Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Error in proving of the formula the sum of squares Given formula
$$
\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}
$$
And I tried to prove it in that way:
$$
\sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n
$$
$$
\int (n^2+n)\ \text d n=\frac{n^3}{3}+\frac{n^2}{2}+C
$$
But
$$
\frac{n^3}{3}+\frac{n^2}{2}+C $$ i... | Here's an interesting approach using the summation of binomial coefficients.
First, note that
$$\sum_{i=1}^n {i+a\choose b} = {{n+a+1}\choose {b+1}}$$
and also that
$$i^2=2\cdot \frac{(i+1)i}{1\cdot 2}-i=2{{i+1}\choose 2}-{i\choose 1}$$
Hence
$$\begin{align}
\sum_{i=1}^ni^2 &=\sum_{i=1}^n \left[2{{i+1}\choose 2}-{i\ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 2
} |
Brain teaser Strategic choice... $X$ and $Y$ are playing a game. There are $11$ coins on the table and each player must pick up at least $1$ coin, but not more than $5$. The person picking up the last coin loses. $X$ starts. How many should he pick up to start to ensure a win no matter what strategy $Y$ employs?
| A player facing $n$ coins can force a win iff there exists $1\le k \le 5$ such that a player facing $n-k$ coins cannot escape a loss.
Clearly, $n=1$ is a lost position.
Therefore $n=2, 3, 4, 5, 6$ are won positions.
Therefore $n=7$ is a lost position.
Therefore $n=8, 9,10,11,12$ are won positions.
One readily sees that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
Find polynomial whose root is sum of roots of other polynomials We have two numbers $\alpha$ and $\beta$.
We know that $\alpha$ is root of polynomial $P_n(x)$ of degree $n$ and $\beta$ is root of polynomial $Q_m(x)$ of degree $m$.
How do you find polynomial $R_{n m}(x)$ which has root equal to $\alpha+\beta$ without fi... | If you have two polynomials, say $P(x)$ and $Q(y)$ then, their Resultant, $\operatorname{Res}(P,Q)$ is the determinant of their Sylvester matrix. It equals to
$$
\operatorname{Res}(P,Q)=\prod_{P(\alpha)=0, \ P(\beta)=0,}(\alpha-\beta),
$$
the product of all the differences of their roots.
Now if you consider the two va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?
Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$?
For reasonable spaces $X$, e.g. CW-complexes, one has $H^1(X,\mathbb{Z}) \cong [X,\mathbb{T}]$, the group of homotopy classes of... | By the universal coefficient theorem $H^1(X)\cong\operatorname{Hom}(H_1(X),\Bbb Z)$ as $H_0$ is always free. Morphism groups to torsion-free groups are always torsion-free.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/890482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Perpendicular line that crosses specific point? I know the coordinates of line AB, I also have the coordinates of a point called C.
I need to find the coordinates of the start of a line that is perpendicular to AB and that would cross point C. (Point D)
Also the coordinates of point A are always (0, 0)
| Given $A(0,0)$ and $B(x_B, y_B)$, we have $$\nabla \vec {AB} = \frac{y_B-y_A}{x_B-x_A} = \frac{y_B-0}{x_B-0} = \frac{y_B}{x_B} $$ and since $\nabla \vec {CD} \times \nabla \vec {AB} = -1$, given $CD\ \bot \ AB $: $$ \nabla\vec {CD} = -\frac{x_B}{y_B}$$ Hence the equation of $\vec {CD}$ would be $y-y_C = -\frac{x_B}{y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Can functions be defined by relations? So let us say that for whatever reasons, we are not allowed to use function symbols in first-order logic. Then can we define and use a function only by relations?
| Usually functions are defined in the language of set theory. Here is a possible definition using only FOL:
Let $D$ and $C$ be unary predicates. Let $F$ by a binary predicate.
$F$ is said to be a functional relation with domain predicate $D$ and codomain predicate $C$ if and only if:
*
*$\forall x,y:[F(x,y)\implies ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
If $ \frac {z^2 + z+ 1} {z^2 -z +1}$ is purely real then $|z|=$? If z is a complex number and $ \frac {z^2 + z+ 1} {z^2 -z +1}$ is purely real then find the value of $|z|$ .
I tried to put $ \frac {z^2 + z+ 1} {z^2 -z +1} =k $ then solve for $z$ and tried to find |z|, but it gets messy and I am stuck.
The answer giv... | Let's come up with a more interesting question (which might be what you intended to ask).
Find an $\alpha \in \mathbb{R}$ such that $\forall z \in \mathbb{C}: |z| = \alpha \implies \dfrac{z^2+z+1}{z^2-z+1} \in \mathbb{R} \cup \{\infty\}$.
I'll show that $\alpha = 1$ works. If $|z| = 1$, then $z = e^{i\theta}$ for s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Let $X_1, X_2, ...$ be a sequence of real-valued random variables.
Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$
Attempt:
Suppose $X_n \xrightar... | Hint:
$X_n \xrightarrow P 0$ mean $\forall \epsilon >0, P(|X_n| > \epsilon) \to 0$.
$\frac{|X_n|}{|X_n|+1} = \frac{|X_n|}{|X_n|+1} 1_{|X_n| > \epsilon} + \frac{|X_n|}{|X_n|+1} 1_{|X_n| \leq \epsilon} \leq 1_{|X_n| > \epsilon} + \epsilon$
$\frac{|X_n|}{|X_n|+1} = \frac{|X_n|}{|X_n|+1} 1_{|X_n| > \epsilon} + \frac{|X_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
The other ways to calculate $\int_0^1\frac{\ln(1-x^2)}{x}dx$
Prove that
$$\int_0^1\frac{\ln(1-x^2)}{x}dx=-\frac{\pi^2}{12}$$
without using series expansion.
An easy way to calculate the above integral is using series expansion. Here is an example
\begin{align}
\int_0^1\frac{\ln(1-x^2)}{x}dx&=-\int_0^1\frac{1}{x}... | Using the dilogarithm $\mathrm{Li}_2\;$ and the particular values for $0,1,-1\;$you get:
$$\int_0^1\frac{\ln(1-x^2)}{x}dx= \int_0^1\frac{\ln(1-x)(1+x)}{x}dx=
\int_0^1\frac{\ln(1+x)}{x}dx + \int_0^1\frac{\ln(1-x)}{x}dx=
-\mathrm{Li}_2(-x)\Big{|}_0^1 - \mathrm{Li}_2(x)\Big{|}_0^1
=\frac{\pi^2}{12}-\frac{\pi^2}{6} = -\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 1
} |
Integrate $\left[\arctan\left(x\right)/x\right]^{2}$ between $-\infty$ and $+\infty$ I have tried to calculate
$$
\int_{-\infty}^{\infty}\left[\arctan\left(x\right) \over x\right]^{2}\,{\rm d}x
$$
with integration by parts and that didn't work.
I looked up the indefinite integral and found it contained a polylogarithm... | You can use the following way to evaluate. It is pretty neat and simple. Let
$$ I(a,b)=\int_{-\infty}^\infty\frac{\arctan(ax)\arctan(bx)}{x^2}dx. $$
Clearly $I(0,b)=I(a,0)=0$ and $I(1,1)=I$. Now
\begin{eqnarray}
\frac{\partial^2I(a,b)}{\partial a\partial b}&=&\int_{-\infty}^\infty\frac{1}{(1+a^2x^2)(1+b^2x^2)}dx\\
&=&\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
} |
Ratios as Fractions I’m having trouble understanding how fractions relate to ratios. A ratio like 3:5 isn’t directly related to the fraction 3/5, is it? I see how that ratio could be expressed in terms of the two fractions 3/8 and 5/8, but 3/5 doesn’t seem to relate (or be useful) when considering a ratio of 3:5.
