Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Centroid of a Triangle on a inscribed circle $AB$ is the hypotenuse of the right $\Delta ABC$ and $AB = 1$.
Given that the centroid of the triangle $G$ lies on the incircle of $\Delta ABC$, what is the perimeter of the triangle?
| I agree with the answer by @Jack D'Aurizio, but I just wanted to suggest a quicker way not involving too much algebra and no trigonometry:
Firstly we can establish, by consideration of equal tangents to the incircle, that if $P$ is the perimeter of the triangle and $r$ is the radius of the incircle, then $$P=1+1+r+r\Ri... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove the digital root of a square can be anything other than $2, 3, 5, 6, 8$? The digital root is the sum of the digits, unless that has more than one digit, so then you add up the digits again, until arriving at a single digit, e.g., $28$ -> $2 + 8 = 10$ -> $1 + 0 = 1$.
For what the digital root of a square can be, I... | First of all, $1001^2$ is not going to break the pattern you've observed so far, being $1002001$ and thus having a digital root of $4$. Squares with a digital root of $4$ follow or precede squares with a digital root of $1$, like $1000^2$.
But you're right in general to distrust the evidence of a thousand examples. In ... | {
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Is the following a conic section All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I find it:
$$
\begin{align}
\mathbf{r}^\mathrm{T} \left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{... | No, these are not conic sections in general.
Your first equation forces $z = 0$. For \begin{align*}
\mathbf{r}_1 = \mathbf{v}_2 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \\
\mathbf{r}_2 = \mathbf{v}_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \text{,}
\end{align*}
with $c = 0$, your second equation reduces to ... | {
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Marbles Combinations problem Martin’s bag of marbles contains two red, three blue and five green marbles. If he reaches in to pick some without looking, how many different selections might he make?
I do not know how to approach this question. It asks if he were to pick some marbles. What does that mean?
| Martin has $3$ choices for how many red. For he can choose $0$ or $1$ or $2$. For each of these choices, he has $4$ choices for how many blue, and then $\dots$.
Remark: We assumed that marbles of the same colour are indistinguishable. We also assumed that the choice of no marbles is allowed. That is a matter of interp... | {
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10% lower and 10% higher of 100 I'm confused about how to get the $10\%$ higher and $10\%$ lower of $100$. I'm alone I don't know if my idea is correct. My idea is $10\%$ higher of $100$ is $110$, then the $10\%$ lower of $100$ is $90$, then the range $10\%$ higher and $10\%$ lower of $100$ is between $90$ to $110$?
| Yes.
But maybe it is less confusing to say 10% lower/higher than 100, or 10% below/above 100.
| {
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Understanding Primitive roots I am trying to find a single primitive root modulo $11$. The definition in my textbook says "Let $a$ and $n$ be relatively prime integers with ($a \neq 0$) and $n$ positive. Then the least positive integer $x$ such that $a^x\equiv1\pmod{\! n}$ is called the order of $a$ modulo... | One cannot in general find primitive roots without trying, but it usually does not take many trials. (If your prime $p$ is so large that factoring $p-1$ is a problem, then just testing whether a given number is a primitive root may be a stumbling block, but that is a different matter.)
In the given case, you can just w... | {
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Compact Hausdorff spaces are normal I want to show that compact Hausdroff spaces are normal.
To be honest, I have just learned the definition of normal, and it is a past exam question, so I want to learn how to prove this:
I believe from reading the definition, being a normal space means for every two disjoint closed ... | Outline: Start with proving regularity i.e. for $x\in X$ and a closed subset $A\subset X$ not containing $x$, there are disjoint open subsets $U,V\subset X$, such that $x\in U$ and $A\subset V$. For this, note that $A$ is compact, as a closed subset of a compact space. Since $X$ is Hausdorff, for every $a\in A$ there a... | {
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Summation functions for wall clock, 10AM, 11AM and 12PM tips needed For a recreational purposes I'm fine tuning my wall clock sheet and like to ask about tips how to esthetically modify the summation function for 10, 11 and 12. Below is the image of the final result:
I have checked sigmas in wolframalpha so that they ... | For 10, $\sum_{i=1}^{1+1}i(i+i)$ has the same effect with one fewer $i$.
For 11, $\sum_{i=1-1}^{1+1}(i+i)^i$ violates the "three 1s" rule instead (and requires the fairly common $0^0=1$ convention), but you may like it better than explicitly using 11.
For 12, $\sum_{i=-1}^{1+1}i(i+i)$ works.
| {
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"timestamp": "2023-03-29T00:00:00",
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Direct proof of inequality between arithmetic and harmonic mean I need to prove inequality from the title.
I know that it follows from $H_n \leq G_n \leq A_n \leq Q_n$, where $H_n, G_n, A_n, Q_n$ are harmonic, geometric, arithmetic and quadratic means of $n$ real numbers, but for some purpose, I need to prove directly... | So we have to prove that $a_i>0$ gives:
$$\frac{a_1+\ldots+a_n}{n}\leq \frac{n}{\frac{1}{a_1}+\ldots+\frac{1}{a_n}}\tag{1}$$
but by Titu's lemma:
$$\left(\frac{b_1^2}{a_1}+\ldots+\frac{b_n^2}{a_n}\right)\left(a_1+\ldots+a_n\right)\geq (b_1+\ldots+b_n)^2 \tag{2}$$
so $(1)$ trivially follows by taking $b_i=1$. $(2)$ can ... | {
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Finding some rational points on elliptic curves
If I am considering an elliptic curve, for example
$$y^2=x^3-2$$
$$\text{Edit: and } y^2=x^3+2$$
over $\mathbb Q$, how to find rational points?
What possibilities do we have to calculate some of the rational points on it? Are there even possibilities for calculat... | The simplest way is to use existing methods in computer algebra systems, e.g. if you use the online Magma calculator here there are now awfully sophisticated algorithms there for this sort of thing. To learn more you could read the relevant section in the Magma handbook here
In the case of the first of your curves, if ... | {
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"timestamp": "2023-03-29T00:00:00",
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Approximation Reasoning I can't understand one step in the following problem. We start with a function $f(x)=x^\alpha$ on the interval $(0,1)$ where $\alpha>0$ is a constant. We pick two points $x_1<x_2$ from this interval so that we have $x_1^\alpha=x_2^\alpha-L$, where $L>0$ depends on $x_1$ and $x_2$. Upon rearrangi... | Presumably, the first-order approximation is performed with respect to $L$, not $x_2$ (which is consistent with the definition of $L$ as the difference between the powers of two close numbers) . In that case, the taylor expansion gives
$$x_1 = x_{2} - \frac{L x_{2}}{\alpha x_{2}^{\alpha}}\text{ ,}$$
which is the indee... | {
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Minimum value of $\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\frac{24}{5\sqrt{5a+5b}}$ Let $a\ge b\ge c\ge 0$ such that $a+b+c=1$
Find the minimum value of $P=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\dfrac{24}{5\sqrt{5a+5b}}$
I found that the minimum value of $P$ is $\dfrac{78}{5\sqrt{15}}$ when $a=b=\dfrac{3}{8};c... | You can use calculus to find this.
$f(a,b) = \sqrt{\frac{a}{1-a}} + \sqrt{\frac{b}{1-b}} + \frac{24}{5\sqrt{5(a + b)}}$
Now, let us first find the critical point, $(x,y)$, of this function, where $f_a(x,y) = 0$ and $f_b(x,y) = 0$.
Taking partial derivatives and setting them to 0, you will see that the critical point is... | {
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Given $26$ balls - $8$ yellow, $7$ red and $11$ white - how many ways are there to select $12$ of them? I'm interested in knowing and understanding the solution to the following problem:
given $26$ balls - $8$ yellow, $7$ red and $11$ white - how many ways are there to select $12$ of them (all balls of the same colou... | Forget for a while about the fact that the numbers of each colour are limited. If there were at least $12$ of each colour, the problem would be straightforward Stars and Bars. The answer would be $\binom{12+3-1}{3-1}$.
From this we must subtract the "bad" choices that involve using more balls of a given colour than ar... | {
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Difference between topology and sigma-algebra axioms. One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically but is there a way... | An easy way to get a feeling for this is to consider basic examples.
For example, let $X=\{1, 2, 3\}$.
