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 |
|---|---|---|---|---|
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Martin Brandenburg | 1,650 | <p>Lagrange's Theorem has some applications to elementary number theory:</p>
<p>1) Wilson's Theorem says that for a prime $p$ we have $(p-1)! \equiv -1 \pmod p$. Proof: By Lagrange's Theorem applied to $\mathbb{F}_p^*$ we have $X^{p-1}-1 = \prod_{a \in \mathbb{F}_p^*} (X+a)$ in $\mathbb{F}_p[X]$. Now let $X \mapsto 0... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Makoto Kato | 28,422 | <p>A lucid proof of the quadratic reciprocity law can be obtained by the theory of cyclotomic number fields.</p>
<p><a href="https://math.stackexchange.com/questions/174858/discriminant-of-the-quadratic-subfield-of-the-cyclotomic-number-field-of-an-odd">Discriminant of the quadratic subfield of the cyclotomic number f... |
453,295 | <p>I wanna show that the non-zero elements of $\mathbb Z_p$ ($p$ prime) form a group of order $p-1$ under multiplication, i.e., the elements of this group are $\{\overline1,\ldots,\overline{p-1}\}$. I'm trying to prove that every element is invertible in the following manner:</p>
<blockquote>
<p><strong>Proof (a)</s... | Ben Grossmann | 81,360 | <p>$\overline{0}$ is the equivalence class of $0$, that is, the multiples of $p$. We can use this because multiplication and addition are well defined ring operations in $\mathbb Z_n$. The precursor to this proof is to show that addition and multiplication over the integers modulo $n$ are, in fact, well defined (i.e. g... |
1,511,078 | <p><strong>Show that the product of two upper (lower) triangular matrices is again upper (lower) triangular.</strong></p>
<p>I have problems in formulating proofs - although I am not 100% sure if this text requires one, as it uses the verb "show" instead of "prove". However, I have found on the internet the proof belo... | Muthu Kumar N P | 390,986 | <p>Let the two upper triangular matrices be A(nxn) and B(nxn). And let C=AB.</p>
<p>Now, for matrix multiplication, $$C_{ij}=\sum_{r=1}^n A_{ir}B_{rj}$$</p>
<p>Since, A and B are upper triangular matrices,</p>
<p>$A_{ir}=0$ , when i>r</p>
<p>$B_{rj}=0$ , when r>j</p>
<p>So, For an element of C (i.e., $A_{ir}B_{rj}... |
762,651 | <p>I have to prove that "any straight line $\alpha$ contained on a surface $S$ is an asymptotic curve and geodesic (modulo parametrization) of that surface $S$". Can I have hints at tackling this problem? It seems so general that I am not sure even how to formulate it well, let alone prove it. Intuitively, I imagine ... | Community | -1 | <p>This answer is very late, but this is another way to view the problem:</p>
<p>We are trying to prove $a^2 > b^2$, or $a^2-b^2>0$.</p>
<p>Using the identity: $a^2-b^2$ = $(a+b)(a-b)$,</p>
<p>we are equivalently trying to prove: </p>
<p>$(a+b)(a-b) > 0 $</p>
<p>Since $0<a<b$, </p>
<p>$(a+b) > 0... |
3,407,489 | <p><span class="math-container">$\neg\left (\neg{\left (A\setminus A \right )}\setminus A \right )$</span></p>
<p><span class="math-container">$A\setminus A $</span> is simply empty set and <span class="math-container">$\neg$</span> of that is again empty set. Empty set <span class="math-container">$\setminus$</span... | hamam_Abdallah | 369,188 | <p>The emptyset is subset of every set <span class="math-container">$ E $</span> because you cannot find an element which is in the emptyset and not in the set <span class="math-container">$ E$</span>.
<span class="math-container">$$\text{ Proof}$$</span></p>
<p>Let <span class="math-container">$ x$</span> be an arbit... |
1,218,582 | <p>I was presented with the function $max (|x|,|y|)$ which should output a maximum value of given two.... I can only suppose this one creates some body in $\mathbb{R}^3$ but how do you sketch it and what does it mean in $\mathbb{R}^3$? for that matter in $\mathbb{R}^2$ I cant really imagine it also.. </p>
| David | 119,775 | <p>The function by itself does not specify any region of $\Bbb R^2$. If you meant $f(x,y)=constant$ then consider for example
$$\max(|x|,|y|)=1\ .$$
This can be written as
$$x=\pm1\ ,\ -1\le y\le1\qquad\hbox{or}\qquad y=\pm1\ ,\ -1\le x\le1$$
which is a square with vertices at $(\pm1,\,\pm1)$.</p>
|
108,331 | <p>I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the <a href="http://en.wikipedia.org/wiki/Prime_gap#Lower_bounds">prime number counting business</a>, somewhat unsettling. What is the reason for them?</p>
<p>Has it maybe to do with the series expansion of the logar... | Gerry Myerson | 8,269 | <p>If you are asking, why do you find it unsettling that logarithms occur in Number Theory, I'm afraid you will have to ask a psychiatrist. </p>
<p>If you are asking, why are there logarithms in Number Theory, consider the following naive effort to find the number of primes up to $N$: </p>
<p>There are $N$ integers u... |
108,331 | <p>I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the <a href="http://en.wikipedia.org/wiki/Prime_gap#Lower_bounds">prime number counting business</a>, somewhat unsettling. What is the reason for them?</p>
<p>Has it maybe to do with the series expansion of the logar... | tomcuchta | 1,796 | <p>One reason that it occurs in analytic number theory has to do with the <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>:</p>
<p>$$\zeta(s) = \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^s},$$</p>
<p>where $s = \sigma + it$ is a complex number <a href="http://en.wikipedia.org/wi... |
107,525 | <p>Say I have two random variables X and Y from the same class of distributions, but with different means and variances (X and Y are parameterized differently). Say the variance converges to zero as a function of n, but the mean is not a function of n. Can it be formally proven, without giving the actual pdf of X and Y... | Jean-Victor Côté | 24,376 | <p>As long as the means are different, when variances go to zero, overlap goes to nothing.</p>
|
4,195,399 | <p>Given that <span class="math-container">$a,b,c > 0$</span> are real numbers such that <span class="math-container">$$\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}\le 1,$$</span> prove that <span class="math-container">$$\frac{1}{b+c+1}+\frac{1}{c+a+1}+\frac{1}{a+b+1}\ge 1.$$</span></p>
<hr />
<p>I first rewrote... | Calvin Lin | 54,563 | <p>Applying Jensens to <span class="math-container">$ f(x) = \frac{ x} { (a+b+c+1) - x } $</span>, we have <span class="math-container">$$ 1\geq \sum \frac{a}{b+c+1} = f(a) + f(b) + f(c) \geq 3 f ( \frac{a+b+c } { 3} ) = 3 \times \frac{ a + b + c } { 2a + 2b + 2c + 3 } \Rightarrow a + b + c \leq 3.$$</span></p>
<p>Ap... |
4,415,559 | <p>I am trying to find the root of <span class="math-container">$f(x)=ln(x)-cos(x)$</span> by writing an algorithm for bisection and fixed-point iteration method. I am currently using python but whenever I'm running it using either of the two methods, it prints out "math domain error". I guess this is due to ... | José C Ferreira | 1,029,870 | <p>You don't need to change the function <span class="math-container">$f(x)$</span>. Choose <span class="math-container">$a=0.1$</span> and <span class="math-container">$b=e$</span> and you will find that <span class="math-container">$f(a)<0$</span> and <span class="math-container">$f(b)>0$</span>. The bisection ... |
1,880,090 | <p>The solution states that the ball of radius $\epsilon >0$ around a real number $x$ always contains the non-real number $x+i\epsilon/2$. </p>