Many ... | In general I would not compare ratios with fractions, because they are different things. As rschwieb mentioned in the comments: when you have apples, lemmons and oranges, you can say that the ratio is $3:4:5$. The only "link" you can make to fractions is that you can now say that this the same as saying the ratio is $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How to make this bet fair? A person bets $1$ dollar to $b$ dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. How to find the value of $b$ so that the bet will be fair?
My effort:
There are a total number of ${52 \choose 2} = 26 \cdot 51$ ways... | It is correct.
So the probability of getting different suits is $\dfrac{13}{17}$.
So the odds should be $\dfrac{4}{17} : \dfrac{13}{17}$ or $4 : 13$ or $1:3.25$.
There are other ways of getting the same result.
For example, given the first card, there are $12$ other cards of the same suit and $39$ cards of other suit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluating a limit using L'Hôpital's rule I know that it can be also evaluate using Taylor expansion, but I am intentionally want to solve it using L'Hôpital's rule:
$$ \lim\limits_{x\to 0} \frac{\sin x}{x}^{\frac{1}{1-\cos x}} =
\lim\limits_{x\to 0}\exp\left( \frac{\ln(\frac{\sin x}{x})}{1-\cos x} \right)$$
Now, fr... | After rearranging the quotient a bit we can apply L'hopitals rules $2$ more times to get the answer:
$$\lim_{x\to0}\frac{x\cos(x)-\sin(x)}{x\sin^{2}(x)}\underbrace{=}_{\text{l'hopital}}\lim_{x\to0}\frac{-x\sin(x)}{\sin^{2}(x)+2x\sin(x)\cos(x)}=\lim_{x\to0}\frac{-x}{\sin(x)+2x\cos(x)}$$
$$\underbrace{=}_{\text{l'hopital... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Construction of an equilateral triangle from two equilateral triangles with a shared vertex Problem
Given that $\triangle ABC$ and $\triangle CDE$ are both equilateral triangles. Connect $AE$, $BE$ to get segments, take the midpoint of $BE$ as $O$, connect $AO$ and extend $AO$ to $F$ where $|BF|=|AE|$. How to prove tha... | Embed the construction in the complex plane.
Let $\omega=\exp\left(\frac{\pi i}{3}\right),B=0,C=1,E=1+v$.
Then $A=\omega$ and $D=1+\omega v$, hence $F=B+E-A$ implies:
$$ F = 1-\omega+v,$$
hence:
$$ \omega F = \omega -\omega^2 + \omega v = 1+\omega v = D, $$
so $BFD$ is equilateral.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/891319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Constructing two tangents to the given circle from the point A not on it I'm trying to complete Level 21 from euclid the game: http://euclidthegame.com/Level21/
The goal is to construct two tangents to the given circle from the point A not on it.
So far I've figured that the segments from B to the tangent points must ... | If a line from $A$ intersects a circumference in two points $P$ and $Q$ then it holds that $AP·AQ = {AT}^2$ where $T$ is a point of the circumference so that $\overline{AT}$ is tangent to the circumference.
This is called power of a point
Create such a line (for instance $\overline{AB}$). Name the points of intersectio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A math line interpretation From the text of the question posed here: "How many whole numbers less than 2010 have exactly three factors?" this statement is made:
If there is no fourth factor, then that third factor must be the
square root of the number. Furthermore, that third factor must be a
prime, or there would... | It is easier than the other answers suggest.
If $k|n$ then also $\frac nk|n$, so the divisors of $n$ come in pairs unless $k=\cfrac nk$ or equivalently $n=k^2$
So the only numbers which have an odd number of divisors are squares.
Every positive integer greater than $1$ has the two divisors $1$ and $n$.
A positive integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Evaluate Left And Right Limits Of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ At $0$ Evaluate Left And Right Limits Of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ At $0$
The graph of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ appears to have a jump discontinuity at $0$ and I want to calculate the left and right limits of $f(x)$ to show there is a dis... | A start: Use $\cos 2x=1-2\sin^2 x$. One needs to be careful when finding the square root of $2\sin^2 x$. It is $\sqrt{2}|\sin x|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/891598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Prove that: If $X$ is a topological space and $A$ and $B$ are two subsets of $X$ then,$Cl(A) \cup Cl(B) = Cl(A \cup B) $ Prove that: If $X$ is a topological space and $A$ and $B$ are two subsets of $X$ then,$Cl(A) \cup Cl(B) = Cl(A \cup B) $ where $Cl(H)$ means the closure f the subsets $H $of $X$.
I was able to prove ... | A very useful lemma here is the fact that $Cl(A)$ is the minimal closed set containing $A$ (Verify for yourself!).
Assume for the sake of contradiction that the two sets are not equal. We find that, just as Michael said, $Cl(A) \cup Cl(B)$ is closed. In addition, we must have $A \cup B \subset Cl(A) \cup Cl(B)$. Combin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Converges Or Diverges: $\sum _{n=1}^{\infty }\:e^{-\sqrt{n}}$ Converges Or Diverges:
Attempt:
int_1^∞ e^(-sqrt(n)) dn
t = -sqrt(n); dt = -dn/(2*sqrt(n));
int -2*sqrt(n)/[-2*sqrt(n)] * e^(-sqrt(n)) dn
int (2t)*(e^t) dt
u = 2t; du = 2 dt; dv = e^t dt; v = e^t
(2t)*(e^t) - int 2*e^t
2t*(e^t) - 2*(e^t)
2 * (e^t) ... | Converges by comparison with $~\displaystyle\sum_{n=1}^\infty e^{-2\ln n}~=~\sum_{n=1}^\infty\frac1{n^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/891746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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2 of 3 dice are selected randomly and thrown. What is the probability that one of the dice shows 6 1 red die with faces labelled 1, 2, 3, 4, 5, 6.
2 green dice labelled 0, 0, 1, 1, 2, 2.
Answer: 1/9
Please can you show me how to get the answer. I'm confused about joining the events of choosing 2 of 3 dice vs. getting t... | Out of the three possible choosings, two contain the die with a $6$. (Chance $2/3$)
If the die is selected, then there is a chance in six to get a six. (Chance $1/6$)
The total chance is:
$$\frac{2}{3}·\frac{1}{6} = \frac{2}{18} = \frac{1}{9}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/891823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why? In the MSE-question in a comment to an naswer Michael Hardy brought up the following well known limit- expression for the Euler-gamma
$$ \lim_{n \to \infty} \left(\sum_{k=1}^n \frac 1k\right) - \left(\int_{t=1}^n \frac 1t dt\r... | For Q1. the proof just relies on summation by parts.
For Q2., you can evaluate
$$S_k = \sum_{n=1}^{+\infty}\frac{1}{n(n+1)\ldots(n+k)} = \frac{1}{k!}\sum_{n=1}^{+\infty}\frac{1}{n\binom{n+k}{k}}$$
by exploiting partial fractions decomposition and the residue theorem, or just the wonderful telescoping trick $\frac{1}{n(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proving or disproving inequality $ \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y} \ge x + y + z $
Given that $ x, y, z \in \mathbb{R}^{+}$, prove or disprove the
inequality
$$ \dfrac{xy}{z} + \dfrac{yz}{x} + \dfrac{zx}{y} \ge x + y + z $$
I have rearranged the above to:
$$ x^2y(y - z) + y^2z(z - x) + z^2x(x - y) \ge 0 ... | Holder's inequality:
$$u\cdot v \leq |u||v|$$
for any vectors $u,v$.