A topological space $(X, )$ could be for constructed by choosing for example $=\{∅,\{1, 2\},\{2\},\{2,3\},X\}$.
But this is as far from a $σ$-algebra as you can get since in fact no complement of any set in $$ is in ... | {
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"timestamp": "2023-03-29T00:00:00",
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Naïve groups, fields and ideals Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right track. I confess I am a little averse to reading about them from the ground up, as my motiv... | While that's actually a pretty clever idea, it's not what ideal means. First of all, Ideals need to be subrings, so if $16$ and $-4$ are in the ideal, then so must $16+(-4)=12$. Secondly, Ideals are meant to be "closed under multiplication." The idea is to generalize the notion of "has a factor of n." So, for any numbe... | {
"language": "en",
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Isomorphism between $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$ I hope you can help me with this:
Show that $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$ are isomorphic, and specify an isomorphism.
Thanks.
| In the beginning I just specify what Rob Arthan meant. There are following three theorems. For simplicity considered vector spaces are over filed $\mathbb{R}.$
1.If $V$ and $W$ are finite dimensional vector spaces then $\dim(V\times W)=\dim(V)+\dim(W).$
Be awere that in category of vector spaces we can use $\times$ a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to check if $x_{100}$ is prime or not? I have $$x_{n}=5x_{n-1}-4x_{n-2}+6$$ and I have found that the $n$-th term is$$x_{n}=-{1\over3}+{7\over12}\cdot4^n-2n$$ I must demonstrate if $x_{100}$ is a prime number or not. How should I begin? I must find if $x_{100}$ is divisible by $3$ or not.
| $$x_n=-{1\over3}+{7\over3}4^{n-1}-2n$$
$x_n$ is an integer for all $n\ge1$.
Proof:
$$7\cdot4^{n-1}\equiv1\cdot1^{n-1}=1\mod 3$$
So fractional part of $x_n=-{1\over3}+{1\over3}=0$.
$x_{100}$ is divisible by $3$.
Proof:
$$7\cdot4^{100-1}\equiv7\cdot64^{33}\equiv7\cdot1^{33}=7\mod 9\\
200\equiv2\mod 3$$
So fractional... | {
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Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator? $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$.
I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then t... | Since $1$ is obviously smaller than all the rest, we can skip it.
Check the rest by taking a pair of numbers, comparing them and proceeding with the larger one:
*
*Take $2^{1/2}$ and $3^{1/3}$
*Raise them both to the power of $6$ (the LCM of $2$ and $3$)
*Since they are both positive, their order will be preserve... | {
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"timestamp": "2023-03-29T00:00:00",
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Quick way to solve the system $\displaystyle \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} = \frac{65}{36}$, $xy-x+y=118$. Consider the system
$$\begin{aligned} \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} & = \frac{65}{36}, \\ xy -x +y & = 118. \end{aligned}$$
I have solved... | Consider the equation $$\left( \dfrac{3}{2} \right)^{x-y} - \left( \dfrac{2}{3} \right)^{x-y} = \dfrac{65}{36}.$$ Let $u=2^{x-y}$ and $v=3^{x-y}$ then we have $36u^2+65uv-36v^2=0.$ Hence $$u=\dfrac49v,\,\,\,\text{or}\,\,\,\,u=-\dfrac94v$$ For real solutions, take the first one and then $\dfrac{u}{v}=\left(\dfrac23\righ... | {
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In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least one ball? In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least 1 ball?
I proceeded like this- In the first row choosing one ... | For a given configuration $c$ let $R_c$ denote the multiset counting the number of balls in each row. We have two cases:
First case: $R_c = \{3, 1, 1\}$
For the row of 3 balls, we have only one configuration, i.e. all cells have a ball. For each row of 1 ball, we can put the ball in one of 3 positions. Also, we have 3 ... | {
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Proving binary integers This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with binary integers (For ${0, 1, 2, 3}$ we have the representations $0, 1, 10, 11$), but other than that, the textbook gave no hints really and I'm really not sure about how ... | The base case is $n=0$, which has binary representation $0_2$.
For the induction step, assume that all integers less than $n$ have a binary representation.
Write $n=2m+r$, with $r=0$ or $r=1$. By induction, $m=(b_k \cdots b_1 b_0)_2$. Thus, $n=(b_k \cdots b_1 b_0r)_2$.
This is one example that induction from $n$ to $n+... | {
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Find the cubic equation of $x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$ Find the cubic equation which has a root $$x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$$
My attempt is
$$x^3=2-\sqrt{3}+3\left(\sqrt[3]{(2-\sqrt{3})^2}\right)\left(\sqrt[3]{(2+\sqrt{3})}\right)+3\left(\sqrt[3]{(2-\sqrt{3})}\right)\left(\sqrt[3]{(... | $x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}\\
\implies x^3=2-\color{red}{\sqrt3}+2+\color{red}{\sqrt3}+3(2-\sqrt3)(2+\sqrt3)(\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}})\\
\implies x^3=4+3.1.\color{red}{x}\\
\implies x^3-4-3x=0$
| {
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"timestamp": "2023-03-29T00:00:00",
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Substituting the value $x=2+\sqrt{3}$ into $x^2 + 1/x^2$ My teacher gave me a question which I am not able to solve:
If $x=2+\sqrt{3}$ then find the value of $x^2 + 1/x^2$
I tried to substitute the value of x in the expression, but that comes out to be very big.
| Since everyone else has answered with various shortcut methods, let me just show you how you could simply substitute in directly and get an answer cleanly without things ever getting 'too big'.
We know that $x=2+\sqrt{3}$, so we can square this using the usual $a^2+2ab+b^2$ binomial formula: $x^2=2^2+2(2)(\sqrt3)+(\sqr... | {
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$ \lim_{x \to 0^+} \frac{f(f(x)) }{f^{-1}(x)}$
Suppose that $f \in \mathcal{C} ^1 \ ([0,1])$ and that $\displaystyle\lim_{x \to 0^+} \frac{f(2x^2)}{\sqrt {3}x^2} = 1$.
Find $\displaystyle \lim_{x \to 0^+} \frac{f(f(x)) }{f^{-1}(x)}$.
I don't know were to begin. They don't say if $\lim_{x \to 0^+} f(2x^2) = 0$, so ... | It is easy to verify that $f'(0)=\frac{\sqrt3}2$ and $f(0)=0=f^{-1}(0)=f(f(0))$
So , $\lim_{x\to0} \frac{f(f(x))}{f^{-1}(x)}$=$\lim_{x\to0} \frac{f(f(x))-f(f(0))}{x-0}\frac 1 {\frac{f^{-1}(x)-f^{-1}(0)}{x-0}}$=$\frac{(f'(0))^2}{(f^{-1})'(0)}$
and $(f^{-1})'(y)=\frac1{f'(x)}$ where $y=f(x)$ i.e. $(f^{-1})'(0)=\frac1{f'(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1331606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fibonacci spiral in octopus tentacles. How you happened to notice the presence of the Fibonacci spiral in nature it is really evident. For example, unlike octopuses, squid and cuttlefishes, the nautilus kept its stunning shell, which is well known for its elaborate internal Fibonacci spiral pattern.
Can you recommend a... | You could check the book Self made tapestry from Phillip Ball
An another one wich is pretty classical On form from D'Arcy Wentworth Thompson
but in the first place I suggest you take a look in the following video
https://www.youtube.com/watch?v=ahXIMUkSXX0.
It is great!
You will see that often what you get is not a Fi... | {
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A classical solution of Poisson's equation is also a weak solution Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ u&=&0&&\text{on }\partial \Omega\end{matrix}\right.\tag{1}$$ Using Gauss's ... | Using Green's first identity what you would find is
$$\int_{\Omega} \Delta u \phi = \int_{\partial \Omega} \phi \langle\nabla u, \nu \rangle - \int_{\Omega} \langle\nabla u, \nabla \phi\rangle$$
But $\phi$ is $0$ on the boundary so really you just distribute the Laplacian into two gradients, and pick up a minus sign.
| {
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How to find out how big a ball is? Ok, This is probably a really simple question but. I need to know how I can find out how big a ball is. For example, a tennis ball is 2 1/2 inches big, but how do you find that?
