for $n>N(x)$
$$
1<\sqrt[n]{n}<1+x
$$
Thus, we have
$$
1... |
2,694,122 | <blockquote>
<p>Given that a function $f$ is defined as $$f(x)=1+2x+3x^2+4x^3+...$$. We have to prove that $f$ is continuous on $[0,\frac{1}{8}]$ and evaluate $$\int_{0}^{\frac{1}{8}}f(x)dx$$ </p>
</blockquote>
<p>I am not sure in a particular area. I have proved $f$ is continuous on $[0,\frac{1}{8}]$. But the evalu... | operatorerror | 210,391 | <p>Another approach: Evaluate the series first, then integrate
$$
f(x)=\sum_{n=0}^\infty (n+1)x^n=\sum_{n=0}^\infty nx^n+
\sum_{n=0}^\infty x^n\\
=x\sum_{n=0}^\infty nx^{n-1}+\frac{1}{1-x}\\
=x\frac{1}{(1-x)^2}+\frac{1}{1-x}
$$
Now both integrating and checking continuity (it's the sum of two rational functions, neithe... |
3,477,532 | <p>In the cyclic quadrilateral <span class="math-container">$ABCD$</span>, <span class="math-container">$AB:BC:CD:DA=1:9:9:8$</span>, <span class="math-container">$AC$</span> intersects <span class="math-container">$BD$</span> at <span class="math-container">$P$</span>, what is <span class="math-container">$S_{\triangl... | Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/9UAPd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9UAPd.png" alt="enter image description here"></a></p>
<p>Note,</p>
<p><span class="math-container">$$\frac{PB}{PD}=\frac{S_{\triangle ABC}}{S_{\triangle ACD}}
= \frac{\frac12\cdot AB\cdot BC\cdot \sin B}{\... |
2,268,225 | <p>I want to investigate the convergence behavior of $$\int_{0}^{\infty} \cos(x^r)\, dx \hspace{5mm} \textrm{and} \hspace{5mm} \int_{\pi}^{\infty} \left(\frac{\cos x}{\log x}\right)\arctan\lfloor x\rfloor dx.$$ My theoretical tools are Abel's test and Dirichlet's test:
Say I have an integral of the form $$\int_{a}^{b... | Hypergeometricx | 168,053 | <p>$$\begin{align}
S&=\qquad \frac 14+\frac 28+\frac 3{16}+\frac 4{32}+\frac 5{64}+\cdots\tag{1}\\
2S&=\frac 12+\frac 24+\frac 38+\frac 4{16}+\frac 5{32}\cdots\tag{2}\\
(2)-(1):\qquad\\
S&=\frac 12+\frac 14+\frac 18+\frac 1{16}+\frac 1{32}\cdots\\
&=\frac {\frac 12}{1-\frac 12}\\
&=\color{red}1
\end... |
2,240,616 | <p>Let $f:[0,1] \to \mathbb{R}$ be a function such that for every $a \in [0,1)$ and $b \in (0,1]$ the one-sided limits $$f(a^+)=\lim _{x\to a^+}f(x) \in \mathbb{R}$$ $$f(b^-)=\lim _{x \to b^-} f(x) \in \mathbb {R}$$ exist. </p>
<p>A) Show that $f$ is bounded. </p>
<p>B) Does $f$ necessarily achieve its maximum at som... | Community | -1 | <p>Such an $f$ is called a regulated function. The space of step functions is dense in the space of regulated functions for $\|\cdot\|_{\infty}$, so $\exists \phi$ a step function $[0,1] \to \Bbb R$ such that $\|\phi - f\|_{\infty} \le 1$. Thus $\|f\|_{\infty} \le 1 + \|\phi\|_{\infty} < \infty$.</p>
<p>For $B$, co... |
4,517,063 | <p>Prove that there are exactly <span class="math-container">$8100$</span> different ways of distributing <span class="math-container">$4$</span> indistinguishable black marbles and <span class="math-container">$6$</span> distinguishable coloured marbles ( none of them black) into <span class="math-container">$5$</span... | Cathedral | 933,575 | <p>We can break this problem down into simply arranging the distinguishable marbles in 5 distinguishable boxes(such that no box gets more than 2 marbles), and then using the indistinguishable ones a "filler" of sorts.</p>
<p>We can do this by considering that the distinguishable marbles can be arranged in 3 w... |
211,297 | <p>Let $ \mathbb{F} $ be an uncountable field. Suppose that $ f: \mathbb{F}^{2} \rightarrow \mathbb{F} $ satisfies the following two properties:</p>
<ol>
<li>For each $ x \in \mathbb{F} $, the function $ f(x,\cdot): y \mapsto f(x,y) $ is a polynomial function on $ \mathbb{F} $.</li>
<li>For each $ y \in \mathbb{F} $, ... | Ewan Delanoy | 15,381 | <p>As shown is Gerry Myerson’s answer, the answer is NO when $\mathbb F$ is countably infinite.</p>
<p>The answer is YES when $\mathbb F$ is uncountable, however.</p>
<p>Sketch of proof : since there are only countably many degrees, the polynomials will share a common degree on an uncountable set. This bound on the d... |
2,417,029 | <p>I have been told that a line segment is a set of points.
How can even infinitely many point, each of length zero, can make a line of positive length?</p>
<p>Edit:
As an undergraduate I assumed it was due to having uncountably many points.
But the Cantor set has uncountably many elements and it has measure $0$.</p>
... | Chappers | 221,811 | <p>This may seem rather a strange thing to say, but I don't think it's helpful to think of lines as made up of points: the "lininess" of a line is an inherent property that points don't have, so it has some extra qualities that points don't, such as length.</p>
<p>The real numbers are basically the answer to the quest... |
2,417,029 | <p>I have been told that a line segment is a set of points.
How can even infinitely many point, each of length zero, can make a line of positive length?</p>
<p>Edit:
As an undergraduate I assumed it was due to having uncountably many points.