Let $\mathbf u=(1/x,1/y,1/z)$ and $\mathbf v=(xz,xy,yz)$. Then show $|v| = xyz |u|$ and thus $$|\mathbf u||\mathbf v| = xyz\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)= \dfrac{xy}{z} + \dfrac{yz}{x} + \dfrac{zx}{y} $$
and:
$$\mathbf u\cdot\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/891997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Evaluating $\int \tan^{1/3}(\theta) d \theta$ I know $\int \tan^{1/3}\theta d \theta$ is a non integrable but wolfram alpha does it easily. And can anyone explain how a fuction is non integrable? My argument is that there has to be a function which represents the area under a graph. It's just that we do not know it. P... | I will just leave here the exact solution.\begin{align}
\int (\tan x)^\frac13\,dx=&\frac14 \bigg[-2\sqrt{3} \tan^{-1}\left(\sqrt{3}-2(\tan x)^\frac13 \right)-2\sqrt{3}\tan^{-1}\left(\sqrt{3}+2(\tan x)^\frac13 \right)\\
&-2\log\left(\tan^\frac23x+1\right)+\log\left(\tan^\frac23x -\sqrt{3}(\tan x)^\frac13+1 \right)\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/892087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How can I find a $k$ and a $n_0$? Find $k$ such that
$$(\lg n)^{\lg n}= \Theta (n^k), k \geq 2$$
That's what I did so far:
$$(\lg n)^{\lg n}=\Theta(n^k) \text{ means that } \exists c_1,c_2>0 \text{ and } n_0 \geq 1 \text{ such that } \forall n \geq n_0: \\ c_1 n^k \leq (\lg n)^{\lg n} \leq c_2n^k$$
How can I continue... | I find that $\ln n$ is a gross exponent, and so I want to simplify the expression. You're right when you say we want to find conditions so that
$$ c_1 n^k < \ln n ^{\ln n} < c_2 n^k,$$
but if we take logs everywhere, then this is the same as finding conditions so that
$$\ln c_1 + k \ln n < \ln n (\ln \ln n) < \ln c_2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/892160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Is there a geometric interpretation for lower order terms in sum of squares formula? So we know that $\sum_{k=1}^n k^2 = n(n+1)(2n+1)/6$ and if we think about making a square layer out of $n^2$ unit cubes, and then placing a square layer of $(n-1)^2$ unit cubes on top of the first layer centered upon the center, and so... | You want to talk about
$$\frac{n(n+1)(2n+1)}6=\frac{n^3}3+\frac{n^2}2+\frac n6$$
I'd prefer to think of the pyramid layers not as centered on one another, but aligned in one corner of the plane. That way they form a regular integer grid, which makes things a bit easier for me. The layer $k$ is represented by the square... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/892259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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How to solve this IVP? Could you please help me solve this IVP?
A certain population grows according to the differential equation:
$$\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{P}{20}\left(1 − \frac{P}{4000}\right) $$
and the initial condition $P(0) = 1000$. What is the size of the population at time $t = 10$?
The answer... | $\textbf{Given}$
$\dfrac{dP}{dt} = \dfrac{P}{20}\left(1-\dfrac{P}{4000}\right)$, $P(0) = 1000$.
$\textbf{Find}$
$P(10)$
$\textbf{Analysis}$
$\dfrac{dP}{dt} = \dfrac{P}{20}\left(1-\dfrac{P}{4000}\right)$.
First divide out $P\left(1-\dfrac{P}{4000}\right)$ from the RHS and multiply $dt$ over from the LHS:
$$
\dfrac{dP... | {
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"source": "stackexchange",
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How do I determine if I should submit a sequence of numbers to the OEIS? When I search a sequence in the OEIS and it's not there, it gives me a message saying "If your sequence is of general interest, please submit it using the form provided and it will (probably) be added to the OEIS!"
But I'm not always sure if I sho... | I would submit any sequence related to typical analytic maneuvers and forms that lie in the intersection of diverse fields of mathematics. For example, in dynamical systems, nonlinear PDEs, complex calculus, and differential geometry, the Schwarzian derivative frequently pops up and can be represented quite simply in t... | {
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"source": "stackexchange",
"question_score": "7",
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Solving exponential equation $e^{x^2+4x-7}(6x^2+12x+3)=0$ How would you find $x$ in:
$e^{x^2+4x-7}(6x^2+12x+3)=0$
I don't know where to begin. Can you do the following?
$e^{x^2+4x-7}=1/(6x^2+12x+3)$
and then find $ln$ for both sides?
| $e^{x^2+4x-7}(6x^2+12x+3)=0 \Rightarrow e^{x^2+4x-7}=0 \text{ or } \ 6x^2+12x+3=0$
$$\text{It is known that } e^{x^2+4x-7} \text{ is non-zero }$$
therefore,you have to solve :
$$6x^2+12x+3=0$$
The solutions are:
$$x=-1-\frac{1}{\sqrt{2}} \\ x=-1+\frac{1}{\sqrt{2}}$$
| {
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"source": "stackexchange",
"question_score": "3",
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What's $\sum_{k=0}^n\binom{n}{2k}$? How do you calculate $\displaystyle \sum_{k=0}^n\binom{n}{2k}$? And doesn't the sum terminate when 2k exceeds n, so the upper bound should be less than n?
EDIT: I don't understand the negging. Have I violated a rule of conduct or something?
| $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{... | {
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How to arrive at coordinates using 2D rotation matrix multiplication. I'm having a little trouble understanding 2D rotation matrices. I apologize in advance, as I have probably missed something really obvious! (My mathematics isn't brilliant!)
Ok, so I have the following matrices (the brackets should be joined vertical... | If you multiply your matrix with the vector, you get the results immediately. There are no steps in between. This is how matrix multiplication works.
| {
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"source": "stackexchange",
"question_score": "1",
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About integrating $\sin^2 x$ by parts This is about that old chestnut, $\newcommand{\d}{\mathrm{d}} \int \sin^2 x\,\d x$.
OK, I know that ordinarily you're supposed to use the identity $\sin^2 x = (1 - \cos 2x)/2$ and integrating that is easy. But just for the heck of it, I tried using the $u$-$v$ substitution method ... | In the second application of the $u$-$v$ method, there may be an error: If $\newcommand{\d}{\mathrm{d}} \d v = \cos x\,\d x$, then $v = \sin x$ and not $-\sin x$. Hence the $u$-$v$ method overall would produce
$$\int \sin^2 x\,\d x = −\sin x\cos x + \sin x\cos x + \int \sin^2 x\,\d x$$
resulting in circular reasoning: ... | {
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"source": "stackexchange",
"question_score": "2",
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Integral Involving Dilogarithms I came across the identity
$$\int^x_0\frac{\ln(p+qt)}{r+st}{\rm d}t=\frac{1}{2s}\left[\ln^2{\left(\frac{q}{s}(r+sx)\right)}-\ln^2{\left(\frac{qr}{s}\right)}+2\mathrm{Li}_2\left(\frac{qr-ps}{q(r+sx)}\right)-2\mathrm{Li}_2\left(\frac{qr-ps}{qr}\right)\right]$$
in a book. Unfortunately, as... | Among various ways to do it, this one is simple :
| {
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Projection of one 2D-rectangle onto another 2D-rectangle I have the following:
http://i.imgur.com/iwOzmxa.png
The center of the blue box is related to the center of the black box. For example, such a relationship could be described such that the black box would be a game's main screen where the red dot is the player, ... | Assuming that you want the co-ordinates in the blue and black rectangles to be proportionately the same across and down the rectangles, you get $ \frac{X_{blue} - X_S}{X_{S2}-X_S}=\frac{X_{black}-X_L}{X_{L2}-X_L}$ and similarly with $Y$, so
$$X_{blue} = X_S + \left(X_{black}-X_L\right)\frac{X_{S2}-X_S}{X_{L2}-X_L}$$
$... | {
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Can this log question be simplified? $ { 2^{log_3 5}} - {5^{log_3 2}}.$
I don't know any formula that can apply to it or is there a formula??