Though, for reference, the explanation and answer to this question needs to be as simple as it can possibly... | Make a cylinder of paper that is large enough to enclose the ball. Put the ball on a table and close the cylinder around the ball. Mark where the paper's (verticle) edge meets the other side of the ball while the cylinder is perpendicular to the table. This allows you to measure the circumference. Then the division ... | {
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Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$ So here the problem goes:
Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$.
This is a very interesting word problem that I came across in an old textbook of mine. So I know it's got something to do with
If $x = 1,$ then we have $1 + \frac 11 = 2,$
and clearly, if $x... | Since we are assuming that $x$ is positive, multiplying by $x$ gives the equivalent inequality
$$x^2+1\geq 2x,\quad \forall x>0.$$
Rearranging, this is
$$(x-1)^2\ge 0,\quad \forall x>0,$$
which is obvious.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Max/min problem, find max area of sector of circle.
A sector of a circle has fixed perimeter. For what central angle $θ$ (in radians) will the area be greatest?
First I put together an equation for the perimiter of the sector being $p$ which I assume as a constant since it is fixed:
$$p = rθ + 2r$$
Since p is a cont... | The area $A$ of a sector is $A = (1/2)(R^2)(\theta)$.
Substitute $R = {P \over {(\theta + 2)}}$ where $P$ is perimeter in above equation then differentiate w.r.t $\theta$. Then equate it to zero to find theta for which Area is Max.
Leave $P$ as it is. Don't put it equal to 1. Treat it as you treat constants.
Note: I am... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1332141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability between multiple dice and single die Can some one explain what is the difference between rolling three dice together and rolling a single die three times? What is the probability that the sum equals 4 for both cases?
| In probability event(theoretically), we usually assume ideal condition.
The probability of a die with result from 1 to 6 is 1/6 regardless what happens to other dies either thrown at the same time or thrown few seconds later. We assume each event of throwing a die is independent, therefore each event is also independe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When is the image of a null set a null set? I came upon this question here which contains the following statement:
It is easy to prove that if $A \subset \mathbb{R}$ is null (has
measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz
then $f(A)$ is null. You can generalize this to $\mathbb{R}^n$ wit... | No, there is a continuous function from the unit interval that sends the Cantor set onto the unit interval. Since the Cantor set is null, the contradicts your conjecture.
See https://en.m.wikipedia.org/wiki/Cantor_function
Since all continuous functions are measurable, your guess after the question is also wrong.
Abso... | {
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"source": "stackexchange",
"question_score": "4",
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Advanced techniques needed to solve a difficult integral. I am looking to solve the following integral.
$\int_{0}^{\infty} \frac{1-\cos(ax)}{x^2}e^{bx} dx$.
I have made an attempt using the differentiation under the integral sign method and I got the following:
$b\ln(b)-\frac{1}{2}b-\frac{1}{2}b\ln(b^2+a^2)-\frac{1}{2... | $b \lt 0$ for convergence, so rewrite for $b=-c$ as
$$2 \int_{-\infty}^{\infty} dx \frac{\sin^2{(a x/2)}}{x^2} e^{-c x} \theta(x) $$
where $\theta(x) = 0$ when $x \lt 0$ and $1$ when $x \gt 0$.
We can use Parseval's equality to evaluate this integral. Parseval states that, if $f$ and $g$ have respective Fourier tra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Reflections in Euclidean plane Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the counterclockwise rotation of $\frac{\pi}{2}$ and $S: \mathbb{R}^2 \to \mathbb{R}^2$ be the reflection w.r.t. the line $x+3y=0$. There exists a reflection $R$ such that $T^{-1}ST=R$? Is there a canonical way to find which is the line w.r.t. we ... | suppose $T^{-1}ST $ fixes a line $l.$ then $$T^{-1}ST(l) = l \implies ST(l) = T(l). $$ that is $T(l)$ is fixed by $S$ and we know that only lines fixed by $S$ is the line $3x+y = 0.$ therefore $T(l)$ is $y-3x = 0$ and $$ T^{-1}ST \text{ is a reflection on the line } y - 3x = 0.$$
| {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Determining the shock solutions to a PDE. I'm confused by the question below. Particularly, sketching the base characteristics at the discontinuities in $u(x,0)$ and thus finding the shock solutions. Some advice would be appreciated.
Problem
My Attempt
Using the method of characteristics I find that $t=\tau$ as $t(0)=... | Let $F(p,z,y):=p_2-z^2p_1$, where $p(s)=\nabla u(y(s))$, $z(s)=u(y(s))$, and $y(s)=(x(s),t(s))$ ($s$ is just a parameter). Note then that $F=0$. The characteristics of the equation are given by $y(s)$ for various values $s\in\mathbb{R}$. The Method of Characteristics (see Lawrence C. Evans book on PDES I think chapt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Law of large numbers for nonnegative random variables I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. Show that
$$\mathbb{P}\bigg(\lim_{n\to\infty}\frac{X_1+X_2+\ldots+X... | For fixed $k \in \mathbb{N}$ set $Y_n := X_n \wedge k$. Then $(Y_n)_{n \in \mathbb{N}}$ is a sequence of iid random variables, $Y_k \in L^1$. By non-negativity and the strong law of large numbers
$$\begin{align*} \liminf_{n \to \infty} \frac{X_1+\ldots+X_n}{n} &\geq \liminf_{n \to \infty} \frac{Y_1+\ldots+Y_n}{n} \\ &=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does $ABC=D\implies \det(ABC)=\det(D )$? $${\color{brown}{\text{Question I am trying to solve:}}}$$
Let $A,B$ and $X$ be 7 x 7 matrices such that $\det A=1$, $\det B=3$ and
$$A^{-1}XB^{t}=-I_7$$
where $I_7$ is the 7 x 7 identity matrix. Calculate $\det X$.
$$\color{brown}{------------------------------------}$$
The way... | I might be missing something here, but as far as I know:
$a = a' \Rightarrow f(a) = f(a')$
for all sets $A,B$; $a,a'\in A$ and functions $f : A\to B$.
| {
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How do you find the minterm list of a boolean expression containing XOR? Let's say I have a boolean expression, such as F1 = x'y' ⊕ z .
How do I go about finding the minterm list for that expression?
The method I've tried is to take each term, such as x'y' and z, then fill in the missing values with all possibiliti... | For an expression with just three variables, the usual way is to write a truth table, depict it as Karnaugh map and find a minimal set of terms which cover all 1entries in the map.
xy
00 01 11 10
+---+---+---+---+
0 | 1 | 0 | 0 | 0 |
z +---+---+---+---+
1 | 0 | 1 | 1 | 1 |
... | {
"language": "en",
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Population changes with time Question
*
*The population of a certain community is known to increase at a rate proportional to the number of people present at time $t$. If the population has doubled in 5 years, how long will it take to triple? To quadruple?
*Suppose it is known that the population of the community i... | Since the population increases at a rate proportional to the number of people present at time t (read: the rate of change of $P$ is proportional to $P$.)
$$\frac{\mathrm{d}P}{\mathrm{d}t} = kP$$
Separating the variables and integrating yields
$$\int \frac{1}{P} \, \mathrm{d}P = \int k \, \mathrm{d}t$$
So that we get
$$... | {
"language": "en",
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Finding $ \lim_{x\to 2}\frac{\sqrt{x+2}-\sqrt{2x}}{x^2-2x} $ I'm kind of stuck on this problem, I could use a hint.
$$
\lim_{x\to 2}\frac{\sqrt{x+2}-\sqrt{2x}}{x^2-2x}
$$
After some algebra, I get
$$
{\lim_{x\to 2}\frac{x+2 - 2x}{x(x-2)-\sqrt{x+2}+\sqrt{2x}}}
$$
EDIT above should be:
$$
\lim_{x\to 2}\frac{x+2 - 2x}{x(x... | rewrite it in the form
$$\frac{(\sqrt{x+2}-\sqrt{2x})(\sqrt{x+2}+\sqrt{2x})}{x(x-2)(\sqrt{x+2}+\sqrt{2x})}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1333110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\sup(\frac{1}{A})=\frac{1}{\inf A}$
Given nonempty set $A$ of positive real numbers, and define $$\frac{1}{A}=\left\{z=\frac{1}{x}:x\in A \right\}$$ Show that $$\sup\left(\frac{1}{A}\right)=\frac{1}{\inf A}$$
let $\sup\left(\frac{1}{A}\right)=\alpha$ and $\inf A = \beta$. Apply the definition of supremum... | The crucial facts here are that if $x \in A$, then $x>0$ and the function
$x \mapsto {1 \over x}$ reverses order in $A$, that is, if $x,y \in A$, then $x<y$ iff${1 \over x } > {1 \over y}$.