<p>I don't understand the answer, for every number $x \in \mathbb{R}$ there is an open ball, right? For every $x \in \mathbb{R}$ there is an $r>0$ such that I can form a... | PMar | 358,074 | <p>To decide if $\mathbb{R}$ is open in $\mathbb{C}$, you must use the topology of $\mathbb{C}$, not of $\mathbb{R}$. That is, you must take $B_r(x)\subset \mathbb{C}$.</p>
|
21,262 | <p><strong>Bug introduced in 9.0 and fixed in 11.1</strong></p>
<hr>
<p><code>NDSolve</code> in Mathematica 9.0.0 (MacOS) is behaving strangely with a piecewise right hand side. The following code (a simplified version of my real problem):</p>
<pre><code>sol = NDSolve[{x'[t] ==
Piecewise[{{2, 0 <= Mod[t... | Michael E2 | 4,999 | <p>Taking the OP's clue that starting at <code>x[10^-100] == 0</code> solves the problem, we can try explicitly setting the derivative value at <code>t == 0</code>. The following produces a consistent and correct result:</p>
<pre><code>solME2 = First@ NDSolve[{
x'[t] == Piecewise[{
{2, t == 0},
{2, 0 &l... |
206,723 | <p>Can any one explain why the probability that an integer is divisible by a prime $p$ (or any integer) is $1/p$?</p>
| MikeEVMM | 865,089 | <p>Although <a href="https://math.stackexchange.com/a/206729/865089">Stephen Stadnicki's answer</a> is much more rigorous, I only made sense of it with the following argument:</p>
<p>If <span class="math-container">$p$</span> is dividing a number <span class="math-container">$i$</span> in <span class="math-container">$... |
105,040 | <p>This question in stackExchange remained unanswered. </p>
<p>Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)$ what is the cardinality of $C_{M_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M_n(\mathbb F)$? There a... | Geoff Robinson | 14,450 | <p>You asked this on Math Stack Exchange too. One messy case is when $A$ is unipotent. Some cases of tat are dealt with in a famous paper of P. Hall and G. Higman on "Reduction Theorems for Burnside's Problem" (Proceedings of London Mathematical Society, 1956). The centralizer of a semisimple matrix is relatively easy... |
1,189,216 | <p>Wikipedia and other sources claim that </p>
<p>$PA +\neg G_{PA}$</p>
<p>can be consistent, where $\neg G_{PA}$ is the Gödel statement for PA.</p>
<p>So what is the error in my reasoning?</p>
<p>$G_{PA}$ = "$G_{PA}$ is unprovable in PA"</p>
<p>$\neg G_{PA} $</p>
<p>$\implies$ $\neg$ "$G_{PA}$ is unprovable in P... | Hanno | 81,567 | <p>The last step is the problematic one: You are given a formula $\varphi$ and try to prove, in the (finitistic!) metatheory, that $\textsf{PA}\vdash "\textsf{PA}\vdash\varphi"$ implies $\textsf{PA}\vdash\varphi$. While the implication "$\Leftarrow$" is fine (a proof of $\varphi$ from $\textsf{PA}$ could be explicitly... |
300,745 | <p>If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?</p>
<p>($a$ and $b$ being finite numbers).</p>
<p>I tried proving and disproving it. Couldn't find an example for a non-bounded image. </p>
<p>Is there any basic proof or counter example for any of the cases?</p>
<p>Thanks a mi... | Community | -1 | <p>There is a $\delta >0$ such that for all $x,y\in (a,b)$ with $|x-y|\leq \delta$ we have $|f(x)-f(y)|\leq 1$. Let $p=\min (\delta,\frac{b-a}{3})$. Then $f$ is continuous on $I=[a+p,b-p]$. Hence $f$ is bounded on $I$. In addition, $f$ is bounded by $|f(a+p)|+|f(b-p)|+1$ on $(a-b)-I$. This means $f$ is bounded on $(... |
355,454 | <p>Let <span class="math-container">$p$</span> be an odd prime. The <span class="math-container">$\mathbb F_p$</span> cohomology of the cyclic group of order <span class="math-container">$p$</span> is well-known: <span class="math-container">$\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$</span> where <span... | Nicholas Kuhn | 102,519 | <p>If <span class="math-container">$P$</span> is the group of order <span class="math-container">$p^3$</span> and exponent <span class="math-container">$p$</span>, its mod <span class="math-container">$p$</span> cohomology ring is known to have its depth = its Krull dimension = rank of a maximal elementary abelian subg... |
355,454 | <p>Let <span class="math-container">$p$</span> be an odd prime. The <span class="math-container">$\mathbb F_p$</span> cohomology of the cyclic group of order <span class="math-container">$p$</span> is well-known: <span class="math-container">$\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$</span> where <span... | Drew Heard | 16,785 | <p>I think the answer you want can be found in the following article:</p>
<pre><code>AUTHOR = {Leary, I. J.},
TITLE = {The mod-{<span class="math-container">$p$</span>} cohomology rings of some {<span class="math-container">$p$</span>}-groups},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical ... |
11,244 | <p>In order to evaluate new educational material the contentment of students with this material is often measured. However, just because a student is contented doesn't mean that he/she has actually learned something. Is there any research investigating the correlation between students contentment and the educational qu... | Anschewski | 199 | <p>I would recommend you to specify the term "educational quality". I think there is a study indicating that in German schools, the two educational goal variables of students' motivation and students' mathematical knowledge are negatively correlated across clases, so you have a trade-off there. I guess, students' conte... |
2,634,277 | <p>I am working on some development formulas for surfaces and as a byproduct of abstract theory i get that:
$$\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\frac{1+\sin^2\theta}{(\cos^4\theta+(\gamma\cos^2\theta-\sin\theta)^2)^\frac{3}{4}}d\theta$$
is independent on the parameter $\gamma\in\mathbb{R}$. I thought that there was so... | Yuriy S | 269,624 | <p>This is not an answer, but too long for a comment.</p>
<p>I still think it may be helpful to consider the derivative (I use $x$ as integration variable).</p>
<p>$$J=\frac{d}{d \gamma} I(\gamma)=\int_{-\pi/2}^{\pi/2} \frac{(1+\sin^2 x)(\gamma ~\cos^2 x-\sin x) \cos^2 x~dx}{\left(\cos^4 x+(\gamma~ cos^2 x-\sin x)^2 ... |
520,046 | <blockquote>
<p>Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$.</p>
</blockquote>
<p>I tried
$$x\equiv5\pmod6\\x\equiv4\pmod5\\x\equiv3\pmod4\\x\equiv2\pmod3$$What $x$ value?</p>
| Brian M. Scott | 12,042 | <p>HINT: Notice that your congruences are equivalent to the following ones:</p>
<p>$$\left\{\begin{align*}
x\equiv-1\pmod6\\
x\equiv-1\pmod5\\
x\equiv-1\pmod4\\
x\equiv-1\pmod3
\end{align*}\right.$$</p>
<p>In other words, $x+1$ is divisible by $6,5,4$, and $3$. What’s the smallest positive integer with that property?... |
520,046 | <blockquote>
<p>Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$.</p>
</blockquote>
<p>I tried
$$x\equiv5\pmod6\\x\equiv4\pmod5\\x\equiv3\pmod4\\x\equiv2\pmod3$$What $x$ value?</p>
| vijay_pavani.123 | 118,683 | <p>The problem can be solved by Chinese Remainder Theorem (CRT) or by taking lcm of divisors. Here divisors are 3, 4, 5 and 6; and the 1 must be subtracted? can you guess why we are subtracting 1? Or you can ask me if you need any clarifications.</p>
|
1,987,387 | <p>I don't remember any method to compute the closed from for the following series.
$$ \sum_{k=0}^{\infty}\binom{3k}{k} x^k .$$</p>
<p>I tried by putting $\binom{3k}{k}$ in Mathematica for different $k$ and asking for the generating function it deliver a complicated formula which is the following.