Even a hint will be helpful.
| $ { 2^{\log_3 5}} - {5^{\log_3 2}}$
$= 2^{\log_3 5} - 5^{\frac{\log_5 2}{\log_5 3}}$ (change of base)
$= 2^{\log_3 5} - (5^{\log_5 2})^{(\frac{1}{\log_5 3})}$ (using $a^{\frac{b}{c}} = {(a^b)}^{\frac{1}{c}}$)
$= 2^{\log_3 5} - 2^{\frac{1}{\log_5 3}}$
$= 2^{\log_3 5} - 2^{\frac{\log_5 5}{\log_5 3}}$ ($\because \log_5 5... | {
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How do elements of $\mathbb{R}(xy,x+y)$ look like? I have problems with determining what are typical elements of such field $\mathbb{R}(xy,x+y)$
In one indeterminate it is easier as
$\mathbb{R}(x)=\Bigl\{\frac{f(x)}{g(x)}, g(x)\neq 0, f(x),g(x)\in\mathbb{R}[x]\Bigr\} $
where
$f(x)=a_{0}+a_{1}x+a_{2}x^2+\dots+a_{n}x^{... | This is the field of all rational functions symmetric in $x$ and $y$. So for example it contains
$x^3+y^3+5x^2y+5xy^2 -x-y$ since this is symmetric in $x$ and $y$.
$x+y$ and $xy$ are the elementary symmetric functions and a basic theorem of algebra is the every symmetric rational functions is a rational function of the... | {
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Continuity of function where $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function which satisfies $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$ is continuous at $x=0$, then it is continuous at every point of $\mathbb{R}$.
So we know $\forall \epsilon > 0 ~~\exis... | To show continuity at $x$ simply notice that:
$$\lvert f(x+h)-f(x)\rvert=\lvert f(x)f(h)-f(x)\rvert=\lvert f(x)\rvert\lvert f(h)-1\rvert$$
and notice that $f(0)=f(0+0)=f(0)f(0)$ so either $f(0)=0$ or $f(0)=1$. So either $f$ is identically $0$ and hence continuous or we use continuity at $x=0$.
| {
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How to solve the recurrence relation $T(n) = T(\lceil n/2\rceil) + T(\lfloor n/2\rfloor) + 2$ I'm trying to solve a recurrence relation for the exact function (I need the exact number of comparisons for some algorithm). This is what i need to solve:
$$\begin{aligned}
T(1) &= 0 \\
T(2) &= 1 \\
T(n) & = T(\lceil n/2\rcei... | As mentioned in comment, the first two conditions $T(1) = 0, T(2) = 1$ is incompatible with
the last condition
$$\require{cancel}
T(n) = T(\lfloor\frac{n}{2}\rfloor) + T(\lceil\frac{n}{2}\rceil) = 2\quad\text{ for }
\color{red}{\cancelto{\;\color{black}{n > 2}\;}{\color{grey}{n \ge 2}}}
\tag{*1}$$
at $n = 2$. We will a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to calculate the probability of rolling 6 at least 5 times in a row, out of 50 tries? If I roll the dice 50 times, how do I calculate the chance that I will roll 6 at least 5 times in a row?
Why this problem is hard
*
*With 5 tries this would be easy: take $(1/6)$ to fifth power.
*With 6 tries this is manageable... | There is no straightforward formula for this probability. However it can be computed exactly (within numerical error). You can keep track of the vector of probabilities $\mathbf{p}_t$ that at time $t$ you are in state:
*
*you haven't already rolled 5 6s and at the present time have rolled 0 6s,
*you haven't already... | {
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"source": "stackexchange",
"question_score": "13",
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Any abstract algebra book with programming (homework) assignment? All:
I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application).
Can anyone recommend an abstract algebra book which cover Galois theory (and its applications)?
I have ... | All books listed in the comments above are listed in my answer to the question Novel approaches to elementary number theory and abstract algebra, so I am placing a CW-answer here to remove this question from the unanswered queue.
| {
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There always exists a subfield of $\mathbb C$ which is a splitting field for $f(x)$ $\in$ $Q[X]$? So I've been studying field theory on my own, and I just started learning about splitting fields. Based on my understandings if a polynomial, $f(x)$ $\in$ $Q[X]$, then there should be always a subfield of $\mathbb C$ which... | Yes. Every polynomial in $\Bbb C[x]$ has a root in $\Bbb C$ (fundamental theorem of algebra), which means that every polynomial has a linear factor; after dividing we get another polynomial in $\Bbb C[x]$ of smaller degree, which also has a root, which means we get another linear factor we can divide by, and we can con... | {
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How to understand uniform integrability? From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
| Many arguments in probability use truncation: take a random variable $X$ and define $X^k := X1_{|X| \leq k}$. This allows us to handle "most of $X$" on a compact set $K:= [-k,k]$ if $X$ is integrable, as $E|X-X^k| < \epsilon$ for sufficiently large $k$. In other words,
$$\lim_{k \rightarrow\infty} E|X1_{|X| > k}| = 0.$... | {
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Find $ \int_0^2 \int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy\hspace{1mm}dx$ I need the approximation to four decimals
Not sure how to start or if a closed form solution exists
All Ideas are appreciated
| In Maple environment:
[> s:= Int(sqrt(5*x^2+8*x*y+5*y^2+1), x = 0 .. 2);
/2 (1/2)
| / 2 2 \
s:= | \5 x + 8 x y + 5 y + 1/ dx
|
|
/0 ... | {
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Elementary algebra problem Consider the following problem (drawn from Stanford Math Competition 2014): "Find the minimum value of $\frac{1}{x-y}+\frac{1}{y-z}+ \frac{1}{x-z}$ for for reals $x > y > z$ given $(x − y)(y − z)(x − z) = 17.$"
Method 1 (official solution): Combining the first two terms, we have
$\frac{x−z}{(... | In the second solution, when $=$ in $\ge$ is satisfied, $a=b=c$. But then, $abc=17$ and $a+b=c$ makes the contradiction.
| {
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Cauchy-Goursat theorem
I understand the solution to this question via the use of the following corollary
However for $k+1=0$ you get :
$\displaystyle \oint_0^{2\pi}id\theta$
Why am I not able to apply the cauchy-goursat theorem to get that the integral=0?
| There's no contradiction, you just proved $z^{-1}$ doesn't have a primitive function on any open set containing $\gamma_1$.
$(z^n)'=nz^{n-1}$ gives you a primitive function of $z^{n-1}$ only for $n\ne0$, so $z^{-1}$ is excluded. You could think of using the principal branch of the complex logarithm, but that's not even... | {
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Conversion of rotation matrix to quaternion We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & m_{21}&m_{22}\end{bmatrix}}_{3\times 3}\tag 1 $.
How do I accurately calculate quaternion $q = ... | The axis and angle are directly coded in this matrix.
Compute the unit eigenvector for the eigenvalue $1$ for this matrix (it must exist!) and call it $u=(u_1,u_2,u_3)$. You will be writing it as $u=u_1i+u_2j+u_2k$ from now on. This is precisely the axis of rotation, which, geometrically, all nonidentity rotations have... | {
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Logic behind dividing negative numbers I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative number will give us a positive number.
Given the following equations:
*
*$\frac... | Division can be thought of (in an algorithmic sense) as repeated subtraction. The question "what is 6/2" is exactly equivalent to the question "how many times must one subtract 2 from 6 to reach 0".
How many times must one subtract 2 from 18 to reach 0? Of course it's 9.