We have $\sup_{x' \in A} {1 \over x'} \ge {1 \over x}$ for all $x \in A$.
Now let $x_n \in A$ such that $x_n \to \inf A$. Then thi... | {
"language": "en",
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"source": "stackexchange",
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Number of ways selecting 4 letter words
The number of ways of selecting 4 letters out of the letters MANIMAL
A. 16
B. 17
C. 18
D. 19
I have made three different cases. Including 1 M, 2 M and none of the M.
So it is 6C3/2 + 5C3 + 4C3 which doesn't meet any of the option.
| the answer should be for MNIMAMAL.,
$(1+x)^3(1+x+x^2)(1+x+x^2+x^3)$ $=$ $x^8+5x^7+12x^6+19x^5+22x^4+19x^3+12x^2+5x+1$
hence,the answer will be 22
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Primitive recursion and $\Delta^0_0$ Until recently I assumed that primitive recursive relations are exactly $\Delta^0_0$ (i.e. bounded) ones, but I learned they're different (the former is a proper superclass of the latter).
I have questions regarding the difference between the two:
*
*I have some intuition about p... | It's many years after the original question was asked, but I saw this recently and think I can give a more satisfactory answer than the current one.
The $\Delta^0_0$ functions can be thought of as functions in a programming language with "only for loops, not while loops," but only if we cannot change the values of non-... | {
"language": "en",
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Value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$ How can we find the value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$?
Note that $\sin^{-1}$ is the inverse sine function.
I'm asking for the solution $x$ for this equation.
Please work out the solution.
| Let $\sin^{-1}x=y\implies-\dfrac\pi2\le y\le\dfrac\pi2$
$\implies y+\sin^{-1}(1-\sin y)=\dfrac\pi2-y$
$\implies\sin^{-1}(1-\sin y)=\dfrac\pi2-2y$
$\implies1-\sin y=\sin\left(\dfrac\pi2-2y\right)=\cos2y=1-2\sin^2y$
$\implies2\sin^2y-\sin y=0$
Observation :
For real $\sin^{-1}x, -1\le x\le1$
and for real $\sin^{-1}(1-x),... | {
"language": "en",
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Smooth function, which separates between a closed and a open set. Let $M$ be a smooth manifold, $O\subseteq M$ an open subset and $B\subseteq M$ a closed subset, such that $closure(O)\subseteq interior(B)$
I think there must exist a smooth function $f\colon M\rightarrow \mathbb{R}$ such that $0\le f\le 1$, $f|_{M-B}\eq... | One reference was supplied by John in a comment, with a correction by Jack Lee:
Proposition 2.26 in Introduction to Smooth Manifolds by John M. Lee's, first edition. (Also, Proposition 2.25 of the second edition).
Note that you could just as well consider the disjoint closed sets $\overline{O}$ and $\overline{M\setm... | {
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Learning by the Moore method - Books for self-study I recently read about the Moore method for learning mathematics (Moore method Wikipedia) and wanted to apply it to my own learning (undergraduate level). However, I am unable to find any books that follow such a method or something similar.
Specifically, I'm looking f... | Coursera has a course called Introduction to mathematical thinking by Stanford math professor Keith Devlin. He has a book by the same name you can buy it on Abebooks.com, it's print to order for about 12 bucks. Or you can audit the course for free. As an introduction they have a video on inquiry-based learning which is... | {
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$f(x)=2x-e^x<0, \forall x \in \mathbb{R}$ The question is quite simple, but I'm finding some trouble doing it...
Prove that the function $f(x)=2x-e^x$ is negative, i.e., $f(x)<0, \forall x \in \mathbb{R}$.
Thanks for the help.
| The function $e^x$ is strictly increasing and $y = e\cdot x$ is a tangent to $y = e^x$ at $x = 1$, so $e^x \geq e\cdot x > 2x$ for positive $x$. For nonpositive $x$: $e^x > 0 \geq 2x$. Finally, $e^x > 2x$
| {
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Triple fractions I've got this simple assignment, to find out the density for a give sphere with a radius = 2cm and the mass 296g. It seems straightforward, but it all got hairy when i've got to a fraction with three elements(more precisely a fraction divided by a number actually this was wrong, the whole point was tha... | to use your "introduce 1 to the denominator" approach, you should instead put that 1 with $a$:
$\frac{a}{\frac{b}{c}}=\frac{\frac{a}{1}}{\frac{b}{c}}=\frac{\frac{ac}{c}}{\frac{b}{c}}=\frac{ac}{b}$
(for the same reasons that the others point out)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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consider the following subsets of complex plane $$\Omega_1=\left\{c\in\Bbb C:\begin{bmatrix}1&c\\\bar c&1\\ \end{bmatrix}\text{ is non-negative definite } \right\} $$
$$\Omega_2= \left\{c\in\Bbb C:
\begin{bmatrix}
1 & c & c \\
\bar c & 1 & c \\
\bar c & \bar c & 1 \\
\end{bmatrix... | *
*Note that the matrices are Hermitian, so it is enough to check if the eigenvalues $\lambda\geq 0$ are non-negative, or equivalently,
$$\mu~:=~1-\lambda~\leq~ 1.$$
*The characteristic polynomials read
$$ p_1(\lambda) ~=~\mu^2-|c|^2, $$
and
$$ p_2(\lambda) ~=~\mu^3+|c|^2(2{\rm Re}(c) -3\mu), $$
respectively.
*Def... | {
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"source": "stackexchange",
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Finding the quotient ring $\mathbb{Z}[i]/(4+i)$
Find the quotient ring $\mathbb{Z}[i]/(4+i)$ by identifying elements with the lattice points in the square generated by $4+i$.
I know that $N(4+i) = 17$.
Therefore, $4+i$ is irreducible.
Now here's where I am stuck - lattice points:
$I = (4+i)$ so $Z = 4+i$
$$(m+ni)(4+... | As you have already noted, $4+ \Bbb i$ is irreducible; $\Bbb Z [ \Bbb i ]$ is a Euclidean domain, therefore a principal ideal domain (PID), and we know that in a PID the ideal generated by an irreducible element is maximal; hence, the ideal $(4+ \Bbb i)$ is maximal, so the quotient ring $\Bbb Z [ \Bbb i ] / (4+ \Bbb i)... | {
"language": "en",
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Rock and weight level weight problem? Given 5 rocks of different weight and level scales in 7 tests determine the order the rock by weight.
So I have 5 rock that is 120 possible ways of ordering them. The rock are named
(A,B,C,D,E)
So my first step is
$A<B$ then I do
$ C<D $
and I then I do
$ C<A$
This gives me the f... | First we measure $A > B$ and $C > D$. Next we measure the two heavier ones and determine that $A > C$. This conveniently leaves us with only $3$ arrangements for the stones $A$, $B$, $C$ and $D$. Namely: $ABCD$, $ACBD$ and $ACDB$.
Note that stone $C$ is nicely in the middle, either as no. $2$ or no. $3$. Therefore we ... | {
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example of a continuous function that is closed but not open Give an example of a continuous function $f:\mathbb{R} \to \mathbb{R}$ that is closed but not open.
$ f(x)=x^2$ is continuous and not open but It's not closed. What is an example?
Thanks in advance.
| As several people have noted, there are much simpler examples, but in fact the squaring map $f:\Bbb R\to\Bbb R:x\mapsto x^2$ is closed.
Let $F$ be a closed set in $\Bbb R$. Let $F_0=F\cap[0,\to)$; $F_0$ is closed in $[0,\to)$, and $f\upharpoonright[0,\to)$ is a homeomorphism of $[0,\to)$ onto itself, so $f[F_0]$ is a ... | {
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Surface integral: Cone cut by a cylinder Ok I've got this exercise from Apostol I'm trying to do:
"The cylinder $x²+y²=2x$ cuts out a portion of a surface S from the upper nappe of the cone x²+y²=z². Compute the value of the integral:
$$\int\int_S(x^4-y^4+y^2z^2-z^2x^2+1)dS$$
Ok, what I've done so far is choosing a par... | The points $(x,y,z)$ on the surface satisfy $x^2+y^2 \le 2x$, $z^2 = x^2+y^2$, $z = 0$.