$$ \frac{2\cos[\f... | epi163sqrt | 132,007 | <blockquote>
<p><em>Hint:</em> A closed form can be found by means of the <em><a href="https://en.wikipedia.org/wiki/Lagrange_inversion_theorem" rel="nofollow noreferrer">Lagrange Inversion Formula</a></em>.</p>
<p>An answer based upon this method is given at <em><a href="https://math.stackexchange.com/questions... |
22 | <p>By matrix-defined, I mean</p>
<p>$$\left<a,b,c\right>\times\left<d,e,f\right> = \left|
\begin{array}{ccc}
i & j & k\\
a & b & c\\
d & e & f
\end{array}
\right|$$</p>
<p>...instead of the definition of the product of the magnitudes multiplied by the sign of their angle... | BBischof | 16 | <p>Assuming you know the definition of orthogonal as "a is orthogonal to b iff $a\cdot b=0$ then we could calculate $(a \times b)\cdot a = a_1(a_2b_3-a_3b_2)-a_2(a_1b_3-a_3b_1)-a_3(a_1b_2-a_2b_1)=0$ and $(a \times b)\cdot b-0$, so the cross product is orthogonal to both. As Nold mentioned, if the two vectors a and b l... |
500,931 | <p>I have this theorem which I can't prove.Please help.</p>
<p>"Show that every positive rational number $x$ can be expressed in the form $\sum_{k=1}^n \frac{a_k}{k!}$ in one and only one way where each $a_k$ is non-negative integer with $ a_k≤ k − 1$ for $k ≥ 2$ and $a_n>0$."</p>
<p>I think the ONE way is <a href... | Brian M. Scott | 12,042 | <p>HINT: For existence I find it easiest to work backwards. Express the fractional part of $x$ as a fraction $\frac{c}d$ in lowest terms, and let $n$ be minimal such that $d\mid n!$. Write $x$ as $\frac{c}{n!}$, and write $c=q_0n+a_n$, where $0\le a_n<n$. Then the fractional part of $x$ is</p>
<p>$$x-\lfloor x\rflo... |
1,800,519 | <blockquote>
<p>Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an exact form? </p>
</blockquote>
<p>I think it is but I've been unable to prove it, so any help would be gre... | YannickSSE | 238,069 | <p>In general this is not true. Recall that
$$
L_X(\omega) = i_x d\omega + d i_x\omega
$$
where you see that the right part is exact and the left part mustn't be. As an example for your case take $N$ a manifold with a non exact form $\mu$
and let $\omega$ be a 0-form (function) on $\mathbb{R}$ and define $M=N\times\m... |
354,100 | <p>Does the expression <span class="math-container">$$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n,$$</span> which gives the volume of an <span class="math-container">$n$</span>-dimensional ball of radius <span class="math-container">$R$</span> when <span class="math-container">$n$</span> is a nonnegative integer... | Anixx | 10,059 | <p>I will answer regarding the dimension <span class="math-container">$-1$</span>.</p>
<p>An example of such space is a set of periodic lattices on a real line.</p>
<p>Indeed, you can see that the <a href="https://en.wikipedia.org/wiki/Hausdorff_dimension" rel="nofollow noreferrer">Hausdorff dimension</a> of a periodic... |
1,044,009 | <p>I have two numbers $N$ and $M$.
I efficiently want to calculate how many pairs of $a$,$b$ are there such that $1 \leq a \leq N$ and $1 \leq b \leq M$ and $ab$ is a perfect square.</p>
<p>I know the obvious $N*M$ algorithm to compute this. But i want something better than that. I think it can be done in a better tim... | Rory Daulton | 161,807 | <p>If you have a list of prime numbers up to $\max(M,N)$ here is one way to do it.</p>
<p>For each increasing $a$ from $1$ through $N$, find the prime decomposition of $a$. Multiply the primes (using exponent $1$) that have an odd exponent in the decomposition--call the result $c$. If $a \cdot c<M$ then $c$ is your... |
876,209 | <p>I am confused on how to write a formal proof for sum notations. How would I write a formal proof for this example?</p>
<p>Prove that $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$ Prove that for any $\alpha \in \{0, 2\}^\mathbb{N}$ that $$0 \le \sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \le 3.$$</p>
| Shawn O'Hare | 161,325 | <p>The series $S:=\sum_{k=0}^{\infty} \frac{2}{3^k}$ is a geometric series with first term $a=2$ and multiplier $r=1/3$, therefor
$$
\lim_{n \to \infty} \sum_{k=0}^n \frac{2}{3^k} = \frac{a}{1-r} = \frac{2}{2/3} = 3.
$$</p>
<p>Now let $T:=\sum_{k=0}^{\infty} \frac{\alpha(k)}{3^k}$. For any $k \in \mathbb N$ it's clea... |
1,557,733 | <p>I have a function $f(\mathbf{u}, \Sigma)$ where $\mathbf{u}$ is a $p \times 1$ vector and $\Sigma$ is a $p \times p$ real symmetric matrix (positive semi-definite).</p>
<p>I somehow successfully computed the partial derivatives $\frac{\partial f}{\partial \mathbf{u}}$ and $\frac{\partial f}{\partial \Sigma}$.</p>
... | Gyumin Roh | 292,972 | <p>(ii) should be fixed. It would be larger than $\frac{3p}{8}$.</p>
<p>Let $AD, BE, CF$ be the medians, and let the centroid be $G$.</p>
<p>WLOG, let the two medians be $m_a, m_b$.</p>
<p>From the Triangle Inequality on $\triangle AGB$, we have $$\frac{2}{3}m_a+\frac{2}{3}m_b > c$$</p>
<p>Summing this cyclicall... |
3,338,885 | <p>I want to prove this theorem using Binomial theorem and I've got trouble in understanding 3rd step if anyone knows why please explain :) Prove that sum: </p>
<p><span class="math-container">$\sum_{r=0}^{k}\binom{m}{r}\binom{n}{k-r}=\binom{m+n}{k}$</span> </p>
<p>1st step:<br>
<span c... | Wuestenfux | 417,848 | <p>In the 3rd step, <span class="math-container">$$\sum_{k=0}^{m+n} {m+n\choose k} y^k = \sum_{i=0}^m {m\choose i} y^i\sum_{i=0}^n{n\choose j} y^j =\sum_{k=0}^{m+n} \sum_{i,j\atop i+j=k}{m\choose i}{n\choose j} y^k.$$</span>
This gives the desired equation.</p>
|
3,338,885 | <p>I want to prove this theorem using Binomial theorem and I've got trouble in understanding 3rd step if anyone knows why please explain :) Prove that sum: </p>
<p><span class="math-container">$\sum_{r=0}^{k}\binom{m}{r}\binom{n}{k-r}=\binom{m+n}{k}$</span> </p>
<p>1st step:<br>
<span c... | drhab | 75,923 | <p>Let <span class="math-container">$a_i\in\mathbb R$</span> denote a constant for <span class="math-container">$i=0,1,\dots,n$</span>.</p>
<p>The function <span class="math-container">$\mathbb R\to\mathbb R$</span> prescribed by <span class="math-container">$x\mapsto\sum_{i=0}^na_ix^i$</span> is the same as the funct... |
1,821,849 | <p>Let $K/F$ be a field extension and $L_1,L_2$ subfields of $K$ such that $L_1$ and $L_2$ have finite degree over $F$. </p>
<p>Does $L_1 \cong L_2$ imply $[L_1 : F ]=[L_2 : F]$? Obviously, if the isomorphism fixes $K$ (which isn't always necessarily true) the result holds. The result even holds if $F$ is of finite de... | Hmm. | 227,501 | <p>This is indeed a very subtle question. If you assume the standard $F$-vector space structures on both $L_1$ and $L_2$, this need not be true in general. For example, let us take $L_1=\mathbb{C}(X^2)$, $L_2=\mathbb{C}(X)$, and take $F=\mathbb{C}(X^2)$. Clearly the map $X^2 \to X$ gives you an isomorphism from $L_1$ t... |
216,532 | <p>How do I find the limit of something like</p>
<p>$$ \lim_{x\to \infty} \frac{2\cdot3^{5x}+5}{3^{5x}+2^{5x}} $$</p>
<p>?</p>
| glebovg | 36,367 | <p>Note that</p>