How many times must one subtract -2 from 18 to... | {
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How to calculate length and area for this curve? $C : x^{2/3} + y^{2/3} = 1$
I'm stuck, so any tip will be helpful
Thanks in advance!
| The solution for $y$ is
$$y(x) = (1 - x^{2/3})^{3/2}$$
The area $A$ is given by
$$A=4\int_0^1 y(x) dx=\frac{3\pi}{8}$$
The length $L$ of the curve is given by:
$$ L=4\int_0^1dx\sqrt{(y'(x))^2+1}=6$$
| {
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Find the value of $\sum_{k=1}^{n} k \binom {n} {k}$ I was assigned the following problem: find the value of $$\sum_{k=1}^{n} k \binom {n} {k}$$ by using the derivative of $(1+x)^n$, but I'm basically clueless. Can anyone give me a hint?
| Notice that $\displaystyle S = \sum_{k=0}^{n} k \binom {n} {k} = \sum_{k=0}^{n}(n-k)\binom {n} {n-k} = n\sum_{k=0}^{n}\binom {n} {n-k}-S$
so, as $\displaystyle \binom {n} {n-k}=\binom {n} {k}$ we have $\displaystyle 2S = n\sum_{k=0}^{n}\binom {n} {k} = n2^n$ and so $\displaystyle S = n2^{n-1}.$
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that a linear matrix transformation is bijective iff A is invertible. Suppose a linear transformation $T: M_n(K) \rightarrow M_n(K)$ defined by $T(M) = A M$ for $M \in M_n(K)$.
Show that it is bijective IFF $A$ is invertible.
I was thinking then that I could show that it is surjective. So suppose there exists a
... | By the rank-nullity theorem $T$ is bijective if and only if $T$ is injective if and only if $T$ is surjective.
For the injectivity: we have
$$T(M)=T(N)\iff AM=AN$$
*
*If $A$ is invertible then $AM=AN\implies M=N$ and then $T$ is injective.
*If $A$ isn't invertible so let $N=M+(x\; x\;\cdots\; x)$ where $x\in\ker A,... | {
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Simplify $(7-2i)(7+2i)$. Found difference between mine and solution guide's and didn't know why. I looked up the solution guide and found out:
$(7-2i)(7+2i)$
$=49-(2i)^2$
$=49+4$
$=53$
Why the unknown "$i$" just disappeared$?$ I supposed it might be:
$(7-2i)(7+2i)$
$=49-(2i)^2$
$=49-4i$
does it?
May someone tell me whi... | We have $(7-2i)(7+2i)=49+14i-14i-4i^2=49-4i^2.$ Note that $i=\sqrt{-1}\implies i^2=(\sqrt{-1})^2=-1.$ Therefore,
$$(7-2i)(7+2i)=49-4i^2=49-4(-1)=49+4=53$$
| {
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Showing Galois Group is Abelian I'm having trouble showing that $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is Abelian. First I want to be able to show that $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is Galois, but I'm also not sure how to do this. Any help is appreciated!
| It's the splitting field for the polynomial $x^n-1$, hence it is Galois by the characterization of normal extensions as splitting fields for a set of polynomials.
To see it is abelian, it is most direct to show that
$$\text{Gal}\left(\Bbb Q(\zeta_n)/\Bbb Q\right)\cong (\Bbb Z/n\Bbb Z)^\times$$
by showing there is a m... | {
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Divergence of a recursive sequence If $(x_n)$ isthe sequence defined by $x_1=\frac{1}{2}$ and $x_{n+1}=\sqrt{x_n^2 +x_n +1}$, show that $\lim x_n = \infty$
Ive tried a couple of things but none of them helped. Ive tried to suppose, by contradiction, that the sequence is bounded, find a lower sequence that goes to infin... | The sequence is increasing. If it were bounded above, it would have a limit $L$. Then
$$L=\lim_{n\to\infty} x_{n+1}=\lim_{n\to\infty} \sqrt{x_n^2+x_n+1}=\sqrt{L^2+L+1}$$ would give $L=\sqrt{L^2+L+1}$, which is impossible, since $L\gt 0$.
| {
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For recurrence T(n) = T(n − a) + T(a) + n, prove that T(n) = O(n^2 ) complexity I have been looking over this question for hours now, and can't seem to work it out. It's a question regarding the complexity of sorting algorithms
Assume that $a$ is constant and so is $T(n)$ for $n ≤ a$. For recurrence $T(n) = T(n −
a) + ... | To show that $T(n) = O(n^2)$, we will prove by induction on $n$ that there exist constants $c, n_0$ such that for all $n \geq n_0$, we have that:
$$
T(n) \leq cn^2
$$
Base Case: I'll let you do this part.
Induction Hypothesis: Assume that the claim holds for all $n' < n$.
It remains to prove that the claim holds for $n... | {
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How prove $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$ How prove that if $x, y \in (0,\sqrt{\frac{\pi}{2}})$ and $x \neq y$, then $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}... | the right question is :
$(\ln{\dfrac{1-\sin{xy}}{1+\sin{xy}}})^2 \le \ln{\dfrac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\dfrac{1-\sin{y^2}}{1+\sin{y^2}}}$
$f(x)=\ln{\dfrac{1-\sin{x}}{1+\sin{x}}}\le 0 , f(0)=0$
WLOG $y=ax,a\ge1 \implies $the inequality $\iff (f(ax^2))^2 \le f(x^2)f(a^2x^2) \iff (f(ax))^2 \le f(x)f(a^2x) \iff \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/894759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Sum of cubes of roots of a quartic equation $x^4 - 5x^2 + 2x -1= 0$
What is the sum of cube of the roots of equation other than using substitution method? Is there any formula to find the sum of square of roots, sum of cube of roots, and sum of fourth power of roots for quartic equation?
| Easy way to do this is to take the first derivative of the polynomial and divide it by the polynomial itself.
f ( x ) = x^4 - 5x^2 + 2x - 1 = 0
f ' ( x ) = 4x^3 - 10x + 2
Now do f ' ( x ) / f ( x ) = ?
(x^4 + 0x^3 - 5x^2 + 2x - 1 ) ) ......4x^3 + 0x^2 - 10x + 2
.......- 0....+5........- 2..+ 1
4 / 1 = 4 so that is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/894826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Which number remains alive? There are $100$ people standing in a circle numbered from $1$ to $100$. The first person is having a sword and kills the the person standing next to him clockwise i.e $1$ kills $2$ and so on. Which is the last number to remain alive? Also if $1$ kills both the person standing next to him. Wh... | Here is the solution in Python : Not an elegant mathematical one ; but since I did not know the mathematically
from itertools import cycle
NUMBER = 100
people = list(range(1, NUMBER + 1))
dead = []
print "Runing"
print people
people_list = cycle(people)
while len(people) != 1:
tolive = next(people_list)
if t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/894912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 9,
"answer_id": 7
} |
Why is $\sin \theta$ just $\theta$ for a small $\theta$? When $\theta$ is very small, why is sin $\theta$ taken to be JUST $\theta$?
| It's not just $\theta$.
What you observe is the fact that $\sin \theta$ and $\theta$ approach zero from either side of the number line at a pretty similar rate. This can be best demonstrated with a graph.
You can see that they are about to overlap just at zero. So when $\sin \theta$ is approaching $0$ for some very v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/895015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
What is the probability that no married couples are among the chosen? Eight married couples are standing in a room.
4 people are randomly chosen.
What is the probability that no married couples are among the chosen?
| Another way to figure this out would be to choose 4 different married couples to provide a chosen person (there are $\left(\begin{array}{c} 8\\ 4\end{array}\right)=70$ ways to do this). Then pick one member of each of the 4 chosen couples (there are $2^4=16$ ways to do this). So altogther there are $70\cdot 16=1120$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/895237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
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Prove that a function $f(n)$ counting the number of odd divisors multiplicative How can I show that $f(n)$ is multiplicative, where $f(n)$ represents the number of the divisors of n in the form $2k + 1$?
I'm studying algebra and I came across some questions on multiplicative functions (which should be number theory th... | First note $f(1)=1$, then let compute $f(2^k)= 1$.