But the Cantor set has uncountably many elements and it has measure $0$.</p>
... | Readin | 278,069 | <p>If you have a full period to use, and your students are bright, you could explain the why integers are countably infinite while irrationals are uncountably infinite. Once you'v done that, you can point out how adding up the lengths of a countably infinite number of points does indeed get you zero (0+0+0..=0) but tha... |
1,072,302 | <p>let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?</p>
<blockquote>
<blockquote>
<p>(a). If $g$ is continuous, then $f\circ g$ is continuous.</p>
<p>(b). If $f$ is continuous, then $f\circ g$ is continuous.</p>
<p>(c). If... | ajotatxe | 132,456 | <p>If $f$ is strictly increasing, we can guarantee that $f$ has, at most, countably infinitely many points of discontinuity. But strict monotonicity does not imply continuity at all. Consider $f(x)=x/2$ for $x<1/2$ and $f(x)=(1+x)/2$ for $x\ge1/2$.</p>
|
1,072,302 | <p>let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?</p>
<blockquote>
<blockquote>
<p>(a). If $g$ is continuous, then $f\circ g$ is continuous.</p>
<p>(b). If $f$ is continuous, then $f\circ g$ is continuous.</p>
<p>(c). If... | Ben Grossmann | 81,360 | <p>Only statement (c) is true.</p>
<p><strong>Hint:</strong> if $f$ is continuous and strictly increasing, it has a continuous (left-)inverse $f^{-1}$. Consider $f^{-1} \circ f \circ g$.</p>
<hr>
<p>Counterexamples, in no particular order:</p>
<ul>
<li>$f(x) = x$, $g(x)$ is discontinuous</li>
<li>$f$ is increasing... |
545,771 | <blockquote>
<p>let sequence $\{a_{n}\}$,such $a_{1}=0,a_{2}=2,a_{3}=5$,and for $n\in N^{+}$,such
$$\begin{cases}
a_{2n}=2n+2a_{n}\\
a_{2n+1}=2n+1+a_{n}+a_{n+1}
\end{cases}$$
How can I find the closed form of $\{a_{n}\}$?</p>
</blockquote>
<p>My try:
we have</p>
<blockquote>
<p>$$a_{2n+1}-a_{2n}=a_{n+1}-a_{n}... | gvo | 78,233 | <p>Take care, if I had one more line, you'll see that we cannot sum like you did : </p>
<p>$$a_{2n+1}-a_{2n}=a_{n+1}-a_{n}+1$$
$$a_{2n}-a_{2n-1}=a_{n}-a_{n-1}+1$$
$$a_{2n-1}-a_{2n-2}=a_{n}-a_{n-1}+1$$ and not
$$a_{2n-1}-a_{2n-2}= a_{n-1}-a_{n-2}+1$$
$\cdots\cdots\cdots$
$$a_{3}-a_{2}=a_{2}-a_{1}+1$$</p>
<p>And the e... |
545,771 | <blockquote>
<p>let sequence $\{a_{n}\}$,such $a_{1}=0,a_{2}=2,a_{3}=5$,and for $n\in N^{+}$,such
$$\begin{cases}
a_{2n}=2n+2a_{n}\\
a_{2n+1}=2n+1+a_{n}+a_{n+1}
\end{cases}$$
How can I find the closed form of $\{a_{n}\}$?</p>
</blockquote>
<p>My try:
we have</p>
<blockquote>
<p>$$a_{2n+1}-a_{2n}=a_{n+1}-a_{n}... | Greg Martin | 16,078 | <p>By calculating the first few dozen values by hand, one notices the pattern that
$$
a_{n+1} = a_n + (k+2) \quad\text{for all } 2^k \le n < 2^{k+1}.
$$
This is quickly verified by induction on $k$ from the original recursion; for example,
$$
a_{2n+1}-a_{2n} = (2n+1+a_n+a_{n+1}) - (2n+2a_n) = 1 + (a_{n+1}-a_n).
$$
F... |
2,938,491 | <p>I have been pondering this question for a little while, and unfortunately the Google has not given me an answer.</p>
<p>I understand that for example you had a table or graph that crosses or touches the x axis at say x= -2, 0, 3 you could form an equation as</p>
<p>f(x) = ax(x+2)(x-3)</p>
<p>Then solve for a and ... | Doug M | 317,162 | <p><span class="math-container">$\sum_\limits{i=0}^n 2(i + 1)^2 = 2\sum_\limits{i=1}^{n+1} i^2$</span></p>
<p>Which gets to the meat of the question, what is <span class="math-container">$\sum_\limits{i=1}^n i^2$</span>?</p>
<p>There are a few ways to do this. I think that this one is intuitive.