Since you set $X(u,v) = v\cos u$, $Y(u,v) = v\sin u$, and $Z(u,v) = v$ we get:
$z \ge 0 \leadsto v \ge 0$
$x^2+y^2 \le 2x \leadsto (v\cos u)^2+(v\sin u)^2 \le 2(v\cos u) \leadsto v^2 \le 2v\cos u \leadsto 0\le v \le 2\cos u$.
In ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Countable and uncountable set.
Which of the following sets of functions are uncountable?
*
*$\{f|f:\Bbb N\to\{1,2\}\}$
*$\{f|f:\{1,2\}\to\Bbb N\}$
*$\{f|f:\{1,2\}\to\Bbb N, f(1)\le f(2)\}$
*$\{f|f:\Bbb N\to\{1,2\}, f(1)\le f(2)\}$
I think 1 and 4 are true. As cardinality of first option is $2^{\aleph_0}=c$ an... | It is a subset of the first, which does not show that it is uncountable. To show that it is uncountable you have to show that given any countable list of elements, you can construct an element not on the list. Alternatively, you can reduce the problem to a previous known result, because the condition $f(1) \le f(2)$ on... | {
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Given a directed graph, give an adjacency list representation of the graph that leads BFS to find the following spanning tree Given a directed graph:
give an adjacency list representation of the graph that leads Breadth first search to find the spanning tree in the left below. And give an adjacency list representation... | I assume that the point of the question is to make one adjacency list for the first directed graph (call it A) such that a BFS of that adjacency list produces the first tree (call it B) and another ordering of the adjacency list for A that produces C.
In other words, the question is about how you order the adjacency li... | {
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Evaluate $ \int _{ 0 }^{ 1 }{ \ln\left(\frac { 1+x }{ 1-x } \right)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$ Problem: Evaluate:
$$\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\frac { 1+x }{ 1-x } \bigg)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$$
On Lucian Sir's advice, I substituted $x=\cos(\theta)$. Thus, the Inte... | Here is how I finally worked it out:
$$\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\dfrac { 1+x }{ 1-x } \bigg)\dfrac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } } $$
Put $x=\cos(\theta)$ to get our integral as :
$$\displaystyle \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \ln\bigg({ \cot }^{ 2 }\bigg(\frac { \theta }{ 2 }\bigg) \bigg)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 3
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Finding the sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$ Deteremine the sum of the series
$$\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$$
So I first write down the $n^{th}$ term $a_n=\frac{\frac{n(n+1)}{2}}{n!}=\frac{n+1}{2(n-1)!}$.
So from there I can write the series as $$1+\frac{... | HINT: Split
$$\frac{n+1}{2(n-1)!}= \frac{1}{2(n-2)!}+\frac{1}{(n-1)!}$$
and use the fact that $\sum_{n=0}^{\infty} \frac{1}{n!}=e$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
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Expectation value of number of drawings of increasing sequences of labelled balls from an urn.
An urn contains $n$ balls, labelled from $1$ to $n$. A sequence of drawings with re-insertion is made, until the drawn ball is labelled with a number which is less than or equal to the number of the previous drawing. a. Give... | For the first part, the number of drawings is greater than $k$ if and only if the first $k$ balls drawn were all increasing. You don't have to consider the final drawing, precisely because we're considering the cases where the final drawing is not among the first $k$.
The probability that the first $k$ drawings form an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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About the exact form of a gaussian kernel Traditionally we define a gaussian function at a point x (assuming mean to be 0) as follows
$$g_{\sigma}(x) = \frac{1}{\sqrt{2\pi \sigma^{2}}} \exp\left(\frac{x^{2}}{2\sigma^{2}}\right)$$
In some sources however, the exact form is given as follows :
$$g_{\sigma}(x) = \frac{... | A gaussian function is any function of the form
$$
a\,e^{-b(x-c)^2}.
$$
A gaussian probability distribution with mean $0$ and variance $\sigma^2$ corresponds to the first of your definitions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to find sum of the infinite series $\sum_{n=1}^{\infty} \frac{1}{ n(2n+1)}$ $$\frac{1}{1 \times3} + \frac{1}{2\times5}+\frac{1}{3\times7} + \frac{1}{4\times9}+\cdots $$
How to find sum of this series?
I tried this: its $n$th term will be = $\frac{1}{n}-\frac{2}{2n+1}$; after that I am not able to solve this.
| Let $f(x)=\sum_{n=1}^{\infty} \frac{x^{2n+1}}{ n(2n+1)}$. Then we have $$f'(x)=\sum_{n=1}^{\infty} \frac{x^{2n}}{ n}=-\log(1-x^2).$$ Hence since f(0)=0, the sum is equal to
\begin{align}
s&=-\int_0^1\log(1-x^2)dx\\
&=-2\int_0^{\pi/2}\log(\cos x) \cos x dx\\
&=-2I
\end{align}
To solve this integral, $I$, note first that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 3
} |
Gauss elimination: Difference between partial and complete pivoting I have some trouble with understanding the difference between partial and complete pivoting in Gauss elimination. I've found a few sources which are saying different things about what is allowed in each pivoting. From my understanding, in partial pivot... | You are basically correct. Partial pivoting chooses an entry from the so-far unreduced portion of the current column (that means the diagonal element and all the elements under it). Full pivoting chooses any element from the so far unreduced lower-right submatrix (the current diagonal element and anything below / to th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1334983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 1
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Max/Min problem - Find proportions of a right circular cylinder
Find the proportions of a right circular cylinder of greatest volume which can be inscribed inside a sphere of radius $R$.
There's a poor image I made of what I think it looks like..
Using Pythagoras, I got this:
$$R^2=(\dfrac{h}{2})^2 + r^2$$
$$r^2= R^... | This looks good to me!
Note that you could make your $h/2$ your $r$ - the radius of the cylinder and have $R$ be the radius of the circle and perhaps make what you called $r$ instead $h/2$ or $H$ to have your variables align more with what they are in the diagram. But the choice of variables is always yours!
The impor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Showing that $f$ is $C^\infty$ Question:
Let $f: U \to \mathbb R$ be a continuous function, with $U \subset \mathbb R^2$ open, such that $$(x^2 +y^4)f(x,y) + f(x,y)^3 = 1,\, \,\, \forall (x,y) \in U$$
Show that $f$ is of class $C^\infty$.
Attempt:
Define $F(x,y,z) = (x^2 + y^4)z + z^3 - 1$. Then $$F_z(x,y,z) = (x^... | The discriminant of $P(z) = t z + z^3 - 1$ is $-4 t^3 - 27$, which is nonzero for $t \ge 0$. Thus $\dfrac{\partial}{\partial z} P(z) \ne 0$ whenever $P(z) = 0$ with $t \ge 0$. In particular, this applies when $t = x^2 + y^4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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How to find the partial derivative of $f(x, y) = x^{2} - y^{2}$ with respect to $y.$ Recently, I began exploring the realms of multi-variable calculus, and, already, I have ran into a problem.
I am trying to find the partial derivative of $f(x, y) = x^{2} - y^{2}$ with respect to $y$.
I believe it to be $-2y,$ but I am... | You are correct, since if you derivate with respect to $y$, $x^2$ will count only as a constant, and the derivate of any constant is $0$. Since derivating is additive, you can do it by first derivating $x^2$, and then derivating $y^2$. The first will be $0$, the second is $2y$, therefore your answer is $0-2y=-2y$. Tot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Cosh and Sinh analogs We know that
$$\cosh{x}+\sinh{x}=e^x$$
and that his can be expressed as
$$\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{(e^x+e^x)+(e^{-x}-e^{-x})}{2}=e^x$$
and this works out nicely because the $e^{-x}$ cancel. Now consider a "higher order" cosh equation of the form
$$C(x)=\frac{e^{\omega^0x}... | If you want a symmetrical generalization for $n$th roots then define
$$C_{k,n}(z)=\frac{1}{n}\sum_{\zeta^n=1}\zeta^k e^{\zeta z}.$$
Then $C_{0,2}(z)=\cosh z$ and $C_{1,2}(z)=\sinh z$. Note that $C_{k,n}(z)=C_{0,n}^{(k)}(z)$.