Note that the total number of divisors function is $\tau(n)$ which is multiplicative, so considering if $n=2^km$ with $m$ odd, we can see that $f(n)=f(m)$, so
$$f(2^km)={\tau(n)\over \tau(2^k)}=\tau(m)$$
and we know $\tau$ is multiplicative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/895321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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derivative formula $\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} \wedge\mathbf{r}) = (n-1)\mathbf{a}$ Assume $\mathbf{r}=\mathbf{x}−\mathbf{x}′$ is the position vector in $\mathbb{R}^n$, for constant $\mathbf{a}$, we have
$$\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{... | I would think it is still applicable for traditional vector analysis approach. With $\mathbf{r}\in\mathbb{R}^n$, then
\begin{align}
[\nabla\times(\mathbf{a}\times\mathbf{r})]&=\varepsilon_{ijk}\partial_j[\mathbf{a}\times\mathbf{r}]_k=\varepsilon_{ijk}\partial_j(\varepsilon_{klm}a_{\ell}r_m)\\
&=\varepsilon_{ijk}\vareps... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/895398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Are $i,j,k$ commutative? I am trying to understand quaternions. I read that Hamilton came up with the great equation:
A) $i^2 = j^2 = k^2 = ijk = −1$
In this equation I understand that $i,j,k$ are complex numbers.
Later on, I read that
B) $ij=k$
C) $ji=-k$
So, if $i,j,k$ are complex numbers, and complex number multipl... | Indeed commutativity still holds in the quaternions if your complex number only contains i's, j's or k's. I.e. any number's of the form $a + bi + 0j + 0k$ are commutative to each other, similarly $a + 0i + bj + 0k$ is commutative with other numbers of that form also and same with $a + 0i + 0j + bk$. However commutativi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/895463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine variables that fit this criterion... There is a unique triplet of positive integers $(a, b, c)$ such that $a ≤ b ≤ c$.
$$
\frac{25}{84} = \frac{1}{a} + \frac{1}{ab} + \frac{1}{abc}
$$
Just having trouble with this Canadian Math Olympiad question. My thought process going into this, is:
Could we solve for $\fr... | Factoring, we see
$\displaystyle \frac{25}{84} = \frac{1}{a}(1+\frac{1}{b}(1+\frac{1}{c}))$
And we know the prime factoring of 84 gives $2\times2\times3\times7$ So we know $a,b,$ and $c$ are each going to be multiples of these primes. So we start with finding $a$:
$\displaystyle \frac{25}{84}a = 1+\frac{1}{b}(1+\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/895556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Nil radical of an ideal on a commutative ring This is a problem of an exercise list:
Let $J$ be an ideal of a commutative ring A. Show that $N(N(J))=N(J)$,
where $N(J)=\{a \in A; a^n \in J$ for some $n \in \mathbb{N}\}$.
What I did:
$N(J)=\{a \in A; a^n \in J$ for some $n \in \mathbb{N}\}$
$N(N(J))=\{a \in A; a^k \... | You didn't do anything wrong.
The definition given for the radical of an ideal does not actually yield an ideal for noncommutative rings, because nilpotent elements needn't be closed under multiplication. There are a couple of ways to fix this, at least for the nilradical (radical of the zero ideal), both achieved by r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/895718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Nonintegral element and a homomorphism
Assume $R\subseteq S$ are rings. Choose $x\in S$ nonintegral over $R$. I want to define a homomorphism from $R[x^{-1}]$ to a field which maps $x^{-1}$ to zero.
I was trying to show that $R[x^{-1}]$ is a polynomial ring in one variable. Then I could define my map in this way $\su... | Let $R=\mathbb{Z}$ and $S=\mathbb{Q}$. Then $x=\frac{1}{2} \in \mathbb{Q}$ is not integral over $\mathbb{Z}$ since $\mathbb{Z}$ is integrally closed. But of course $\mathbb{Z}=\mathbb{Z}[x^{-1}]$ which is not the polynomial ring. I guess you need some additional assumptions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/895821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Function with continuous inverse is continuous? If function $\textbf{F}^{-1}(x)$ is an inverse of function $\textbf{F}$ and $\textbf{F}^{-1}(x)$ is continuous. Is it true that $\textbf{F}(x)$ is continuous too?
| The answer is no: take $f^{-1}(x) = e^{ix}$, defined from $[0,2\pi)$ to $\mathbb{S}^1$ (the unit sphere in the plane)
This function is clearly continuous.
Unfortunately, its inverse cannot be continuous since otherwise $[0,2\pi)$ would be compact being the image of the compact set $\mathbb{S}^1$ under a continuous func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/895889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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if $ax-2by+cz=0$ and $ac-b^2>0$ , Prove $zx-y^2\leq0$
for real numbers like $a,b,c,x,y,z$ that $ax-2by+cz=0$ and $ac-b^2>0$ Prove:$$zx-y^2\leq0$$
Additional info: The Proof should be by contradiction.we can use Cauchy , AM-GM and other simple inequalities.
Things I have done so far: as Problem wants a Proof by contra... | Notice that $by=\frac{ax+cz}{2}$, so that
$$
acy^2 \geq b^2y^2=\bigg(\frac{ax+cz}{2}\bigg)^2 \geq ax\times cz
$$
and hence $y^2 \geq xz$ (notice that $ac>0$ because $ac>b^2$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/896079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
matrix differentiation - derivative of matrix vector dot product with respect to matrix Given the function
$$f(N) = x_1^T M x_2 $$
where
*
*$x_1 = Nv_1 $
*$x_2 = Nv_2 $
*$x_1, x_2, v_1, v_2$ are vectors with dimension $n \times 1$
*$M$ and $N$ are matrices with dimension $n \times n$
what's the derivative of $... | Generalizing the problem slightly, to use matrix (instead of vector) variables, $X_k = N\cdot V_k$.
We can write the function as
$$ \eqalign {
f &= X_1 : M\cdot X_2 \cr
} $$
Now take the differential and expand
$$ \eqalign {
df &= dX_1 : M\cdot X_2 + X_1 : M\cdot dX_2 \cr
&= d(N\cdot V_1) : M\cdot X_2 + X_1 : M\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/896157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Inequality involving a finite sum this is my first post here so pardon me if I make any mistakes.
I am required to prove the following, through mathematical induction or otherwise:
$$\frac{1}{\sqrt1} + \frac{1}{\sqrt2} + \frac{1}{\sqrt3} + ... + \frac{1}{\sqrt{n}} < 2{\sqrt{n}}$$
I tried using mathematical induction th... | If you want to take a look on the Riemann's sum method :
$\forall n > 1$, we have $$\int_{n-1}^n \frac{dt}{\sqrt{t}} \ge (n-(n-1))\cdot \underset{x \in [n-1,n]}{\min} \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{n}} $$
Hence, $$\frac{1}{\sqrt1} + \frac{1}{\sqrt2} + \frac{1}{\sqrt3} + ... + \frac{1}{\sqrt{n}} \le 1+ \int_{1}^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/896259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
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Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y \vec{\imath} −5x \vec{\j... | well, I think that it is rather a question of getting C in its parametric form.
Your plane is $\Pi :x+y+z=3$.
I will try here to find the orthogonal vectors to the $\Pi$:
let's say that $\underline e_1=(a_1,a_2,0)$ and $\underline{e_2}=(-a_1,a_2,0)$.
Dot product of those vectors with the normal vector of $\Pi$ gives yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/896365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Integration of some floor functions Can anyone please answer the following questions ?
1) $\int\left \lfloor{x}\right \rfloor dx$
2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$
3) $\int_0^2$ $\left \lfloor{x^2+x-1}\right \rfloor$ $dx$
4) $\int_o^\pi$ $\left \lfloor{x(1+\sin(\pi x)}\right \rfloor$
Also can anyo... | The floor function turns continuous integration problems in to discrete problems, meaning that while you are still "looking for the area under a curve" all of the curves become rectangles. In general, the process you are going to want to take will go something like this:
Consider the function (before taking the floor o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/896437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 7,
"answer_id": 2
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Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent? Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$
Attempt:
I got this using Ratio Test:
$$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
| Hint: Use the integral test on $\sum_{k=1}^{\infty}\frac{1}{k^{2}}$.