<a href="https://i.... |
2,843,337 | <blockquote>
<p>If $0 \lt x \lt \dfrac{\pi}{2}$, prove that
$$x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$$</p>
</blockquote>
<p>This question must be done without calculus. First, I tried splitting it into the intervals $(0,\pi/4)$ and $(\pi/4, \pi/2)$, hoping that, $\sin x$ was bound tightly enough on the interva... | mengdie1982 | 560,634 | <p>Let $$\alpha=\sqrt{x^3},~~~~\beta=\sqrt{9-x^3};~~~~u=\sin x,~~~~ v=\cos x; ~~~~~~~~\.$$</p>
<p>You may see that $\alpha,\beta,u,v>0$ when $x \in (0,\pi/2)$, and$$\alpha^2+\beta^2=9;~~~~~u^2+v^2=1.$$</p>
<p>Thus, by Cauchy's inequality,$$x^{3/2}\sin x + \sqrt{9-x^3}\cos x=\alpha u+\beta v \leq \sqrt{(\alpha^2+\b... |
394,489 | <p>$$\int^L_{-L} x \sin(\frac{\pi nx}{L})$$</p>
<p>I've seen something like this in Fourier theory, but I'm still not sure how to approach this integral. Wolfram Alpha gives me the answer, but no method. Integrate by parts? Substitution?</p>
| gt6989b | 16,192 | <p>Integrate by parts letting $u = x, dv = \sin(ax) dx$, and so $du = dx, v = -\cos (ax)/a$ and you have</p>
<p>$$\int x \sin(ax) dx = \int u \cdot dv = uv - \int v \cdot du
= \frac{-x\cos(ax)}{a} + \frac{1}{a} \int \cos(ax) dx,$$</p>
<p>which should be much easier.</p>
|
148,612 | <p>You are given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of the sheet, what is the ratio of length to breadth of the sheet?</p>
<p>This is my first question on this site, so if ... | Community | -1 | <p>$\hskip 2.2in$ <img src="https://i.stack.imgur.com/vLPeY.png" alt="enter image description here"></p>
<p>The above figure was done using grapher on mac osx.</p>
<p>Let $l$ be the length (i.e. the sides $AD$ and $BC$) and $b$ be the breadth (i.e. the sides $AB$ and $CD$). Once you get the diagonal vertices together... |
162,364 | <p>Assuming we have a list below, it has real number elements and complex numbers, how can I quickly find if the list has any Real number that its value is less than 1.0? </p>
<pre><code> lis = {3 Sqrt[354], Sqrt[2962], Sqrt[2746], 3 Sqrt[282], Sqrt[2338],
Sqrt[2146], 3 Sqrt[218], Sqrt[1786], Sqrt[1618], 27 Sqrt[2]... | kglr | 125 | <pre><code>Pick[lis, Internal`RealValuedNumericQ@# && # < 1 & /@ lis]
</code></pre>
<blockquote>
<p>{}</p>
</blockquote>
<pre><code>Pick[lis, Internal`RealValuedNumericQ @ # && # < 5 & /@ lis]
</code></pre>
<blockquote>
<p>{3 Sqrt[2], 1.09545, 1.02956}</p>
</blockquote>
<p><code>Pi... |
407,289 | <p>Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - <span class="math-container">$\mathbb{R}$</span> and <span class="math-container">$\mathbb{C}$</span>?</p>
| Arthur Parker | 436,511 | <p>Finite-dimensional manifolds are locally compact, and the only non-discrete locally compact topological fields are the reals and the complex numbers. So the answer is yes.</p>
|
1,143,161 | <p>My question is inspired by the structure of Royden's <em>Real Analysis</em>, which introduces measure theory and Lebesgue integration for $\mathbb{R}$ in its Part I and then reconstructs large portions of those mathematical apparatuses in greater generality by constructing (versions of) them for general metric space... | PVAL-inactive | 83,337 | <p>For $\Bbb R^n$ with the Euclidean metric the open $\epsilon$-ball around a point $p$, $$B=\{x\in \Bbb R^n\ |\ d(x,p)<\epsilon\}$$
has closure
$$\bar B= \{x\in \Bbb R^n\ |\ d(x,p)\leq \epsilon\}$$ </p>
<p>The analogous statement when $\Bbb R^n$ is replaced by a general metric space is false even for relatively n... |
1,143,161 | <p>My question is inspired by the structure of Royden's <em>Real Analysis</em>, which introduces measure theory and Lebesgue integration for $\mathbb{R}$ in its Part I and then reconstructs large portions of those mathematical apparatuses in greater generality by constructing (versions of) them for general metric space... | spin | 12,623 | <p>$\mathbb{R}^n$ is a complete inner product space where closed balls are compact.</p>
<p>Topologically it is path-connected, separable, locally compact, non-compact and non-discrete.</p>
<p>All of these properties can fail to hold for metric spaces.</p>
|
68,147 | <p>While I was investigating some specific types of prime numbers I have faced with the following infinite sequence :</p>
<p>$1,2,8,9,15,20,26,38,45,65,112,244,303,393,560,....$</p>
<p>I tried to find recursive formula using Maple and it's listtorec command, so up to $393$ I got the next output:</p>
<p>$ f(n+3) = ((... | JavaMan | 6,491 | <p>Try the <a href="http://oeis.org/search?q=,2,8,9,15,20,26,38,45,65,112,244,303,393,560&language=english&go=Search" rel="nofollow">Online Encyclopedia of Integer Sequences</a>.</p>
|
4,543,203 | <p>I am going through Achim Klenke's Probability Theory textbook. In Section 7.4, he discusses what it means for two measures to be singular to one another and gives the following example.</p>
<p>Let <span class="math-container">$\Omega = \{0, 1\}^\mathbb{N}$</span> and let <span class="math-container">$(\mathrm{Ber}_p... | Mason | 752,243 | <p>Consider the sequence space under the measure <span class="math-container">$P_r = \text{Ber}(r)^{\otimes \mathbb{N}}$</span>. By the strong law of large numbers, <span class="math-container">$$P_r(\limsup_{n \to \infty}\frac{X_1 + \dots + X_n}{n} = r) = 1.$$</span>
So <span class="math-container">$A_r = \{\limsup_{n... |
183,237 | <blockquote>
<p>Does</p>
<p><span class="math-container">$$\int_{-\infty}^\infty \text{e}^{\ a\ (x+b)^2}\ \text dx=\int_{-\infty}^\infty \text{e}^{\ a\ x^2}\ \text dx\ \ \ \ \ ?$$</span></p>
<p>hold, even if the imaginary part of <span class="math-container">$b$</span> is nonzero?</p>
</blockquote>
<p>What I really wan... | Fabian | 7,266 | <p>Let us write $b= r+it$. The real part of $b$ does not matter as you have already proven yourself. So wlog $r=0$.</p>