In particular, for $n=3$, if $\omega$ is the cube root of unity in the upper half plane then
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
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Prove the group is a direct product Let $G$ be an abelian group of finite order $n = mk$ with gcd$(m,k) = 1$. For $r=m,k$, let $G(r) = \{g \in G: g^r = 1 \}$ . Prove that $G = G(m) \times G(k)$.
| It is not hard to show that $G(m)$ and $G(k)$ are subgroups and intersect each other trivially. We need to show then that $G(m)G(k)=G$. Suppose $g\in G$ is an element; then since $m$ and $k$ are relatively prime we have that there are integers $a,b$ such that $am+bk=1$. Then $g=g^{am}g^{bk}$, and $g^{am}\in G(k)$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Why is the potential function defined differently in physics and calculus? I am very familiar with the concept of a potential function, and potential energy, from calculus-based physics.
For instance, if we have the familiar force field $\mathbf{F} = -mg \,\mathbf{j}$, then a potential function is given by $U = mgy + C... | Recall where the negative sign comes from in physics -- it is simply due to your coordinate system and point of view. The difference is analogous to the difference between work done by gravity and work done on gravity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Product of roots of unity using e^xi Find the product of the $n\ n^{th}$ roots of 1 in terms of n. The answer is $(-1)^{n+1}$ but why? Prove using e^xi notation please!
| The $n $ roots of unity are
$$ r_k = e^{i \frac{2 \pi k }{n}} \, \text{ for }k=0 ...n-1 $$
so
$$\prod_{k=0}^n r_n = \prod_{k=0}^n e^{i \frac{2 \pi k }{n}} =e^{i \frac{2 \pi X}{n}} $$
where $$X = \sum_{k=0}^{n-1} k = \frac{n(n-1)}{2} $$
so $\frac{2 \pi X}{n} = (n-1)\pi$
leaving
$$\prod_{k=0}^n r_n = e^{i(n-1)\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Locally Lipschitz and Gâteaux Derivative if and only if Frechet Derivative Consider $f$ locally Lipschitz. So $f$ is Gâteaux Derivative if and only if $f$ is Frechet Derivative.
PS.: the converse is trivial.
| Seems to me the result is false.
Say $S$ is the unit circle in $\mathbb R^2$. Say $\phi:S\to\mathbb R$ is any smooth function with $\phi(-x)=\phi(x)$. There's a function $f:\mathbb R^2\to\mathbb R$ such that
*
*$f(x)=\phi(x)\quad(x\in S),$
*$f(tx)=tf(x)\quad(x\in\mathbb R^2,t\in\mathbb R).$
This seems like a count... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Infinity indeterminate form that L'Hopital's Rule: $\lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}}$ When I tried to find the limit of
$$
\lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}}
$$
by applying L'Hopital's Rule the order of denominator would increase. What else can I do for it?
| Don't use L'Hospital's rule. It won't work here, and when it works, it is equivalent with Taylor's polynomial at order $1$, which is much less error-prone.
It is a problem of Asymptotic analysis: set $u=\dfrac1x$. Then
$$\lim_{x\to 0^+}\frac{\mathrm e^{-\tfrac1x}}{x^2}=\lim_{u\to+\infty}\frac{u^2}{e^u}=0$$
since a ba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Transitive subgroup of symmetric group $S_n$ containing an $(n-1)$-cycle and a transposition
Suppose $G$ is a transitive subgroup of $S_n$ such that it there exist $\sigma, \tau \in G$ such that $\sigma$ is an $(n-1)$-cycle and $\tau$ is a transposition. Prove that $G = S_n$.
I just don't understand how to mathematic... | Take your subgroup $G$, up to the study of a conjugate $G$ you can assume that the $n-1$-cycle of $G$ is $c=(2,...,n)$. Now if $\tau$ is a transposition in $G$ then $\tau=(i,j)$ with $i\neq j$. Take $\sigma_i\in G$ such that $\sigma_i(i)=1$ (this is where I use the transitivity).
Then I claim that $\sigma_i\tau\sigma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1335994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the infinite sum of a sequence Define a sequence $a_n$ such that $$a_{n+1}=3a_n+1$$ and $a_1=3$ for $n=1,2,\ldots$. Find the sum $$\sum_{n=1} ^\infty \frac{a_n}{5^n}$$ I am unable to find a general expression for $a_n$. Thanks.
| HINT:
Let $a_m=b_m+c\implies b_{n+1}+c=3(b_n+c)+1\iff b_{n+1}=b_n+2c+1$
Set $2c+1=0$ to get
$$a_{n+1}+\dfrac12=3^1\left[a_n+\dfrac12\right]=\cdots=3^r\left[a_{n-r+1}+\dfrac12\right]=3^n\left[a_1+\dfrac12\right]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Chain fixed at two points, how far does it drop down? Not too sure whether this should be in maths or physics, but oh well.
If you have a metal chain of length h metres
and you have 2 points, the distance between them being x metres,
If h is less than x, then the chain will obviously not fit between the 2 points.
If h=... | Hint.
One can prove that the curve that will fit your chain is a catenary.
Based on that, you can find the parameter $a$ of the catenary in order to have its length equal to $h$ knowing the distance $x$ between the two points. You will then be able to compute the height of the drop down.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Product of Matrices I Given the matrix
\begin{align}
A = \left( \begin{matrix} 1 & 2 \\ 3 & 2 \end{matrix} \right)
\end{align}
consider the first few powers of $A^{n}$ for which
\begin{align}
A = \left( \begin{matrix} 1 & 2 \\ 3 & 2 \end{matrix} \right)
\hspace{15mm} A^{2} = \left( \begin{matrix} 7 & 6 \\ 9 & 10 \end{m... | In answer to part 2, the general form is $$M^n=\frac{1}{5}\begin{pmatrix} 2\times 4^n+3 \times(-1)^n & 2\times 4^n-2(-1)^n\\3\times 4^n+3(-1)^{n+1} & 3\times 4^n+2(-1)^{n}\\\end{pmatrix}$$
But you know this already because it has already been posted!
(it just took me longer to write out in MathJax)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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show there exists an integer k such that $2013^k$ ends with '0001' Prove that there exists an integer k so that $2013^k$ ends with '0001'.
we couldn't figure this out.
i thought we might try to prove that we can find an integer m such that $m*10^4 +1 = 2013^k$, but was unable to get any clues.
both hints and similar so... | You have to prove that for some $k$,
$$ 2013^{k}\equiv 1\pmod{10^4} $$
but since $2013=3\cdot 11\cdot 61$ and $10^4=2^4\cdot 5^4$, we have $\gcd(2013,10^4)=1$, so $2013$ is an element of the group $\mathbb{Z}_{/10^4\mathbb{Z}}^*$, and its order is a divisor of $\varphi(10000)=4000$ by Euler's theorem and Lagrange's the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Analysis: Is $A$ dense in $[0,1]$? Let $f_n:[0,1]\to\mathbb R$ define by $f_n(x)=\cos(nx)$. Let $A_n=\{x\mid f_n(x)=0\}$ and $A=\bigcup_{n\in\mathbb N}A_n$. I have shown that $|A|=+\infty$. Do you think that $A$ is dense in $[0,1]$ ? I think that it is, and my proof would go like this:
Suppose by contradiction that it'... | Why approach by contradiction when you can do it directly?
You know explicitly that $\cos(nx) = 0$ when $x = \frac{1}{n} (k+ \frac12) \pi$ for any $k\in \mathbb{Z}$. This means that any $y\in [0,1]$ is within $\pi/n$ distance of a zero of $f_n(x)$. This is enough to show that $y$ is arbitrarily close to elements of $A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of solutions of $x_{0}x_{1} - x_{2}x_{3}$ over $\mathbb{F}_{p^{s}}$ I'm trying to solve the following exercise:
Compute the zeta function of $x_{0}x_{1} - x_{2}x_{3}$ over $\mathbb{F}_{p}$. Well, for this, I need to find $N_{s}$, the number of solutions in the field $\mathbb{F}_{p^{s}}$. What I did:
First, looki... | I computed the number $N_s$ and got the same number as you i.e. $N_s = p^{3s} + p^{2s} - p^s$. I use that there is a bijection between the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$ and the projective solutions of $x_0x_1 - x_2x_3 = 0$ in $\mathbb{P}^3$ with homogeneous coordinates $[x_0 : x_1 : x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there an flat unordered pairing function in ZFC? Is there an unordered pairing function that does not increase rank whenever the max rank is infinite, in ZFC? An unordered pairing function is one such that $f(x,y)=f(z,w)$ iff $(x=z \wedge y=w) \vee (x=w \wedge y=z)$.
| First, let's look at the special case when $x\cap y=\emptyset$.