<p>$$\frac{{2 \cdot {3^{5x}} + 5}}{{{3^{5x}} + {2^{5x}}}} \sim \frac{{2 \cdot {3^{5x}}}}{{{3^{5x}} + {2^{5x}}}} = \frac{2}{{1 + {{\left( {\frac{2}{3}} \right)}^{5x}}}}.$$</p>
<p>So ...</p>
|
1,613,171 | <p>On page $61$ of the book <a href="http://solmu.math.helsinki.fi/2010/algebra.pdf" rel="nofollow">Algebra</a> by Tauno Metsänkylä, Marjatta Näätänen, it states</p>
<blockquote>
<p>$\langle \emptyset \rangle =\{1\},\langle 1 \rangle =\{1\}. H\leq G \implies \langle H \rangle =H$</p>
</blockquote>
<p>where $H \leq ... | hhh | 5,902 | <blockquote>
<p>$\langle \emptyset \rangle=\{1\},\langle 1 \rangle=\{1\}. H\leq G \implies \langle H\rangle=H.$ </p>
</blockquote>
<p><strong>Examples.</strong></p>
<p>$G=\mathbb R^*$ without zero. (G,*) is multiplicative group.</p>
<ol>
<li><p>If you have $\langle \emptyset \rangle =\{\}=\emptyset$, then this $\e... |
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | almagest | 172,006 | <p>As @DavidRicherby implies out in a comment below, one should ideally distinguish carefully the history of the dot notation from the possible reasons for keeping it, retaining it, or modifying its usage. Unfortunately although I am qualified by age (66) to comment on the last 50 years, I am otherwise ill-qualified to... |
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | Ben C | 218,588 | <p>There is also an ambiguity between a decimal fraction with a dot, as in $3.5^2$, and multiplication with a centre dot, as in $3\cdot5^2$, particularly if the latter doesn't have spacing around the dot to give context, as in $3\!\cdot\!5^2$.</p>
<p>In fact some textbooks use a centre dot for decimal fractions, for e... |
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | snulty | 128,967 | <p>I'm looking for sources (see edit), but I would imagine when teaching children to count, add, multiply, you start with integers and addition is symbolised like $1+1=2$. Then you try to teach them that multiplication is short for lots of addition $3+3+3+3=4\times 3$ and $4+4+4=3\times 4$, and theres the obvious simil... |
4,203,704 | <p>Understanding the Yoneda lemma maps.</p>
<p>I'm trying to understand the maps between the natural transformations and <span class="math-container">$F(A)$</span> in the proof of the Yoneda lemma. I've been struggling for a bit to understand the Yoneda lemma, so I'm trying to understand the mapping construction as a r... | qualcuno | 362,866 | <p>Note that since <span class="math-container">$\eta$</span> is a natural transformation from <span class="math-container">$h^A $</span> to <span class="math-container">$F$</span>, by definition <span class="math-container">$\eta_A$</span> is a function (i.e. an arrow in <span class="math-container">$\mathsf{Set}$</sp... |
250,687 | <p>I'm doing a sanity check of the following equation:
<span class="math-container">$$\sum_{j=2}^\infty \frac{(-x)^j}{j!}\zeta(j) \approx x(\log x + 2 \gamma -1)$$</span></p>
<p>Naive comparison of the two shows a bad match but I suspect one of the graphs is incorrect.</p>
<ol>
<li>Why isn't there a warning?</li>
<li>H... | Roman | 26,598 | <p>Turn the sum around to make it non-alternating:</p>
<p><span class="math-container">$$
\sum_{j=2}^{\infty}\frac{(-x)^j}{j!}\zeta(j) =
\sum_{j=2}^{\infty}\frac{(-x)^j}{j!}\left(\sum_{n=1}^{\infty}\frac{1}{n^j}\right) =
\sum_{n=1}^{\infty}\left(\sum_{j=2}^{\infty}\frac{(-x)^j}{j!}\frac{1}{n^j}\right) =
\sum_{n=1}^{\in... |
12,102 | <p>For local field, the reciprocity map establishes almost an isomorphism from the multiplicative group to the Abelian Absolute Galois group. (In global case the relationship is almost as nice). It is tempted to think that there can be no such nice accident. </p>
<p>Do we know any explanation which suggest that there ... | Emerton | 2,874 | <p>The reciprocity map is completely natural (in the technical sense of category theory). For example, if $K$ and $L$ are two local fields,
and $\sigma:K \rightarrow L$ is an isomorphism, then $\sigma$ induces an isomorphism of
multiplicative groups $K^{\times} \rightarrow L^{\times}$ and also of abelian absolute Gal... |
2,495,440 | <p>If $f'(x) + f(x) = x,\;$ find $f(4)$.</p>
<p>Could someone help me to solve this problem ? </p>
<p>The answer is 3 but I don't know why.
<em>with no use of integration or exponential functions</em> and
<em>the function is polynomial</em></p>
| Aggelos Bessis | 348,376 | <p>The only way f is a polynomial is if it is of 1st degree because $ deg( f'(x)+f(x))=deg(x)=1$. Since $degf'(x)<degf(x)$ we obtain that $degf(x)=1$ so you can set $ f(x)=ax+b$ so we get that $ f'(x)=a $. Plug those in and:
$ f'(x)+f(x)=x=> ax+(a+b)=x $</p>
|
624,850 | <blockquote>
<p>Prove: $$x^\alpha-\alpha x \le 1- \alpha \\
\forall x\ge 0 \ , \ 0<\alpha <1$$</p>
</blockquote>
<p>It does resamble Lagrange's MVT, in order to get to the RHS, say $\alpha\in[0,1]\Rightarrow \frac{1-\alpha-0^\alpha-\alpha 0}{1-0}=1-\alpha$.</p>
<p>But I'm not sure what to do about the LHS. ... | daulomb | 98,075 | <p>Let $f(x)=x^\alpha-\alpha x-\alpha+1$ for $x\geq 0$, where $\alpha<1$, so $f'(x)=\alpha x^{\alpha-1}-\alpha$ and $f''(x)=\alpha(\alpha-1)x^{\alpha-2}$. Then $f'(x)=0\Rightarrow x=1$ with $f''(1)=\alpha(\alpha-1)\leq 0$. This implies that $f$ has a local maximum at $x=1$ and so $f(1)=2(1-\alpha)\geq1-\alpha=f(0)$ ... |
624,850 | <blockquote>
<p>Prove: $$x^\alpha-\alpha x \le 1- \alpha \\
\forall x\ge 0 \ , \ 0<\alpha <1$$</p>
</blockquote>
<p>It does resamble Lagrange's MVT, in order to get to the RHS, say $\alpha\in[0,1]\Rightarrow \frac{1-\alpha-0^\alpha-\alpha 0}{1-0}=1-\alpha$.</p>
<p>But I'm not sure what to do about the LHS. ... | Paramanand Singh | 72,031 | <p>The problem is too easy once one rewrites it as $x^{\alpha} - 1 \leq \alpha(x - 1)$. If $f(x) = x^{\alpha}$, then $f'(x) = \alpha x^{\alpha - 1}$ and then by Mean Value Theorem we have $$x^{\alpha} - 1 = f(x) - f(1) = (x - 1)f'(c) = \alpha(x - 1)c^{\alpha - 1}$$ where $c$ is between $1$ and $x$. Clearly we need to c... |
366,096 | <p>Let's consider $J\subset \mathbb R^2$ such that J is convex and such that it's boundary it's a curve $\gamma$. Let's suppose that $\gamma$ is anti-clockwise oriented, let's consider it signed curvature $k_s$. I want to prove the intuitive following fact:</p>
<p>$$
\int\limits_\alpha {k_s } \left( s \right)ds \geqs... | Community | -1 | <p>This is a more formal version of Brian Rushton's answer. Suppose there is a point of negative curvature. Choose $xy$ coordinates so that this point is the origin $(0,0)$, the tangent direction is $x$-axis, and the $y$-axis points inside the convex set. Let $y=f(x)$ be the equation of a part of curve near $(0,0)$. (I... |
694,668 | <p>Let (X,Y) be uniformly distributed in a circle of radius 1. Show that if R is the distance from the center of the circle to (X,Y) then $R^2$ is uniform on (0,1). </p>
<p>This is question from the Simulation text of Prof. Sheldon Ross. Any hints? </p>
| copper.hat | 27,978 | <p>Here is a much more tedious way of doing the same thing:</p>
<p>Let $A \subset (0,1)$ be an open set.