<p>For shifting along the imaginary axis, we have to employ the residue theorem. We have
$$
\begin{align}
\int_{-\infty}^\infty f(x+i t) \,dx&- \int_{-\infty}^\infty f(x)\, dx\\
&=\int_{-\inf... |
481,017 | <blockquote>
<p>Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$.</p>
</blockquote>
<p>Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the text of the problem does not specify further. </p>
<p><strong>Possibly Helpful L... | Christian Blatter | 1,303 | <p>We are looking for maps $f$ satisfying $$f\circ f=h\ ,\tag{1}$$ where $h$ is given by
$$h:\quad x\mapsto h(x):=x^2-2\ .$$
The map $h$ has the two fixed points $p=-1$, $q=2$ with $h'(p)=-2$, $h'(q)=4$.</p>
<p>Writing $x=2+t$ with a new coordinate $t$ the map $h$ assumes the form $$h:\ t\mapsto 4t+t^2\ .\tag{2}$$
By ... |
481,017 | <blockquote>
<p>Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$.</p>
</blockquote>
<p>Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the text of the problem does not specify further. </p>
<p><strong>Possibly Helpful L... | user3791713 | 519,190 | <p>The solution that I have seen, I think due <a href="https://en.wikipedia.org/wiki/John_Horton_Conway" rel="nofollow noreferrer">to John Horton Conway</a>, is
$$f(x) = 2 \cos\Big(\sqrt2 \arccos\frac x2\Big)$$
This was alleged to work for $-2\leqslant x\leqslant 2$.</p>
<p>$$\begin{align*}
f(f(x))&= 2 \cos\Big(\... |
3,671,941 | <p>In <a href="https://sites.math.northwestern.edu/~mpopa/571/chapter3.pdf" rel="nofollow noreferrer">this chapter</a>, Example 2.1 notes that we can write <span class="math-container">$\mathbb{Z}_p$</span> as <span class="math-container">$$p\mathbb{Z}_p\cup(1+p\mathbb{Z}_p)\cup\cdots\cup (p-1+p\mathbb{Z}_p).$$</span> ... | Lubin | 17,760 | <p>I think that in this case, a very good way to think is that <span class="math-container">$p\Bbb Z_p$</span> is a <em>subgroup</em> of <span class="math-container">$\Bbb Z_p$</span>, indeed of index <span class="math-container">$p$</span>, since <span class="math-container">$\Bbb Z_p/p\Bbb Z_p\cong\Bbb F_p$</span>, t... |
28,166 | <p>With the initial conditions: $a>b>0$;</p>
<p>I need to find $$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}.$$</p>
<p>I tried to block the equation left and right in order to use the Squeeze (sandwich, two policemen and a drunk, choose your favourite) theorem.</p>
| Isaac | 72 | <p>Suppose the limit exists and is $$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}=L.$$ Then,
$$\begin{align}
\log L
&=\log\lim_{n\to\infty}\sqrt[n]{a^n-b^n}
\\
&=\lim_{n\to\infty}\log\sqrt[n]{a^n-b^n}
\\
&=\lim_{n\to\infty}\frac{1}{n}\log(a^n-b^n)
\\
&=\lim_{n\to\infty}\frac{\log(a^n-b^n)}{n}.
\end{alig... |
28,166 | <p>With the initial conditions: $a>b>0$;</p>
<p>I need to find $$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}.$$</p>
<p>I tried to block the equation left and right in order to use the Squeeze (sandwich, two policemen and a drunk, choose your favourite) theorem.</p>
| AD - Stop Putin - | 1,154 | <p>Here is a short solution based on standard inequalities.</p>
<p>Our first inequality is obvious since $b^n>0$
$$(1)\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad a^n-b^n\leq a^n.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$$ </p>
<p>Next we n... |
3,489,376 | <p>Let <span class="math-container">$C$</span> be a category. Then by definition, for very ordered triple <span class="math-container">$A,B,C$</span> of objects, there is a law of composition of morphisms, i.e., a map <span class="math-container">$$Hom_C(A,B)\times Hom_C(B,C)\longrightarrow Hom_C(A,C)$$</span> where <s... | SCappella | 199,074 | <p>No. If <span class="math-container">$Hom_C(A,B)$</span> and <span class="math-container">$Hom_C(B,C)$</span> are inhabited, then <span class="math-container">$Hom_C(A,B) \times Hom_C(B,C)$</span> is inhabited. This forces <span class="math-container">$Hom_C(A,C)$</span> to be inhabited as well. If <span class="math-... |
2,145,412 | <p>If you have a vector line m with equation $r = i + j + 2k + s( 3i + j - k)$
how do you find the normal?</p>
<p>If you presume that normal is $n = ( n i + o j + p k )$ then</p>
<p>$$3n + o -p = 0 $$</p>
<p>but this leaves you with too many options $\left \lbrace (0,0,0), (1,0,3), (0,1,1), ....\right \rbrace$ is i... | Adren | 405,819 | <p>Let $\mathbb{P}$ the set of all prime numbers.</p>
<p>It can be shown that if $2^p-1\in\mathbb{P}$ (which implies that $p\in\mathbb{P}$), then $2^{p-1}(2^p-1)$ is perfect. This was proved by Euclid (and those numbers are sometime called Euclid's numbers).</p>
<p>Conversely, any <em>even</em> perfect number is an E... |
2,525,903 | <blockquote>
<p>Given $x^2+x-a<3$, solve for $a$ so that there is at least one solution $<0$</p>
</blockquote>
<p>I have the idea how to solve this problem but my solution seems weird and unfinished.</p>
<p>Firstly, I move $3$ to the left hand side.</p>
<p>The graphic of the solution is open upwards and has ... | Nosrati | 108,128 | <p>\begin{align}
\int\sec^3x\,dx
&= \int\dfrac{1}{\cos^3x}\,dx \\
&= \int\dfrac{\cos x}{(1-\sin^2x)^2}\,dx \hspace{0.5cm};\hspace{0.5cm} \sin x=u\\
&= \int\dfrac{1}{(1-u^2)^2}\,dx \\
&= \dfrac{u}{2(1-u^2)}+\dfrac{1}{4}\ln\dfrac{1+u}{1-u}+C\\
&= \dfrac{u}{2(1-u^2)}+\dfrac{1}{4}\ln\dfrac{1+u}{1-u}+C\... |
2,525,903 | <blockquote>
<p>Given $x^2+x-a<3$, solve for $a$ so that there is at least one solution $<0$</p>
</blockquote>
<p>I have the idea how to solve this problem but my solution seems weird and unfinished.</p>
<p>Firstly, I move $3$ to the left hand side.</p>
<p>The graphic of the solution is open upwards and has ... | Michael Hardy | 11,667 | <p>Here's one way:
\begin{align}
\int \sec^3 x \, dx = {} & \int (\sec x) \Big( \sec^2 x \, dx\Big) = \overbrace{\int u\, dv = uv - \int v \, du}^{\text{integration by parts}} \\[10pt]