Note that the ratio test is inconclusive in this case.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/896521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Maximum of Three Uniform Random Variables Here is the question, I am studying for exam P, and am using a study guide with a solution guide. I am stumped on this problem, and the solution in the back was very confusing. Any clarity I can get would be tremendously appreciated. Here is the problem:
Three individuals are r... | If $y > 3.1$, then you know that the first two runners are already done. So the probability that the last runner finishes in at most $y$ minutes is the probability that $X_3 \le y$; i.e., $$\Pr[Y \le y] = \Pr[X_3 \le y] = \frac{y-2.9}{3.3-2.9}, \quad 3.1 < y \le 3.3.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/896614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Proof by Induction: $(1+x)^n \le 1+(2^n-1)x$ I have to prove the following by induction:
$$(1+x)^n \le 1+(2^n-1)x$$
for $n \ge 1$ and $0 \le x \le 1$.
I start by showing that it's true for $n=1$ and assume it is true for one $n$.
$$(1+x)^{n+1} = (1+x)^n(1+x)$$
by assumption:
$$\le (1+(2^n-1)x)(1+x)$$
$$= 1+(2^n-1)x+x+(... | $$(1+x)^{n+1}=(1+x)(1+x)^{n}\le(1+x)(1+(2^{n}-1)x)=1+(2^{n}-1)x+x(1+(2^{n}-1)x)$$
$$\underbrace{\le}_{x\le1}1+(2^{n}-1)x+x(1+2^{n}-1)=1+(2^{n}-1)x+2^{n}x$$
$$=1+(2\cdot2^{n}-1)x=1+(2^{n+1}-1)x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/896720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Consecutive Prime Gap Sum (Amateur) List of the first fifty prime gaps:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4.
My conjecture is that the sum of consecutive prime gaps is always prime whenever a prime gap ... | Set $g_n=p_{n+1}-p_n$, where $p_n$ is the series of prime numbers, with $p_1=2$.
Then
$$
p_1+\sum_{i=1}^n g_i=\sum_{i=1}^n g_i+2=p_{n+1}.
$$
So the conjecture is obviously true, but not useful.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/896802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$? It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as:
$$
\begin{align}
f : (a,b) &\to (\alpha,\beta) \\
x &\mapsto \alpha+(x-a)(\beta-\alpha).
\end{align}
$$
It is also possible to biject a bou... | Cantor showed that any two countable densely ordered sets with no first or last element are order isomorphic. One can build the isomorphism using his Back and Forth Method.
If all we want is a bijection, we can more simply note that both sets are countably infinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/896989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Why learn to solve differential equations when computers can do it? I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background.
One question is haunting me.
Why do I need to learn to do complex math operations on paper when m... | So that someone can teach the computer how to do it better.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/897034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "63",
"answer_count": 15,
"answer_id": 13
} |
how to find $(I + uv^T)^{-1}$ Let $u, v \in \mathbb{R}^N, v^Tu \neq -1$. Then I know that $I +uv^T \in \mathbb{R}^{N \times N}$ is invertible and I can verify that $$(I + uv^T)^{-1} = I - \frac{uv^T}{1+v^Tu}.$$ But I am not able to derive that inverse on my own. How to find it actually? That is if am given $A = I +uv^... | If you guess that the inverse has a similar form,
$$(I+uv^T)^{-1}=\alpha I+\beta uv^T\ ,$$
then multiplying out and equating coefficients will solve the problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/897114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Placing symbols so that no row remains empty.
Q.1.The symbols +, + , #, # , *, $ ($6$ in total) are to be placed in the squares of the given figure. Find the number of ways of placing the symbols so that no row (there are $5$ rows) remains empty.
I know combinatorics at undergrad level. This problems are from advanced... | Position would be $(1,1,1,1,2)$ or $(1,1,2,1,1)$ or $(2,1,1,1,1)$ for five rows respectively.
Positions to which can beselected in $\binom 32$ for the row with $2$ elements and $3\times3$ for other two.
Now treating each symbol as a unique entity, arrangement can be done in $6!$ ways. Now to cancel multiples we divide... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/897224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$ For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$.
$k > 0$ is some constant.
And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$.
At the initial time, $y$ follows a parabol... | Your equation can be written as follows:
$$F(p,q,x,t,y) = (p^2+1)q^2 -k = 0, \quad p = y_x, \quad q = y_t,$$
and hence:
\begin{align}
F_p & = 2pq^2, \\
F_q & = 2(1+p^2) q,\\
F_t = F_x = F_y & = 0,
\end{align}
so the Lagrange-Charpit equations read:
$$ \frac{\mathrm{d}x}{2 pq^2 }= \frac{\mathrm{d}t}{2(1+p^2)q} = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/897299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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An ABC soft question about epsilon-delta argument Someone told me that some textbooks present epsilon-delta argument somewhat misleadingly. For example, consider the simplest one: the convergence of the sequence $(1/n)_{1}^{\infty}$
to $0$.
These textbooks may prove this convergence as follows: Let $\epsilon > 0$. Sinc... | " ...what requires to prove is the existence of N for every ϵ>0."
The Archimedean property of real numbers ensures that for every real $\epsilon > 0$ there exists a positive integer $N$ such that
$$N > \frac1{\epsilon}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/897396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Vertical asymptote of $h(x)=\frac{x^2e^x}x$ $$h(x)=\frac{x^2e^x}x$$
The function h is defined above. Which of the following are true about the graph of $y=h(x)$?
*
*The graph has a vertical asymptote at $x=0$
*The graph has a horizontal asymptote at $y=0$
*The graph has a minimum point
A. None
B. 1 and 2 Only
C... | If you are allowed negative values of $x$, then you do get a horizontal asymptote at $y = 0$ for negative $x$ because $x e^x$ increases up to $0$ (but never reaches it) as $x$ becomes larger and larger magnitude negative numbers. Also, if you are allowed negative $x$, then the function has a minimum. The derivative is ... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 2
} |
Complex function defined by contour integral along a smoothly varying path Let $D$ be a domain in the complex plane. Consider the function $F: D\to \mathbb{C}$, defined by
$$
F\left( z \right) = \int_{\mathscr{C}\left( z \right)} {f\left( {z,t} \right)dt} .
$$
Suppose that the path $\mathscr{C}\left( z \right)$ is a co... | Simple special case for intuitions sake: Let $D=\mathbb C$, $f(z,t)=1$, $\gamma_z(t)=g(z)t$, $0\leq t \leq 1$, where the bar denotes complex conjugation. Then $F(z)=g(z)$ so $F$ is analytic if and only if $g$ is analytic.
Suppose now in addition to your stated hypothesis that $z\mapsto \gamma_z'(t)$ and $z\mapsto \gam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/897528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Group with order $|G|=p^2q^2$ and $p\not\mid q-1, p\not\mid q+1$ Group with order $|G|=p^2q^2$ and $p\not\mid q-1, p\not\mid q+1$. $q\not=p$ both prime.
I want to show that there is only one $p$-Sylow subgroup. Let $S_p(G)$ the number of $p$-Sylow subgroups. I know that $S_p(G) \equiv 1 \mod p$ and $S_p(G) \mid q^2$.