For later convenience, let $N = \{(x,0) | x \in (-1,0] \}$.</p>
<p>Let $C_A = \{ x | x_1^2+x_2^2 \in A \}$, and note that $C_A = \{ \sqrt{t} (\cos \theta, \sin \theta) | t \in A, \theta \in (-\pi, \pi] \} \subse... |
357,101 | <p>There exists a minimal subshift <span class="math-container">$X$</span> with a point <span class="math-container">$x \in X$</span> such that <span class="math-container">$x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$</span>?</p>
| Ville Salo | 123,634 | <p>It is well-known that the Chacon substitution <span class="math-container">$\tau$</span> defined by <span class="math-container">$\tau(0) = 0010$</span>, <span class="math-container">$\tau(1) = 1$</span> produces a minimal subshift, when you take the legal words to be the words that appear in some <span class="math-... |
445 | <p>Under what circumstances should a question be made community wiki?</p>
<p>Probably any question asking for a list of something (e.g. <a href="https://math.stackexchange.com/questions/81/list-of-interesting-math-blogs">1</a>) must be CW. What else? What about questions asking for a list of applications of something ... | Tom Boardman | 160 | <p>Okay, so I like Casebash's approach of taking SO as the template but, as Grigory says, there is perhaps more necessity for a divide here*, there is also more scope.</p>
<p>Certainly, as Casebash says, the <strong>broadly discursive</strong> but <strong>well motivated</strong> should be hit with the wiki hammer (and ... |
445 | <p>Under what circumstances should a question be made community wiki?</p>
<p>Probably any question asking for a list of something (e.g. <a href="https://math.stackexchange.com/questions/81/list-of-interesting-math-blogs">1</a>) must be CW. What else? What about questions asking for a list of applications of something ... | Larry Wang | 73 | <p>Some facts relevant to this question:</p>
<p><strong>The effects of making a question/answer Community Wiki:</strong> </p>
<ul>
<li>Lower threshold to edit: only requires 100 rep, compared to
1000 for normal posts </li>
<li>No reputation
is gained for upvotes, nor lost for
downvotes (this also means no -1 rep ... |
3,309,511 | <p>Prove that there exists infinitely many pairs of positive real numbers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> such that <span class="math-container">$x\neq y$</span> but <span class="math-container">$ x^x=y^y$</span>.</p>
<p>For example <span class="math-container">$\tfr... | eyeballfrog | 395,748 | <p>While I like the continuity-based approach of the other answers, you can also get a parameterized set of solutions through algebraic methods.</p>
<p>Let <span class="math-container">$x$</span> be the larger of the two and define <span class="math-container">$a \in (0,1)$</span> by <span class="math-container">$a = ... |
3,978,606 | <p>Question says</p>
<blockquote>
<p>For <span class="math-container">$(C[0,1], \Vert\cdot\Vert_{\infty})$</span>, let <span class="math-container">$B=\{f\in C[0,1] :
\Vert f\Vert_{\infty} \leq 1\}$</span>. Find all <span class="math-container">$f\in B$</span> such that
there exist <span class="math-container">$g,h\in... | Chrystomath | 84,081 | <p>Suppose <span class="math-container">$f$</span> has a point <span class="math-container">$x$</span> such that <span class="math-container">$-1<f(x)<1$</span>. Then, by continuity of <span class="math-container">$f$</span>, there is a whole interval about <span class="math-container">$x$</span>, say <span class... |
990,340 | <p>I have an arguement with my friends on a probability question.</p>
<p><strong>Question</strong>: There are lots of stone balls in a big barrel A, where 60% are black and 40% are white, black and white ones are identical, except the color.</p>
<p>First, John, blindfolded, takes 110 balls into a bowl B; afterwards, ... | Dr. Sonnhard Graubner | 175,066 | <p>we have $7^{17}<2^{51}<3^{34}$</p>
|
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Dan Piponi | 1,233 | <p>I'm surprised this example hasn't been mentioned already:</p>
<p>The 3x3x3 Rubik's cube forms a group.
The 15-puzzle forms a groupoid.</p>
<p>The reason is that any move that can be applied to a Rubik's cube can be applied at any time, regardless of the current state of the cube.</p>
<p>This is not true of the 15... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| André Henriques | 5,690 | <p>Penrose tilings are beautiful objects, with a lot of
<b>symmetry...</b>
but their symmetry group is <b>trivial</b>!</p>
<p>So there's a discrepancy somewhere. The answer is: "groupoids"!
The topological groupoid of symmetries of a Penrose tiling is non-trivial, and contains all the information that you... |
3,780,959 | <p>Consider a connected, unweighted, undirected graph <span class="math-container">$G$</span>. Let <span class="math-container">$m$</span> be the number of edges and <span class="math-container">$n$</span> be the number of nodes.</p>
<p>Now consider the following random process. First sample a uniformly random spanning... | smapers | 306,270 | <p>Approximately sampling according to the effective resistances is done in the sparsification algorithm of Spielman and Srivastava. See Theorem 2 of <a href="https://arxiv.org/abs/0803.0929" rel="nofollow noreferrer">this paper</a>. The complexity has a one-off cost of <span class="math-container">$\tilde{O}(m)$</span... |
1,341,505 | <p>Let U: $\mathbb R$ -> $\mathbb R$ be a concave function, let X be a random variable with a finite expected value, and let Y be a random variable that is independent of X and has an expected value 0. Define Z=X+Y. Prove that $E[U(X)] \ge E[U(Z)]$</p>
<p>I know that $E(X)=E(Z)$, and by Jensen's inequality $U[E(X)] \g... | Ant | 66,711 | <p>Rewrite the thesis as </p>
<p>$$E[u(X)] = \int u(x) dP^X \ge \int u(x + y) dP^X \otimes dP^Y = E[u(X + Y)]$$</p>
<p>Now let's concentrate on the right hand side; in particular we can use fubini tonelli to write </p>
<p>$$\int u(x + y) dP^X \otimes dP^Y = \int \left(\int u(x+y)dP^Y\right) dP^X$$</p>
<p>The inner ... |
3,223,732 | <p>Let <span class="math-container">$X=U\cup V$</span> where <span class="math-container">$U,V$</span> are simply-connected open sets and <span class="math-container">$U\cap V$</span> is the disjoint union of two simply connected sets. We also have the condition that any subspace <span class="math-container">$S$</span>... | 雨が好きな人 | 438,600 | <p>Here’s one suggestion (not rigorous but to give you some intuition):</p>
<p>The kernel is the set of vectors in the domain that are mapped to zero in the codomain. The dimension of the kernel can be thought of as the number of dimensions that get ‘squashed’ by the transformation. By ‘squashed’, I mean, for example,... |
3,845,968 | <p><a href="https://math.stackexchange.com/q/29666/717872">There</a>
<a href="https://math.stackexchange.com/q/11/717872">are</a>
<a href="https://math.stackexchange.com/q/363977/717872">tens</a>
<a href="https://math.stackexchange.com/q/3339682/717872">of</a>
<a href="https://math.stackexchange.com/q/2005492/717872">p... | Community | -1 | <p>The non-terminating notation (either <span class="math-container">$0.9999\cdots$</span> or <span class="math-container">$0.\overline9$</span>) is a disguised limit, namely</p>
<p><span class="math-container">$$\lim_{n\to\infty}\sum_{k=1}^n\frac 9{10^k}$$</span> or <span class="math-container">$$\lim_{n\to\infty}\lef... |
2,204,812 | <p>The solution of the differential equation $\frac{dy}{dx}-xtan(y-x)=1$ will be?</p>
<p>For solving such problems first we should see if the equation is in variable seperable form or not. Obviously here it is not. So I tried to see if it can be made to variable seperable by substitution, but substituting $y-x=z$ woul... | k.Vijay | 428,609 | <p>Substituting $y-x=z$ will give the solution.</p>
<p>let $y-x=z$ then $\dfrac{dy}{dx}=1+\dfrac{dz}{dx}$</p>
<p>Now,
\begin{align*}
\dfrac{dy}{dx}-x\cdot\tan(y-x)&=1\\
\Rightarrow1+\dfrac{dz}{dx}-x\cdot\tan z&=1\\
\Rightarrow\dfrac{dz}{dx}&=x\cdot\tan z\\
\displaystyle\int\cot z\ dz&=\displaystyle\in... |
3,932,803 | <p>I need to prove <span class="math-container">$$ \lim_{x\rightarrow\ 0}\frac{x^2-8}{{x-8}} =1 $$</span> using epsilon-delta definition.