= {} & \sec x \tan x - \int (\tan x) \big( \sec x\tan x\, dx\big) \\[10pt]
= {} & \sec x \tan x - \int(\sec x)(\tan^2 x) \,... |
2,840,091 | <blockquote>
<p>Consider the linear map$:T:\mathbb{R}^3 → \mathbb{R}$ with
$$T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right)=x-2y-3z$$
Find the basis of its kernel.</p>
</blockquote>
<p><strong>My try</strong></p>
<p>Since the plane is the nullspace of the matrix
$$A=\begin{bmatrix} 1 & -2 & -3 ... | Surb | 154,545 | <p>Set $x=1$ and solve $0=x-2y-3z$ for $y$ with $z=0$ and vice versa. Once you have found $y',z'$ satisfying $0=1-2y'$ and $0=1-3z'$, then you know that $f(1,y',0)=f(1,0,z')=0$ and so you have two vectors in the kernel of $f$. Verify that they are linearly independent. Finally, as $\dim(\Bbb R^3)=\dim(Im(f))+\dim(\ker(... |
1,773,060 | <p>Is every (non-trivial) quotient of a Boolean algebra isomorphic to a subalgebra of that Boolean algebra? And conversely is every subalgebra isomorphic to a quotient algebra?</p>
| Doug McLellan | 250,373 | <p>This question gets right to the inner workings of Boolean algebras so I'm surprised nobody has answered it. I don't know the answer in general and would love to see a proof of a positive answer, or a counterexample!</p>
<p>For what it's worth I think I can show the answer is "yes" in both directions for <em>finite<... |
2,497,799 | <h2>Question</h2>
<p>While Solving a recursive equation , i am stuck at this summation and unable to move forward.Summation is </p>
<blockquote>
<p>$$\sum_{j=0}^{n-2}2^j (n-j)$$</p>
</blockquote>
<h2>My Approach</h2>
<blockquote>
<p>$$\sum_{j=0}^{n-2}2^j (n-j) = \sum_{j=0}^{n-2}2^j \times n-\sum_{j=0}^{n-2}
2^... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Consider $$\sum_{i=0}^p i x^i=x\sum_{i=0}^p i x^{i-1}=x\left(\sum_{i=0}^p x^i \right)'$$</p>
|
1,413,955 | <p>I get confused when I put the following three notes together:</p>
<ol>
<li>Power set of any set is a $\sigma$-algebra.</li>
<li>If $X$ is a set and $\Sigma$ is a $\sigma$-algebra over $X$, then the pair $(X, \Sigma)$ is a measurable space.</li>
<li><a href="https://en.wikipedia.org/wiki/Vitali_set" rel="nofollow">V... | supinf | 168,859 | <p>the reason for your confusion could be, that you forgot some conditions for 3.</p>
<p>Vitali sets are a counterexample that there is no measure on all the subsets of $\mathbb R$ with the following conditions:</p>
<ol>
<li>the measure of an open interval (a,b) should be b-a</li>
<li>the measure should be invariant ... |
1,842,448 | <p>$F(x)= 2x^2 -3x$. </p>
<p>find the range of $x$ to check whether the function the is strictly increasing and strictly decreasing.</p>
| haqnatural | 247,767 | <p><strong>Hint</strong> :
Write it as follows $$ 2{ x }^{ 2 }-3x=2{ \left( x-\frac { 3 }{ 4 } \right) }^{ 2 }-\frac { 9 }{ 8 } $$</p>
|
2,127,110 | <p>I am working on this differetiation problem:</p>
<p>$ \frac{d}{dx}x(1-\frac{2}{x})$</p>
<p>and I am currently stuck at this point:</p>
<p>$1\cdot \left(1-\frac{2}{x}\right)+\frac{2}{x^2}x$</p>
<p>Symbolab tells me this simplifies to $1$ but I do not understand how. I am under the impression that;</p>
<p>$1\cdot... | Kanwaljit Singh | 401,635 | <p>$1\cdot \left(1-\frac{2}{x}\right)+\frac{2}{x^2}x $</p>
<p>= $1\cdot 1- 1 \cdot \frac{2}{x} + \frac{2}{x}$</p>
<p>= $1- \frac{2}{x} + \frac{2}{x}$</p>
<p>=$1$</p>
|
588,228 | <p>A,B,C are distinct digits of a three digit number such that </p>
<pre><code> A B C
B A C
+ C A B
____________
A B B C
</code></pre>
<p>Find the value of A+B+C.</p>
<p>a) 16 b) 17 c) 18 d) 19</p>
<... | Community | -1 | <p><em>EDIT</em> I just realized this question was asked over a year ago, my bad. </p>
<p>I assume that the question was a typo, at least I hope it was. </p>
<pre><code> A B C
B C A
C A B
___________
... |
549,505 | <p>I am trying to show the open mapping theorem. As I was trying to prove it, I made the following conjecture:</p>
<blockquote>
<p>Let $X$ be a complete metric space. Let $V_1,V_2,...$ be a sequence of
open sets in $X$ such that: $$V_1\supseteq \overline{V_2}\supseteq
V_2\supseteq \overline{V_3}\supseteq V_3\sups... | Neal | 20,569 | <p>No, the conjecture need not be true. Consider $V_i = (i,\infty)\subset\mathbb{R}$. Then:
$$(1,\infty)\supseteq [2,\infty)\supseteq (3,\infty)\supseteq \cdots$$
but (by the Archimedean principle) their intersection is empty.</p>
<p>If $X$ is bounded but not complete, the conjecture still does not hold: Consider the ... |
549,505 | <p>I am trying to show the open mapping theorem. As I was trying to prove it, I made the following conjecture:</p>
<blockquote>
<p>Let $X$ be a complete metric space. Let $V_1,V_2,...$ be a sequence of
open sets in $X$ such that: $$V_1\supseteq \overline{V_2}\supseteq
V_2\supseteq \overline{V_3}\supseteq V_3\sups... | Stefan Smith | 55,689 | <p>Question 1 : false . take $X = \mathbb{R}$, $V_n = (-\infty, -n)$.</p>
|
1,515,645 | <p>The question is to find the values of a real number $\lambda$ for which the following equation is satisfied for all real values of $\alpha$ which are not integral multiples of $\pi/2$
$${\sin\lambda\alpha\over \sin\alpha}-{\cos\lambda\alpha\over \cos\alpha}=\lambda-1$$</p>
<p>All I could do was to guess some values... | james_d | 248,598 | <p>$$
1111111111111111 = \sum_0^{15} 10^ i
$$</p>
<p>Since 17 is prime, the 16 values $10^0$, $10^1$, ..., $10^{15}$ are distinct and non-zero mod 17. (Otherwise we have $10^m = 17k$ for some integers $k$, $0\leq m < 17$ in which 17 is a factor of the RHS but not of the LHS.)</p>
<p>Thus </p>
<p>$$
11111111111111... |
259,431 | <p>In the book of Richard Hammack, I come accross with the following question:</p>
<blockquote>
<p>There are two different equivalence relations on the set $A = \{a,b\}$.