Here, we can define a disjoint unordered pairing function: $\langle x, y\rangle_d=\{[a, b]: a, b\in x\}\cup\{[c, d]: c, d\in y\}$, where $[\cdot, \cdot]$ is the flat (ordered) pairing function of your choice. The point is that from $\langle x, y\rangle_d$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Calculate the sum $\sum_{n=0}^\infty\frac{1}{(4n)!}$ How to determine the sum $\sum_{n=0}^\infty\frac{1}{(4n)!}$ ?
Do I need to somehow convert (4n)! to (2n)! or in tasks like this, should I get the (4n)! after some multiplying?
Thank you all for your time!
| $cos(x) = \sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}$
$cosh(x) = \frac{1}{2}\sum_{n=0}^\infty\frac{x^{n}}{n!} + \frac{1}{2}\sum_{n=0}^\infty(-1)^n\frac{x^{n}}{n!}$
How to get (4n)! from these (2n)! and n! ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Question about countable and uncountable map correspodence So I was solving the following question:
If $B$ is uncountable with countable subset $A\subset B$, prove that there exists a one-to-one correspondence between $B$ and $B-A$.
So here is how I proved it:
Since $B$ is uncountable, $B-A$ is uncountable, so there... | Let $B$ be an uncountable set and for each $n \in \mathbb N$ let $A_n = \{a_1^n, a_2^n, \ldots \} \subseteq B$ be pairwise disjoint. Let $G = \{g_1,g_2, \ldots\}$ be a countable subset of $B$ disjoint from all $A_n$. Let $\mathbb P = \{p_1,p_2, \ldots\} \subseteq \mathbb N$ be the set of prime numbers. Let $D = \{d_1, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Laplace Transforms of Step Functions The problem asks to find the Laplace transform of the given function:
$$ f(t) =
\begin{cases}
0, & t<2 \\
(t-2)^2, & t \ge 2
\end{cases} $$
Here's how I worked out the solution:
$$\mathcal{L}[f(t)]=\mathcal{L}[0]+\mathcal{L}[u_2(t)(t-2)^2]=0+e^{-2s}\mathcal{L}[(t-2)^2]=e^{-2s}\math... | This seems to be the thing that students in DE have more trouble with than anything else. That formula in the book reads $$\mathcal L[u_c(t)f(t-c)]
=e^{-cs}\mathcal L[f(t)].$$ You're trying to use this to find $\mathcal L[u_2(t)(t-2)^2]$. To fit this into that formula you must have $$u_2(t)f(t-2)^2=u_c(t)f(t-2).$$
But ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1336978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determinant of a matrix with binomial coefficients. Let $n \in\mathbb{N}$ and $A=(a_{ij})$ where
\begin{equation}a_{ij}=\binom{i+j}{i}\end{equation}
for $0\leq i,j \leq n$. Show that $A$ has an inverse and that every element of $A^{-1}$ is an integer.
I have shown that this $n\times n$ matrix is symmetric since,
\beg... | If you take a particular case (say $n=5$) and you consider the LDL or Cholesky decomposition of this matrix you notice something very interesting: ( WA link).
So one should try to prove that our matrix $A$ is the product
$$A = L \cdot L^t$$
where
$L = (\binom{i}{j})_{0\le i,j \le n}$.This translates into the equalities... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Logic Problem with truth tables According to a truth table, if "p is false, and q is false" then "p implies q" is true. However, when studing inverses, we see that the inverse of a conditional statement may or may not be true.
For example,
Statement: If a quadrilateral is a rectangle, then it has two pairs of paralle... | Ok, so you know that $p \rightarrow q$ is true when $p$ and $q$ are false. The inverse $\neg p \rightarrow \neg q$ is also true when $p$ and $q$ are both false.
Your confusion seems to be that you are conflating the above situation with a different claim: $(p\rightarrow q) \rightarrow (q\rightarrow p)$. Notice that th... | {
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Divide a square into different parts This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with geometry, which perhaps yields the shortest, simplest proofs, but other than that, the textbook gave no hints really and I'm really not sure about how to app... | By request, I'm spinning my comment out into an answer. For $n$ even, say $n=2k$, subdivide the square into a $k\times k$ grid of squares. I'll show it for $k=5$, because I think it's easier to visualize when everything renders as true squares rather than with $\dots$:
$$\begin{array}{|c|c|c|c|c|}
\hline
\,\,&\,\,&\,\,... | {
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"timestamp": "2023-03-29T00:00:00",
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Nested Quantifiers Doubt: "If $xy$ is equal to $x$ for all $y$, then $x=0$" If $P(x,y,z)$ represents $xy=z$.
Then represent the following statement using quantifiers,connectives etc.
"If $xy$ is equal to $x$ for all $y$, then $x=0$".
The answer given is $\forall x[ \forall y P(x,y,x)\to x=0]$.
Can't we write the same ... | No, that is not the same at all.
*
*The first expression says is that for any $x$, the statement "For any $y$ we have $xy = x$" implies the statement "$x = 0$".
*The second expression says that for any $x$ and $y$, the statement "$xy = x$" implies the statement "$x = 0$".
The second expression isn't true, and parti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1337418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integral of monomial and logarithm: is this true? $\lim_{k\to -1}\frac{x^{k+1}}{k+1} = \log|x|$ It is well know that:
$$\int x^k \text{d}x = \begin{cases}
\displaystyle\frac{x^{k+1}}{k+1} + c & k \neq -1\\
\\
\log|x| + c & k = -1\end{cases}$$
My guess is:
$$\lim_{k\to -1}\frac{x^{k+1}}{k+1} = \log|x| ???$$
Apparently, ... | I must say its a very out of the box question and OP deserves credit to think in this way. +1 from my end. You are right but you need to express your ideas in concrete manner. You know that indefinite integrals are well "indefinite" and hence not unique. So the antiderivative of a function is always expressed with a $+... | {
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Completion of the proof of theorem 3.3 in Dale Husemoller: Elliptic Curves I want to read the proof of the following theorem:
This is from p.35. But it is not complete there. There is written that:
Can someone tell me where I can find the rest of the proof?
Any other sources are also welcome :)
Thanks in advance
| Claim: Let $E: y^2=x^3+D$ and $p>3$ be a prime. Then, there is no point of order $p$ in $E(\mathbb{Q})$.
Here are some hints. Let $p>3$ be a prime as in the statement of the claim:
*
*If $q$ is a prime such that $q\equiv 2 \bmod 3$, and $q$ does not divide $6D$, then $E(\mathbb{F}_q)=q+1$.
*A prime $q$ that does n... | {
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M/M/1 vs G/G/1 vs G/M/1 I am using queuing theory to model a router.
I have a model that assumes Poisson traffic and I need to modify it as my actual traffic is not Poisson
I want to ask what's the main difference between Poisson arrivals and general arrivals in terms of results ?
ie what are the formulas that don'... | For general inter-arrival times instead of Possion, the formulas are expressed in terms of a general cumulative distribution function $F(x)$ so to use them you need to have a distribution in mind.
Look at Introduction to Probability Models Eleventh Edition by Sheldon M. Ross
for reference.
| {
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An example of a module that is not injective I know that since $\mathbb Z$ is a PID every free module is projective and conversely.
Hence since $\mathbb Q$ is not free as a $\mathbb Z$-module then it is not projective.
But is $\mathbb Q$ an injective $\mathbb Z$ module? Does there exist some similar result like above ... | The injective $\mathbb{Z}$-modules are precisely the divisible abelian groups (in particular, $\mathbb{Q}$ is an injective $\mathbb{Z}$-module). Now, the abelian groups $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ are not divisible, and hence give us examples of non-injective $\mathbb{Z}$-modules.
| {
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Convergence of subsequence of partial sums implies full convergence? Define $S_n = \sum_{j=1}^n a_j$ a sum of real numers.
Suppose there is a subsequence such that $S_{n_j} \to S$ for some $S$, i.e., the infinite sum converges along this subsequence. Does this imply that $S_n$ converges too?