I know I need to show that for every <span class="math-container">$\epsilon >0$</span> there exist a <span class="math-container">$\delta >0$</span> such that if <span class="m... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">$$\frac{x^2-x}{x-8}=x\frac{x-1}{x-8}.$$</span>Now, if <span class="math-container">$|x|<1$</span>, then <span class="math-container">$-1<x<1$</span> and therefore you have two things:</p>
<ul>
<li><span class="math-container">$-9<x-8<-7$</span>, which implies that... |
3,932,803 | <p>I need to prove <span class="math-container">$$ \lim_{x\rightarrow\ 0}\frac{x^2-8}{{x-8}} =1 $$</span> using epsilon-delta definition.
I know I need to show that for every <span class="math-container">$\epsilon >0$</span> there exist a <span class="math-container">$\delta >0$</span> such that if <span class="m... | Barry Cipra | 86,747 | <p>We have</p>
<p><span class="math-container">$$\left|x^2-x\over x-8\right|=|x|\left|x-1\over x-8\right|\le|x|{|x|+1\over||x|-8|}$$</span></p>
<p>Now the main thing that causes a problem is the smallness of the denominator if <span class="math-container">$|x|\approx8$</span>. But since we're interested in the limit as... |
3,932,803 | <p>I need to prove <span class="math-container">$$ \lim_{x\rightarrow\ 0}\frac{x^2-8}{{x-8}} =1 $$</span> using epsilon-delta definition.
I know I need to show that for every <span class="math-container">$\epsilon >0$</span> there exist a <span class="math-container">$\delta >0$</span> such that if <span class="m... | CopyPasteIt | 432,081 | <p>We've solved the problem on scrap paper and now present a solution devoid of the process details.</p>
<p>It is easy to show that</p>
<p><span class="math-container">$\quad \large |x| \lt \frac{1}{6} \implies |\frac{x-1}{x-8}| \lt \frac{1}{7}$</span></p>
<p>For <span class="math-container">$\varepsilon \gt 0$</span>... |
94,501 | <p>The well-known Vandermonde convolution gives us the closed form <span class="math-container">$$\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$$</span>
For the case <span class="math-container">$r=s$</span>, it is also known that <span class="math-container">$$\sum_{k=0}^n (-1)^k {r \choose k} {r \choose n... | Robert Israel | 8,508 | <p>According to Maple, the answer is ${s\choose n}{{}_2F_1(-r,-n;\,s-n+1;\,-1)}$
(of course we must assume $s \ge n$ for this to make sense).</p>
|
3,238,914 | <p>When is the <a href="https://en.wikipedia.org/wiki/Euler_line" rel="nofollow noreferrer">Euler line</a> parallel with a triangle's side?</p>
<p>I have found that a triangle with angles <span class="math-container">$45^\circ$</span> and <span class="math-container">$\arctan2$</span> is a case.</p>
<p>Is there any o... | Blue | 409 | <p>The figure shows <span class="math-container">$\triangle ABC$</span> with orthocenter <span class="math-container">$P$</span>, circumcenter <span class="math-container">$Q$</span>, and circumradius <span class="math-container">$r$</span>. For a non-equilateral, the Euler line is determined by <span class="math-conta... |
2,930,292 | <p>I'm currently learning the unit circle definition of trigonometry. I have seen a graphical representation of all the trig functions at <a href="https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/unit-circle-definition-of-trig-functions/a/trig-unit-circle-review" rel="nofollow noreferrer">khan academ... | KM101 | 596,598 | <p>If you complete the diagram with all the right triangles, you would notice all of them are actually similar, meaning pairs of corresponding sides produce equivalent ratios. I’m not exactly sure how to explain this considering there aren’t any points in your diagram, so here’s a link to the proof I used.<br>
<a href... |
2,845,085 | <p>Find $f(5)$, if the graph of the quadratic function $f(x)=ax^2+bx+c$ intersects the ordinate axis at point $(0;3)$ and its vertex is at point $(2;0)$</p>
<p>So I used the vertex form, $y=(x-2)^2+3$, got the quadratic equation and then put $5$ instead of $x$ to get the answer, but it's wrong. I think I shouldn't hav... | Matheus Nunes | 523,585 | <p>Intersect the ordinate in $(0;3)$ implies: $$3=f(0)=a0^2+b0+c=c$$
If you use $x=\frac{-b}{2a}$, with $x$ the vertex, you get
$$2=-b/2a \implies b=-4a$$
Evolve $f(2)=0$
using the equation of the form given in the problem, then you will have
$$0=a2^2+b2+3 \implies 2b=-3-4a = -a+b \implies b=-3$$
So, $$-3=-4a \impli... |
1,057 | <p>Suppose a finite group has the property that for every $x, y$, it follows that </p>
<p>\begin{equation*}
(xy)^3 = x^3 y^3.