Describe them.</p>
</blockquote>
<p>OK, I found that the solution is,</p>
<p>$$R_1 = \{(a,a),(b,b),(a,b),(b,a)$$
and
$$R_2 = \{(a,a),(b,b)\}$$<... | amWhy | 9,003 | <p>A relation is an equivalence relation if and only if it is reflexive, symmetric, and transitive. Your first two relations are indeed equivalence relations.</p>
<blockquote>
<p>A relation $R$ is reflexive on a set $A$ if and only if for all $x \in A, (x, x) \in R$.</p>
</blockquote>
<p>In $R_3$, we do not have th... |
1,679,383 | <p>Find numbers $\alpha,\beta, \gamma \in \mathbb{C}$ so that the integral $$\int_{-1}^{1}|x^3-\alpha-\beta x-\gamma x^2|^2dx$$ is minimal.
Here I want to apply Hilbert space and projection lemma by considering the integral as the distance square between a point and a closed linear subspace. So how can I start my work?... | Friedrich Philipp | 303,983 | <p>As the Hilbert space, choose the span of the functions $x^0,x^1,x^2,x^3$ (on $(-1,1)$), equipped with the scalar product
$$
\langle f,g\rangle = \int_{-1}^1 f(x)\overline{g(x)}\,dx.
$$
The subspace $M$ that you want to project onto should be the span of $x^0,x^1,x^2$. In this setting you are looking for that functio... |
2,711,785 | <p>Having some issues with this proof. Assume we've already proven addition, etc.</p>
<p>Definition of multiplication:</p>
<p>$a \times S(b) = a \times b + a$ (the "definition of multiplication")</p>
<p>$a \times 0 = 0$ (the "zero property of multiplication")</p>
<p>First, some supporting proofs:</p>
<p><strong>Cl... | Mauro ALLEGRANZA | 108,274 | <p><em>Long comment</em></p>
<p>Last <strong>Claim</strong> : the induction is on $b$ with $S(a)$ fixed. </p>
<p><em>Basis</em> : $b=0$. It's Ok. </p>
<p><em>Induction hypothesis</em> : $S(a)b=ab+b$ and we want to show that $S(a)S(b)=aS(b)+S(b)$. </p>
<p>Thus: $S(a)S(b)=S(a)b+S(a)$ (by axiom for mult) $=(ab+b)+S(a)... |
1,113,934 | <p>How to prove that $\mathrm{Hom}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$ ?</p>
<p>(We treat it as field homomorphisms. )</p>
<p>I know that $\mathrm{Aut}(\mathbb{R})=\{\mathrm{id}\}$ and $\mathrm{Mon}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$.</p>
| Olórin | 187,521 | <p>Let $\varphi : \mathbf{R}\to \mathbf{R}$ be a morphism of fields. The kernel of $\varphi$ is an ideal of the field $\mathbf{R}$, and a (commutative) field $k$ has only two ideal, $(0)$ and $k$, and the kernel of $\varphi$ can't obviously be equal to whole $\mathbf{R}$, as then we would have $\varphi(1)=0$ and at the... |
1,583,540 | <p>Is it possible to find an example of an one-sided inverse of a function? other than matrix?</p>
<p>I am trying to find such an example but having no luck. Anybody got an idea about it?</p>
| BrianO | 277,043 | <p>Examples are abundant. Some basic facts may help you find many:</p>
<ol>
<li><p><em>If $\:f\colon X\to Y$ has a left inverse $h\colon Y\to X$ ($h\circ f = id_X$), then $f$ is a injection and $h$ is a surjection.</em></p></li>
<li><p><em>If $\:f\colon X\to Y$ has a right inverse $g\colon Y\to X$ ($f\circ g = id_Y$),... |
2,278,120 | <p>In one of my algorithm courses, there is this: </p>
<blockquote>
<p>A subset $S$ of vertices in a directed graph $G$ is
strongly connected if for each pair of distinct
vertices ($v_i$, $v_j$) in $S$, $v_i$ is connected to $v_j$ and
$v_j$ is connected to $v_i$.</p>
</blockquote>
<p>And then the following ex... | vadim123 | 73,324 | <p>Also, "connected" does not mean by a directed edge, but by a directed <a href="https://en.wikipedia.org/wiki/Path_(graph_theory)" rel="noreferrer">path</a>.</p>
|
195,432 | <p>Mathematica 12 has a new function <code>FunctionCompileExportLibrary</code>, which can export a pure function as a <code>.dll</code> file like this:</p>
<pre><code>FunctionCompileExportLibrary["function.dll", Function[
Typed[arg, "MachineInteger"], arg + 1]]
</code></pre>
<blockquote>
<p><code>"...\\function.... | yode | 21,532 | <p>As the ilian's answer, I make a note for Visual Studio user here to supplement:</p>
<p><strong>source.cpp file</strong></p>
<pre class="lang-cpp prettyprint-override"><code>#include<iostream>
using namespace std;
extern "C" int Main(int);
int main() {
cout << Main(3) << endl;
return 0;
}
... |
2,656,095 | <p>I want to solve the following PDE using canonical form:</p>
<p>$$xu_{xx} + 2x^2u_{xy} = u_x - 1$$</p>
<p>I have found the characteristic curves $\xi = y$ and $\eta = y-x^2$. By computing $u_{xx}$ and $u_{xy}$, I got the following (hopefully correct) relation :
$$4x^3u_{\xi\eta} = -1$$
How do I get the general solu... | ab123 | 454,871 | <p>I have now solved it by putting $x^3 = (\xi - \eta)^{3/2}$ and then integrating the canonical equation will give the given solution.</p>
<p><strong>Edit(Complete solution):</strong>
Comparing the given PDE with the general form $A(x, y)u_{xx} + B(x, y)u_{xy} + C(x, y)u_{yy} = Φ(x, y,u,u_x,u_y)$ we find
$$A = x, B... |
2,656,095 | <p>I want to solve the following PDE using canonical form:</p>
<p>$$xu_{xx} + 2x^2u_{xy} = u_x - 1$$</p>
<p>I have found the characteristic curves $\xi = y$ and $\eta = y-x^2$. By computing $u_{xx}$ and $u_{xy}$, I got the following (hopefully correct) relation :
$$4x^3u_{\xi\eta} = -1$$
How do I get the general solu... | Aleksas Domarkas | 562,074 | <p>$$xu_{xx} + 2x^2u_{xy} -u_x =- 1$$</p>
<p>Equation is linear. Then
$$u=u_h+u_p$$</p>
<p>where $u_p$ is is particular solution(we can take $\; u_p=x\;$),</p>
<p>$u_h$ is solution homogeneus equation $xu_{xx} + 2x^2u_{xy} -u_x =0$.</p>
<p>After change $\quad \xi = x^2-y,\quad \eta = y\quad$ we get
$$u_{\xi\eta}=0,... |
2,126,299 | <p>My goal is to write $\mathbb{Q}(\pi^3+\pi^2, \pi^8+\pi^5)$ as $\mathbb{Q}(f(\pi))$ where $f \in \mathbb Q(X)$.</p>
<p>I have $(\pi^3+\pi^2)^2 = \pi^6+2\pi^5+\pi^4$ and then $2\pi^8 - \pi^6 - \pi^4$ is in my field. But I don't think it generates it. </p>
<p>Thank you for your help!</p>
| Ewan Delanoy | 15,381 | <p>Let $a=\pi^3+\pi^2$ and $b=\pi^8+\pi^5$.