It seems like it should, si... | No, if e.g. $a_j = (-1)^j$ then $S_{2k} = 0$ for all $k$ but the series itself does not converge.
| {
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How to prove that the cross product of a countable and uncountable set is uncountable? so my question is, how can you prove that
${\Bbb Z}$ x ${\Bbb R}$ is uncountable?
So far I have tried proving that there is an uncountable subset of ${\Bbb Z}$ x ${\Bbb R}$ without luck and I'm not really sure what I can do
Thanks... | If $\mathbb{Z}\times\mathbb{R}$ were countable, we could pick a surjection $\pi=(\pi_1, \pi_2)$: $\mathbb{N}\longrightarrow\mathbb{Z}\times\mathbb{R}$. But then $\pi_2$: $\mathbb{N}\longrightarrow\mathbb{R}$ would be a surjection, so $\mathbb{R}$ would be countable.
| {
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Is Keno a fair game? This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with probability, which perhaps yields the shortest, simplest proofs, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any... | P(win) = P(choose 3 correct #s) = ${20\choose 3}/{80\choose 3} = \frac{57}{4108}$
P(break even) = P(choose 2 correct and 1 wrong #) = ${20\choose 2}\cdot{60\choose 1}/{80\choose 3} = \frac{570}{4108} $
P(lose) = $1 - \frac{57}{4108} - \frac{570}{4108} = \frac{3481}{4108}$
Let x dollars be the net winnings if all 3 #s a... | {
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prove continuity Let $ f:\Bbb R \to \Bbb R $ satisfy the property $ f(x+y)=f(x)+f(y)$ for all $x,y$ in $ \Bbb R $
I have to show that
1)$f(0)=0 , f(-x)=-f(x),$ for all $x$ in $\Bbb R$, and $f(x-y)=f(x)-f(y)$ $y$ in $\Bbb R$
2) If $f$ is continuous at $x=a$ then $f$ is continuous at every point in of $\Bbb R$
I can pro... | If we take $x_{0}\in\mathbb{R}
$ we have $$\lim_{x\rightarrow x_{0}}\left(f\left(x\right)-f\left(x_{0}\right)\right)=\lim_{x\rightarrow x_{0}}f\left(x-x_{0}\right)
$$ and now if we put $$ x-x_{0}=y-a
$$ the limit is equal to $$=\lim_{y\rightarrow a}f\left(y-a\right)=\lim_{y\rightarrow a}\left(f\left(y\right)-f\left(... | {
"language": "en",
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Halmos, Finite-Dimensional Vector Spaces, Sec. 7, Ex. 8 As a linear algebra refresher, I am working through the above cited text (2nd ed., 1958). The exercise asks to determine the number of subsets of $\{0,1\}^3$ the are bases of $\mathbf{C}^3$ as a vector space over $\mathbf{C}$. My approach was basically a somewha... | Here's an attempt to invoke theory to solve this problem.
We have the following theorem.
Theorem. Let $V$ be an $n$-dimensional vector space over a field $\Bbb R$ with $p$ elements. Then the number of linearly independent subsets of $V$ consisting of $m$ elements is
$$
\frac{1}{m!}\prod_{k=0}^{m-1}\left(p^n-p^k\right)\... | {
"language": "en",
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Equivalent Definitions of Prime Subfield I found two definitions for a prime subfield $K$ of a field $F$.
1. Wolfram $-$ $K$ is the subfield of $F$ generated by the multiplicative identity $1$ of $F$.
2. ProofWiki $-$ $K$ is the intersection of all the subfields, say $\{ K_c \}$, of $F$.
I am aware that in the firs... | Let's write $K$ for the field "generated by $1$" and $L$ for the intersection of all subfields of $F$.
It should be clear that $L\subseteq K$. This is because $K$ is a subfield of $F$, and is therefore one of the fields in the intersection that created $L$.
What does it mean to be "generated by $1$", and why is $K\subs... | {
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Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+...+S $. Let
$$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$
By the usual notation for sum of sets let
$$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1),
(x_2,y_2)\in S \} $$
Let
$$ nS\overs... | Yes, brute force adding $10^5n$ points in a Monte-Carlo'esk fashion tells you what the solution's gonna be.
In Mathematica, you get one such point via
With[{x = RandomReal[{-1, 1}]}, {x, Sqrt[1 - x^2]}]
Here for $n=1,2,3,4$:
The way the picture naturally becomes sparse when it comes to addition of multiple very $y-$s... | {
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Primary decomposition of modules - uniqueness proof
Let $M$ be $A$-module, $A$ commutative ring, and $N$ submodule and let
$$N=Q_1\cap\dots\cap Q_r=Q'_1\cap \dots \cap Q'_s$$ be reduced primary decompositions of $N$. Then $r=s$. The set of primes belonging to $Q_1,\dots,Q_r$ and $Q'_1,\dots,Q'_r$ is the same.
It is... |
If $N=Q_1\cap\cdots\cap Q_n$ is a reduced primary decomposition, and $P_i=\sqrt{Q_i}$, then $\operatorname{Ass}(M/N)=\{P_1,\dots,P_n\}$.
We have an exact sequence $0\to M/N\to\bigoplus_{i=1}^n M/Q_i$. Then $\operatorname{Ass}(M/N)\subseteq\operatorname{Ass}(\bigoplus_{i=1}^n M/Q_i)$. But $\operatorname{Ass}(\bigoplus... | {
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Mean value theorem for integration in two dimensions The mean value theorem for integration says that, if $G$ is a continuous real-valued function defined over an interval, $G: [a,b] \to \mathbb{R}$, then the mean value of G on the interval is achieved as a certain point of the interval, i.e:
$$\exists x_0\in[a,b]: G(x... | In addition to muaddib's answer, I found the following specific counter-example. The domain is the square $[0,2\pi]\times[0,2\pi]$, and the function is:
$$G(x,y)=[sin(x+y), cos(x+y)]$$
Then, by symmetry it is easy to see that the integral of $G$ over the domain is $(0,0)$.
However, there is no point in which $G(x,y)=(... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$e^{i\theta}$ versus $\cos\theta+i\sin\theta$ I am teaching an basic university maths course, and have been thinking about the complex numbers part. Specifically, I was wondering why I should include Euler's formula in my course.
This led me to the following "big list" question, which I thought interesting.
What can y... | Complex Analysis uses a lot the logarithm. As you said with $e^{i\theta}$ you can talk more easily for the argument. The important thing is that a branch of the argument exists iff exist a branch of the logarithm.
Therefore, exponential goes with logarithm. Also of the expressions $z^a$ are exponential forms...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1338792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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For $n \geq 2$, there is no $\phi : S^n \to S^1$ such that $\phi(-x) = -\phi(x)$.
Show that for $n\geq 2$ there is not any function $\phi: S^n \rightarrow S^1$ such that $\phi(-x)=-\phi(x)$.
I have no idea about how to solve this problem. It is quite similar to Borsuk-Ulam Theorem.
Any hint?
Thanks for help!
| The Borsuk-Ulam Theorem states that for any continuous odd function $f : S^n \to \mathbb{R}^n$, there is $p \in S^n$ such that $f(p) = 0$.
If such a map $\phi$ existed, then $f = i\circ\phi$ would be a continuous odd map $S^n \to \mathbb{R}^n$ (here $i$ denotes the inclusion $S^1 \to \mathbb{R}^n$). By Borsuk-Ulam, th... | {
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Show that $(r!)^2 ≡ (−1)^{r−1} \pmod p,\ r = (p-1)/2$ I need to prove that if p is an odd prime and
$r = (p-1)/2$ then
$(r!)^2 ≡ (−1)^{r−1} \pmod p$
I think it has something to do with gauss's lemma
https://en.wikipedia.org/wiki/Gauss%27s_lemma_(number_theory)
but I tried a lot and couldn't find a way to break it . ... | Hint: We have $p-1\equiv -1\pmod{p}$, $p-2\equiv -2\pmod{p}$, $p-3\equiv -3\pmod{p}$, and so on up to $\frac{p+1}{2}\equiv -\frac{p-1}{2}\pmod{p}$.
It follows that
$$(p-1)!\equiv (-1)^{(p-1)/2}\left(\left(\frac{p-1}{2}\right)!\right)^2\pmod{p}.$$
Now use Wilson's Theorem.
| {
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