\end{equation*}</p>
<p>How do you prove that it is abelian?</p>
<hr>
<p>Edit: I recall that the correct exercise needed in addition that the order of the group is not divisible by 3.</p>
| Jorge Miranda | 446 | <p>On the other hand, if the order of your group is not a multiple of 3 then it must be abelian!</p>
<p>You can read a proof <a href="http://groupprops.subwiki.org/wiki/Cube_map_is_endomorphism_iff_abelian_%28if_order_is_not_a_multiple_of_3%29" rel="noreferrer">here</a></p>
|
879,886 | <p>If one number is thrice the other and their sum is $16$, find the numbers.</p>
<p>I tried,
Let the first number be $x$ and the second number be $y$
Acc. to question </p>
<p>$$
\begin{align}
x&=3y &\iff x-3y=0 &&(1)\\
x&=16-3y&&&(2)
\end{align}
$$</p>
| Disha Sidhwani | 665,944 | <p>Let the first number be <span class="math-container">$x$</span>
And second which is thrice be <span class="math-container">$3y$</span></p>
<p>Acc to ques.. </p>
<p><span class="math-container">$x=3y\ldots \textrm{equation 1}$</span></p>
<p><span class="math-container">$x+y=16\ldots\textrm{equation 2}$</span></p>
... |
2,918,091 | <p>Suppose I want to find the locus of the point $z$ satisfying $|z+1| = |z-1|$</p>
<p>Let $z = x+iy$</p>
<p>$\Rightarrow \sqrt{(x+1)^2 + y^2} = \sqrt{(x-1)^2 + y^2}$ <br/>
$\Rightarrow (x+1)^2 = (x-1)^2$ <br/>
$\Rightarrow x+1 = x-1$ <br/>
$\Rightarrow 1= -1$ <br/>
$\Rightarrow$ Loucus does not exist</p>
<p>Is my a... | Peter Szilas | 408,605 | <p>Consider the points $A (-1,0)$ and B $(1,0)$ in the complex plane.</p>
<p>Let $z$, call it $C$, be any point in the complex plane.</p>
<p>In $\triangle ABC$ length $BC = |z-1$|, length $AC = |z+1|$.</p>
<p>Since $|z-1|=|z+1|$, $ \triangle ABC$ is isosceles with base $AB$.</p>
<p>Hence the locus of $z$ is the pe... |
447,484 | <p>I am just learning about differential forms, and I had a question about employing Green's theorem to calculate area. Generalized Stokes' theorem says that $\int_{\partial D}\omega=\int_D d\omega$. Let's say $D$ is a region in $\mathbb{R}^2$. The familiar formula to calculate area is $\iint_D 1 dxdy = \frac{1}{2}\int... | Bill Cook | 16,423 | <p>You <strong>can</strong> use $\int_{\partial D} x\,dy$ to compute area in this context. The "familiar formula" does have a more symmetric look to it -- maybe that's why you find it more familiar. </p>
<p>There are infinitely many formulas like this that work. In general you need two functions $P$ and $Q$ such that ... |
1,644,905 | <p>How to simplify the following equation:</p>
<p>$$\sin(2\arccos(x))$$
I am thinking about:</p>
<p>$$\arccos(x) = t$$</p>
<p>Then we have:</p>
<p>$$\sin(2t) = 2\sin(t)\cos(t)$$</p>
<p>But then how to proceed?</p>
| KR136 | 186,017 | <p>So, you have reasoned that the expression is equivalent to: </p>
<p>$2\sin(\arccos(x))\cos(\arccos(x))$</p>
<p>Because $\arccos(x) ≡ \cos^{-1}(x)$, this is equivalent to </p>
<p>$2x\sin(\arccos(x))$</p>
<p>$\sin(\arccos(x))$ is in fact equivalent to $\sqrt{1+x^2}$ by identity. This can be shown by the Pythagorea... |
3,873,521 | <p>Going through Christof Paar's book on cryptography. In his chapter on DHKE, he has the following</p>
<p><a href="https://i.stack.imgur.com/xQQPc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xQQPc.png" alt="enter image description here" /></a></p>
<p>The book doesn't seem to have the proof for <... | halrankard2 | 819,436 | <p>You've worded the question in a way that suggests the book <em>does</em> have the proof of part 2. But, just in case, the proof is that any <span class="math-container">$a\in G$</span> generates a (cyclic) subgroup of <span class="math-container">$G$</span> of size <span class="math-container">$ord(a)$</span> and, m... |
3,873,521 | <p>Going through Christof Paar's book on cryptography. In his chapter on DHKE, he has the following</p>
<p><a href="https://i.stack.imgur.com/xQQPc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xQQPc.png" alt="enter image description here" /></a></p>
<p>The book doesn't seem to have the proof for <... | Oliver Kayende | 704,766 | <p>Each translation <span class="math-container">$gH:=\{gh:h\in H\}$</span> of a <span class="math-container">$G$</span> subgroup <span class="math-container">$H$</span> is called a <span class="math-container">$\it{left}\;H\;coset$</span>. The left <span class="math-container">$H$</span> cosets all have the same size ... |
3,014,670 | <p>I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors. </p>
<p><em>I've done some initial research, and reviews these Stack questions: <a href="https://math.stackexchange.com/quest... | John L Winters | 603,347 | <p>All rational people are not lakers.</p>
<p>The opposite converse is equally valid.</p>
<p>The converse that you posit is not equally valid.</p>
|
3,014,670 | <p>I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors. </p>
<p><em>I've done some initial research, and reviews these Stack questions: <a href="https://math.stackexchange.com/quest... | user | 505,767 | <p>Indicating with <span class="math-container">$L$</span> the set of lakers <span class="math-container">$l$</span> and with <span class="math-container">$\Pi$</span> the set of irrational people <span class="math-container">$\pi$</span>, the first statement is equivalent to</p>
<p><span class="math-container">$$\for... |
3,873,433 | <p>Is it true that for all square, complex matrices A, B
<span class="math-container">$$
\left\|AB\right\|_p\leq\left\|A\right\|\left\|B\right\|_p$$</span></p>
<p>where <span class="math-container">$\left\| .\right\|_p$</span> refers to the Schatten p-norm and <span class="math-container">$\left\| .\right\|$</span> ref... | user1551 | 1,551 | <p>One can use the minimax principle for singular values to prove that <span class="math-container">$\sigma_k(AB)\le\sigma_1(A)\sigma_k(B)$</span> for each <span class="math-container">$k$</span>. The inequality in question now follows directly.</p>
|
3,181,696 | <p>I am attempting to solve the integral of the following...</p>
<p><span class="math-container">$$\int_{0}^{2 \pi}\int_{0}^{\infty}e^{-r^2}rdr\Theta $$</span></p>
<p>So I do the following step...</p>
<p><span class="math-container">$$=2 \pi\int_{0}^{\infty}e^{-r^2}rdr$$</span></p>
<p>but then the next step is to s... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$s=-r^{2}$</span> gives <span class="math-container">$ds=-2rdr$</span> so <span class="math-container">$dr =-\frac 1 {2r} ds$</span>. Also, as <span class="math-container">$r$</span> increases from <span class="math-container">$0$</span> to <span class="math-container">$\infty$</span>, <... |
3,181,696 | <p>I am attempting to solve the integral of the following...</p>
<p><span class="math-container">$$\int_{0}^{2 \pi}\int_{0}^{\infty}e^{-r^2}rdr\Theta $$</span></p>
<p>So I do the following step...</p>
<p><span class="math-container">$$=2 \pi\int_{0}^{\infty}e^{-r^2}rdr$$</span></p>
<p>but then the next step is to s... | DINEDINE | 506,164 | <p>Do you really need substitution. We already know the antiderivative of <span class="math-container">$re^{-r^2}$</span> and it is <span class="math-container">$-e^{-r^2}\over 2$</span></p>
|
57,232 | <p>Given a Heegaard splitting of genus $n$, and two distinct orientation preserving homeomorphisms, elements of the mapping class group of the genus $n$ torus, is there a method which shows whether or not these homeomorphisms, when used to identify the boundaries of the pair of handlebodies, will produce the same $3$-m... | Peter Humphries | 3,803 | <p>Expanding on Matt's answer, it is possible to show without too much difficulty (see <a href="http://www-personal.umich.edu/~hlm/math775/ch18.pdf">here</a>, Exercise 3 of section 18.2.1) that if $(a,q) = 1$, then
$$\sum_{n \leq x}{\mu(n) e^{2\pi i an/q}} = \sum_{d \mid q} \frac{\mu(d)}{\varphi(q/d)} \sum_{\chi \pmod{... |
3,956,913 | <p>For this equation :</p>
<blockquote>
<p><span class="math-container">${ (x^2 - 7x + 11)}^{x^2 - 13x +42}=1$</span></p>
</blockquote>
<p>The integer solutions of <span class="math-container">$x$</span> found by WolframAlpha using inverse (logarithmic) function are <span class="math-container">$ 2 , 5 , 6 , 7 .$</span... | trancelocation | 467,003 | <p>You can use the binomial theorem but in a slightly adopted manner.</p>
<p>Consider</p>
<p><span class="math-container">$$\left(\frac{(n+1)^{\frac 1{n+1}}}{n^{\frac 1n}}\right)^{n(n+1)}=\frac 1n\left(1+\frac 1n\right)^n$$</span></p>
<p>Now use the binomial theorem on <span class="math-container">$\left(1+\frac 1n\rig... |
798,307 | <p>I have a linear functional from the space of nxn matrices over a field F. The functional satisfies $f(A) = f(PAP^{-1})$ for all invertible $P$ and $A$ an nxn matrix. I'm trying to show that $f(A) = \lambda tr(A)$ for some constant $\lambda$.</p>
<p>So far I have:</p>
<ul>
<li>The linear functionals have basis $f_{... | EPS | 133,563 | <p>I write this as a separate answer, because the method of solution is different from my previous post.</p>
<p>By your first observation, or simply using the Riesz Representation Theorem, one can deduce that there exists a matrix $B$ such that $f(X)=\text{tr }(BX)$. Since $f(X)=f(PXP^{-1})$ and $\text{ tr} AB=\text{t... |
1,419,483 | <p>Can anyone please help me in solving this integration problem $\int \frac{e^x}{1+ x^2}dx \, $?</p>
<p>Actually, I am getting stuck at one point while solving this problem via integration by parts.</p>
| Yash | 246,871 | <p><a href="http://www.wolframalpha.com/input/?i=integrate%28exp%28x%29%2F%281%2Bx" rel="nofollow">http://www.wolframalpha.com/input/?i=integrate%28exp%28x%29%2F%281%2Bx</a>*x%29%29</p>
<p>Read from the link
It cannot be solved using byparts</p>
|
443,475 | <p>I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining
$$ ab = a\cdot b + a \wedge b$$ Yes, it can be done like complex numbers, but what will we lose if we deal with inner product and wedge ... | Muphrid | 45,296 | <p>It is, perhaps, misleading to even call this addition. It is no more (and no less) addition than it is addition to add $5 e_1$ and $3 e_2$. You might say, "Of course we can add those. They're members of the same vector space; you just add corresponding components."</p>
<p>Well, we can do the same thing with mult... |
443,475 | <p>I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining
$$ ab = a\cdot b + a \wedge b$$ Yes, it can be done like complex numbers, but what will we lose if we deal with inner product and wedge ... | user48672 | 138,298 | <p>Well, actually the perspective should be reversed.