Then</p>
<p>$$
\pi=\frac{3b(1-a)+(a^4+3a^3)}{b(a-3)+(3a^3+3a^2)}
$$</p>
<p>so you can take $f(\pi)=\pi$.
How did I obtain this formula ? I tried to find the minimal
polynomial of $\pi$ over ${\mathbb Q}(a,b)$, by applying the Euclidean
algorithm to $X^3+X^2-a$ and $X^8+X^5-... |
2,126,299 | <p>My goal is to write $\mathbb{Q}(\pi^3+\pi^2, \pi^8+\pi^5)$ as $\mathbb{Q}(f(\pi))$ where $f \in \mathbb Q(X)$.</p>
<p>I have $(\pi^3+\pi^2)^2 = \pi^6+2\pi^5+\pi^4$ and then $2\pi^8 - \pi^6 - \pi^4$ is in my field. But I don't think it generates it. </p>
<p>Thank you for your help!</p>
| 1Emax | 324,326 | <p>Let $\alpha = \pi^3+\pi^2$ and $\beta = \pi^8+\pi^5$. It can be shown that if $f = \frac{p}{q}$ where $\gcd(p,q)=1$, we have $$[\mathbb{Q}(\pi):\mathbb{Q}(f(\pi))] = \max\{\deg(p), \deg(q)\}.$$ Therefore $[\mathbb{Q}(\pi):\mathbb{Q}(\pi^3+\pi^2)]=3$, and if $\pi \notin \mathbb{Q(\alpha, \beta)}$, we must have $\math... |
187,511 | <p>There are two questions:</p>
<ol>
<li><p>How to prove that in general </p>
<p>$[\hat{A}(\mathbb HP^m)]_{4m} = 0$</p>
<p>It is possible to verify it for low values of $m$.</p></li>
<li><p>How to prove that in general</p>
<p>$\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$</p>
<p... | Juan Ospina | 61,860 | <p>I think that using the method of Professor Ebert is possible to prove the question 2 according with the following procedure.</p>
<p>For the expression ${\frac {\hat{A}}{\hat{M}}}$ the corresponding $F(x)$ is</p>
<p>$F(x) = \frac{x/2}{\sinh(x/2) \cosh(x/2)} = \frac{x}{\sinh(x)} $</p>
<p>Then we have</p>
<p>$\fra... |
2,111,579 | <p>A question:</p>
<p>If $\displaystyle{m}^{2}={1}-{m},{\quad\text{and}\quad}{n}^{2}={1}-{n},{\quad\text{and}\quad}{n}\ne{m};$</p>
<p>Proof that $\displaystyle{m}^{7}+{n}^{7}+{30}={1}$</p>
<p>Without finding the roots of equation $\displaystyle{x}^{2}+{x}-{1}={0}$.</p>
<p>Is there such a shortcut solution?</p>
| G Cab | 317,234 | <p>Just to show a different approach.<br>
Consider the matrix $\mathbf E$, having $1$ only on the first subdiagonal
$$
\mathbf{E} = \left\| {\,e_{\,n,\,m} = \left\{ {\begin{array}{*{20}c}
1 & {n = m + 1} \\
0 & {n \ne m + 1} \\
\end{array} } \right.\;} \right\| = \left\| {\,\left( \begin{gathered}
0... |
3,411,982 | <p>I know the answer using
Total-none of it is <span class="math-container">$6 = 900−648=252$</span></p>
<p>My doubt is that if we you it like</p>
<p>when unit digit is <span class="math-container">$6 = 9\cdot10\cdot1 = 90$</span></p>
<p>when tens place is <span class="math-container">$6 = 9\cdot1\cdot10 = 90$</span... | Shashi kant | 1,027,485 | <p>your answer is different in second method because in first case when you have counted many no. 2 or 3 times like 666 is counted all three times so that ways counted multiple times must be subtracted and your answer will match</p>
|
2,278,436 | <p>Does there exist integers $a, n > 1$ such that $1 + \frac{1}{1 + a} + \frac{1}{1 + 2a} + ... + \frac{1}{1 + na}$ is an integer? I have no clue how to begin. I've tried to simplify this somehow, but with no effect.</p>
| vrugtehagel | 304,329 | <p>This is not a complete answer (yet). I'm posting it because it may be useful to people who are thinking about this problem.</p>
<hr>
<p>Note that</p>
<p>$$\gcd(1+a,1+ka)=\gcd(1+a,(k-1)a)=\gcd(1+a,k-1)$$</p>
<p>Let $p\mid a+1$ (where $p$ is prime). For $\frac1{1+a}+\frac1{1+2a}+\cdots+\frac1{1+na}$ to be an integ... |
2,278,436 | <p>Does there exist integers $a, n > 1$ such that $1 + \frac{1}{1 + a} + \frac{1}{1 + 2a} + ... + \frac{1}{1 + na}$ is an integer? I have no clue how to begin. I've tried to simplify this somehow, but with no effect.</p>
| sku | 341,324 | <p>Let $X = (1+a)(1+2a)\cdots(1+na)$</p>
<p>Let $Y = \sum_{k = 1}^{n}\frac{X}{1+ka}$</p>
<p>Let $Z = \frac{Y}{X}$</p>
<p>Let $p^r$ be the highest power of a prime $p$ that divides $X$</p>
<p>Let $p^s$ be the highest power of $p$ that divides $1+ka$</p>
<p>So the power of $p$ that divides $Y$ is at least $p^r/p^s =... |