Starting from the geometric product $a b$ having an inverse as mentioned
$$
a a^{-1}=1
$$
one can define 2 identities
$$
a \cdot b= \frac{1}{2} \left(a b + \alpha(a) \, \alpha (b) \right)
$$
where $\alpha$ is the reflection automorphism
$$
a \wedge b= \frac{1}{2... |
635,989 | <p>What counterexample can I use to prove that ($ \mathbb{R}_{[x]}$is any polynomial):</p>
<p>$L :\mathbb{R}_{[x]}\rightarrow\mathbb{R}_{[x]},(L(p))(x)=p(x)p'(x)$ is not linear transformation. I have already proven this using definition but it is hard to think about example. I would be grateful for any help.</p>
| Thomas | 26,188 | <p>How about this?
$$
L(x + x^2) = (x+x^2)(1 + 2x) = \dots
$$
$$
L(x) + L(x^2) = \dots
$$</p>
<hr>
<p>As a side note, if you want to <em>prove</em> that $L$ is not linear, you just have to provide one example where one of the properties fail. You say that you have proved it "using the definition" (I am not sure what ... |
635,989 | <p>What counterexample can I use to prove that ($ \mathbb{R}_{[x]}$is any polynomial):</p>
<p>$L :\mathbb{R}_{[x]}\rightarrow\mathbb{R}_{[x]},(L(p))(x)=p(x)p'(x)$ is not linear transformation. I have already proven this using definition but it is hard to think about example. I would be grateful for any help.</p>
| Michael Hoppe | 93,935 | <p>$L(x+1)=(x+1)\cdot1$, whereas $L(x)+L(1)= x\cdot1+1\cdot0$.</p>
<p>Edit: evidently $L(2x)\neq2L(x)$.</p>
|
4,547,480 | <p>I am working with some data for which I am interested in calculating some physical parameters. I have a system of linear equations, which I can write in matrix form as:</p>
<p><span class="math-container">$$
\textbf{A} \textbf{x} = \textbf{b}
$$</span></p>
<p>where <span class="math-container">$\textbf{A}$</span> is... | Erwin Kalvelagen | 295,867 | <p>You can use Linear Programming with a dummy objective. The usual approach is to use an objective with all zero coefficients:
<span class="math-container">$$\begin{aligned} \min_x \> & 0^Tx \\ & Ax=b \\ & 0 \le x_i \le u_i \end{aligned}$$</span></p>
<p>This will find a feasible solution and then stop.<... |
2,401,281 | <blockquote>
<p>Show that for $\{a,b,c\}\subset\Bbb Z$ if $a+b+c=0$ then $2(a^4 + b^4+ c^4)$ is a perfect square. </p>
</blockquote>
<p>This question is from a math olympiad contest. </p>
<p>I started developing the expression $(a^2+b^2+c^2)^2=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2$ but was not able to find any useful... | Michael Rozenberg | 190,319 | <p>Since $$2(a^2b^2+a^2c^2+b^2c^2)-a^4-b^4-c^4=(a+b+c)(a+b-c)(a+c-b)(b+c-a)=0,$$
we obtain
$$2(a^4+b^4+c^4)=a^4+b^4+c^4+2(a^2b^2+a^2c^2+b^2c^2)=(a^2+b^2+c^2)^2.$$
Done!</p>
|
2,401,281 | <blockquote>
<p>Show that for $\{a,b,c\}\subset\Bbb Z$ if $a+b+c=0$ then $2(a^4 + b^4+ c^4)$ is a perfect square. </p>
</blockquote>
<p>This question is from a math olympiad contest. </p>
<p>I started developing the expression $(a^2+b^2+c^2)^2=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2$ but was not able to find any useful... | achille hui | 59,379 | <p>A systematic way doing this is using <a href="https://en.wikipedia.org/wiki/Newton's_identities" rel="nofollow noreferrer">Newton's identifites</a>.</p>
<p>Let $p_k = a^k + b^k + c^k$ for $k = 1, 2, 3, 4$ and
$$\begin{align}
s_1 &= a + b + c\\
s_2 &= ab+bc+ca\\
s_3 &= abc
\end{align}$$
be the eleme... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Thomas Riepe | 451 | <p><a href="http://www.pbs.org/wgbh/nova/sciencenow/3210/04.html" rel="nofollow" title="PBS docu">This video</a> is less about mathematics, but about a fascinating mathematician in two bodies who helped saving medieval unicorns - students liked it. </p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Konrad Voelkel | 956 | <p>My personal all-time favorite is the Klein Four with their song "Finite Simple Group (of Order Two)"... it has lots of puns on topology in it, but I guess it doesn't teach anything.</p>
<p><a href="https://www.youtube.com/watch?v=BipvGD-LCjU" rel="noreferrer" title="Go to the Klein Four Video on YouTube">Here's the... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Andrew Stacey | 45 | <p>On <a href="http://www.k-3d.org/wiki/Animation_Gallery" rel="nofollow">this page</a> of sample animations using the k3d program there's a short animation of a "flower" blooming which is actually the first part of the sphere eversion.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | GMRA | 135 | <p>The Newton institute in Cambridge tapes alot (all?) of it's lectures, and they can be found on the Institutes webpage. High quality for videos of lectures.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Mitch | 825 | <p>The Institute for Advanced Study tapes some of its <a href="http://video.ias.edu/">lectures</a>. They tend to be very good.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | vonjd | 1,047 | <p>Among the best math videos can be found here:
<a href="http://www.khanacademy.org/" rel="nofollow">http://www.khanacademy.org/</a></p>
<p>(or the youtube-channel: <a href="http://www.youtube.com/khanacademy" rel="nofollow">http://www.youtube.com/khanacademy</a> )</p>
<p>There is everything from counting to solving... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Spinorbundle | 675 | <p>Sir Michael Atiyah: <a href="https://www.youtube.com/watch?v=dToui7IVwBY" rel="nofollow noreferrer">Beauty in Mathematics</a>.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Bob | 13,822 | <p>Documentary about infinite and its implications in mathematics (BBC)<br>
<a href="http://www.youtube.com/watch?v=Cw-zNRNcF90" rel="nofollow">http://www.youtube.com/watch?v=Cw-zNRNcF90</a></p>
<p>As usual, Gregory Chaitin on the history of logic<br>
<a href="http://www.youtube.com/watch?v=HLPO-RTFU2o" rel="nofollow"... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Jesus Martinez Garcia | 1,887 | <p>All the talks of <a href="http://www.maths.ed.ac.uk/~aar/atiyah80.htm" rel="nofollow">Atiyah 80+</a></p>
|
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