4,155,453 | <p>I am trying to evaluate <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^\infty \frac{n}{n^2+k^2}.$$</span> Now I am aware that clearly <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^n \frac{n}{n^2+k^2} = \int_0^1 \frac{1}{1+x^2}dx = \tan^{-1}(1) = \frac{\pi}{4},$$</span> but I do not know ... | Paresseux Nguyen | 758,600 | <p>Let <span class="math-container">$$f(x):= \frac{1}{1+x^2}$$</span>
then
<span class="math-container">$$\sum_{k=1}^{\infty} \frac{n}{n^2+x^2} = \int_{0}^{+\infty} f\left( \frac{ \lceil nx \rceil }{n} \right) dx$$</span>
From this you can do whatever you like to conclude, maybe using DCT while knowing that <span class... |
4,155,453 | <p>I am trying to evaluate <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^\infty \frac{n}{n^2+k^2}.$$</span> Now I am aware that clearly <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^n \frac{n}{n^2+k^2} = \int_0^1 \frac{1}{1+x^2}dx = \tan^{-1}(1) = \frac{\pi}{4},$$</span> but I do not know ... | zhw. | 228,045 | <p>Let <span class="math-container">$f(x)= \dfrac{1}{1+x^2}.$</span> Note that our sum equals</p>
<p><span class="math-container">$$\sum_{k=1}^{\infty}f\left(\frac{k}{n}\right)\frac{1}{n}.$$</span></p>
<p>Now <span class="math-container">$f$</span> decreases from <span class="math-container">$1$</span> to <span class="... |
3,133,152 | <blockquote>
<p><span class="math-container">$g(x)=|\sin{x}-1|+|3-\cos{x}-\sin{x}|+2\sin{x}$</span></p>
<p>Answer: Above equality is simplified to <span class="math-container">$$1-\sin{x}+3-\cos{x}-\sin{x}+2\sin{x}=4-\cos{x}$$</span></p>
</blockquote>
<p><span class="math-container">$$-1 \le\sin{x}\le1$$</span></p>
<p>... | Robert Lewis | 67,071 | <p><span class="math-container">$\vert \sin x - 1 \vert = \vert 1 - \sin x \vert, \; \forall x \in \Bbb R; \tag 1$</span></p>
<p>since</p>
<p><span class="math-container">$1 - \sin x \ge 0, \; \forall x \in \Bbb R, \tag 2$</span></p>
<p><span class="math-container">$\vert \sin x - 1 \vert = \vert 1 - \sin x \vert =... |
868,663 | <blockquote>
<p>Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer.</p>
</blockquote>
<p>I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.</p>
| Community | -1 | <p>This is in answer to the edits (I'm going to start with your second question):</p>
<p><strong>#2: How we know that we can always represent a number by adding factors of base-10 numbers</strong></p>
<p>There is no proof needed. Decimal notation is simply a shorthand for expressing numbers in terms of base-10 number... |
1,860,243 | <p>Let $|G|$ be an abelian group and let $H = \{g \in G : |g| \text{ divides } 12 \}.$ Prove that $H$ is a subgroup of $G$.</p>
<p>I know that I have to show that $a,b \in H \Rightarrow ab^{-1} \in H$ or $(ab \in H \land a^{-1} \in H).$ But I can't figure out how $|a|$ and $|b|$ dividing $12$ relates to $|ab|$ or $|... | Jendrik Stelzner | 300,783 | <p>Notice that $|g|$ divides $12$ if and only if $12g = 0$.
Because the map $f \colon G \to G$, $g \mapsto 12g$ is a group homomorphism it follows that $H = \ker f$ is a subgroup.</p>
|
4,244,565 | <p>In resources like Wikipedia the following equation holds for <span class="math-container">$k>0$</span>.
<span class="math-container">\begin{equation}
\sum_{n=0}^{k} [x^n]=\frac{1-x^{k+1}}{1-x}\hspace{1cm}x\neq1
\end{equation}</span>
If you put the right hand side in a limit it holds for <span class="math-containe... | user247327 | 247,327 | <p>(You mean that |X|= |Y|= m or cardinality of X= cardinality of Y= m. X and Y themselves are sets so not equal to a number.)</p>
<p>I don't know what you mean by "<span class="math-container">$f(x_1)= y_1, y_2,..., y_m$</span>". <span class="math-container">$f(x_1)$</span> is a single element of X (I suppo... |
30,970 | <p>I try to filter sublists of a list which match a pattern. </p>
<pre><code>test = {{"String1", "a"}, {"String2", "b"}, {"String3",
"a"}, {"String4", "a"}};
</code></pre>
<p>The result should be:</p>
<pre><code>result = {{"String1", "a"}, {"String3", "a"}, {"String4", "a"}}
</code></pre>
<p>That means the first e... | M6299 | 7,115 | <p>Another solution:</p>
<pre><code>Select[test, Function[{x}, StringQ[x[[1]]] && x[[2]] == "a"]]
</code></pre>
<blockquote>
<pre><code>{{"String1","a"},{"String3","a"},{"String4","a"}}
</code></pre>
</blockquote>
|
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