... | Note that from $q^2 = kp + 1$ we get that $$kp = q^2 - 1 = (q - 1)(q+1).$$ So $p \mid (q-1)(q+1)$, but as $p$ is prime we must have $p \mid q - 1$ or $p \mid q + 1$; either way, we have a contradiction. Therefore, $S_p(G) \neq q^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/897614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show there exists a value such that each partial sum equals its limit in modulus
For each $n \in \mathbb{N}_0$, and for all $z \in \mathbb{C}$, define $$p_n(z) := \sum^{n}_{k=0} {z^k \over {k!}}.$$
Show that for all $r > 0$ and for all $n \in \mathbb{N}_0$, there exists $z \in \mathbb{C}$ with $|z|=r$ such that $|p_n(... | As stated in the comments, we have that $\|p_n(r)\|=p_n(r)<e^r=\|e^r\|$, hence we just need to prove that:
$$\exists\, z=r\,e^{i\theta}:\quad \|p_n(z)\|^2 > \|e^z\|^2 = e^{2r\cos\theta}\tag{1}$$
then apply a continuity argument. If we prove that:
$$\int_{0}^{\pi}\|p_n(r e^{i\theta})\|^2\sin\theta\,d\theta > \int_{0}^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/897716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Is $\mathbb{R}$ a subspace of $\mathbb{R}^2$? I think I've been confusing myself about the language of subspaces and so on. This is a rather basic question, so please bare with me. I'm wondering why we do not (or perhaps "we" do, and I just don't know about it) say that $\mathbb{ R } $ is a subspace of $\mathbb{ R }^... | This is indeed an important question. No doubt that $\mathbb{R}$ is isomorphic to many subspaces of $\mathbb{R}^2$, or in other words, $\mathbb{R}$ can be embedded in many ways in $\mathbb{R}^2$. The thing is that there isn't any specific subspace in $\mathbb{R}^2$ which is the best one to represent $\mathbb{R}$.
If yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/897775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Find $\delta$ when $\epsilon$ is known (limit problem) I had a wicked time of these limit problems on a quiz, I can't find any examples of this with the $x$ in the denominator. How do you establish the connection between $|x – 1|$ and $|f(x) – 1|$?
Find the largest $\delta$ such that if $0 < |x – 1| < \delta $, then $|... | Start by writing down what $|f(x) - 1| < 0.1$ means and then solve for a range of $x$.
$|f(x)-1| < 0.1$
$-0.1 < f(x)-1 < 0.1$
$0.9 < f(x) < 1.1$
$0.9 < 2-\dfrac{1}{x} < 1.1$
Can you find a range for $x$ that makes this true? That will help you determine what $\delta$ needs to be.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/897860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluation of a sum of $(-1)^{k} {n \choose k} {2n-2k \choose n+1}$ I have some question about the paper of which name is Spanning trees: Let me count the ways. The question concerns about $\sum_{k=0}^{\lfloor\frac{n-1}{2} \rfloor} (-1)^{k}
{n \choose k} {2n-2k \choose n+1}$. Could you recommend me how to prove $\dis... | Since $\binom{m-2k}{n+1}$ is a degree $n+1$ polynomial in $k$ with lead term $\frac{(-2k)^{n+1}}{(n+1)!}$, we get that the multiple forward difference $\Delta_k^{n+1}\binom{m-2k}{n+1}=(-2)^{n+1}$. Multiply both sides by $(-1)^{n+1}$ to get
$$
\begin{align}
2^{n+1}
&=\sum_{k=0}^{n+1}(-1)^k\binom{n+1}{k}\binom{m-2k}{n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/897948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Finding the asymptotes of a general hyperbola I'm looking to find the asymptotes of a general hyperbola in $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ form, assuming I know the center of the hyperbola $(h, k)$. I came up with a solution, but it's too long for me to be confident that I didn't make a mistake somewhere, so I wa... | For a conic, $$ax^2+2hxy+by^2+2gx+2fy+\color{blue}{c}=0$$
which is a hyperbola when $ab-h^2<0$.
Its asymptotes can be found by replacing $\color{blue}{c}$ by $\color{red}{c'}$ where
$$\det
\begin{pmatrix}
a & h & g \\
h & b & f \\
g & f & \color{red}{c'}
\end{pmatrix}
=0$$
That is
$$\color{red}{c'}=\frac{af^2+bg^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Determining if a recursively defined sequence converges and finding its limit Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists.
I prove that the sequence is eventually decr... | Setting
$$
y_n=x_{n+1}-x_n \quad \forall n\in \mathbb{N},
$$
we have
$$
y_{n+1}=x_{n+2}-x_{n+1}=\frac{x_{n+1}+2x_n}{3}-x_{n+1}=-\frac23(x_{n+1}-x_n)=-\frac23y_n.
$$
It follows that
$$
y_n=\left(-\frac23\right)^{n-1}y_1=2\left(-\frac23\right)^{n-1} \quad \forall n \in \mathbb{N}.
$$
Finally, for every $n\in \mathbb{N}$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Geometry, finding the possible values of 'a' $P (a,4)$, $Q (2,3)$, $R (3,-1)$ and $S (-2,4)$ are four points.
If $|PQ| = |RS|$, find the possible values of a
I know this is a pretty basic problem but I'm having a lot of trouble with it, here is my answer:
I found $|RS|$ to be $\sqrt{50}$ so |PQ| = sqrt-2a^2 + 1 = sqrt... | There's two things that have gone wrong here.
The first is that you have incorrectly calculated the horizontal distance (I'll call it $d$) between P and Q. It should be
$$\sqrt{50} = \sqrt{d^2+(4-3)^2}$$
$$50 = d^2+1^2$$
$$d^2=49$$
$$d=7$$
The second is, once you've gotten a distance value, you can then use it to find... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
The limit of complex sequence $$\lim\limits_{n \rightarrow \infty} \left(\frac{i}{1+i}\right)^n$$
I think the limit is $0$; is it true that $\forall a,b\in \Bbb C$, if $|a|<|b|$ then $\lim\limits_{n\rightarrow \infty}\left(\frac{a}{b}\right)^n=0$?
I would like to see a proof, if possible. Thank you
| $\frac{i}{1+i} = \frac{i(1-i)}{(1+i)(1-i)} = \frac{i(1-i)}{2}=\dfrac{e^{\frac{\pi}{2}i}e^{-\frac{1}{4}\pi i}}{\sqrt{2}} = \dfrac{e^{\frac{\pi}{4}i}}{\sqrt{2}}$, so $(\frac{i}{1+i})^n = \dfrac{e^{\frac{n\pi}{4}i}}{(\sqrt{2})^n}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/898296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Integrate $\int \left(A x^2+B x+c\right) \, dx$ I am asked to find the solution to the initial value problem:
$$y'=\text{Ax}^2+\text{Bx}+c,$$
where $y(1)=1$,
I get:
$$\frac{A x^3}{3}+\frac{B x^2}{2}+c x+d$$
But the answer to this is:
$$y=\frac{1}{3} A \left(x^3-1\right)+\frac{1}{2} B \left(x^2-1\right)+c (x-1)+1.$$
Co... | $$\frac{A x^3}{3}+\frac{B x^2}{2}+c x+d$$ where $d$ is a cosntant to fix is equivalent to $$\frac{1}{3} A \left(x^3-1\right)+\frac{1}{2} B \left(x^2-1\right)+c (x-1)+d$$ where $d$ is a constant to fix.
And the second one is surely more handy to apply the initial condition.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/898363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Division of point-value representation polynomials In Cormen's "Introduction to algorithms" is exercise:
"Explain what is wrong with the “obvious” approach to polynomial division using
a point-value representation, i.e., dividing the corresponding y values. Discuss
separately the case in which the division comes out ex... | If you are using a point-value representation of polynomials, then if you have something that isn't a polynomial (e.g. because it is a rational function with nonconstant denominator), then it can't possibly be represented correctly.
However, to go beyond the point your book is trying to make, you can do point-value re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
find formula to count of string I have to find pattern of count of series.
Lenght of series is $2n$. It is neccessary to use exaclty double times every number in range $[1...n]$ and all of neighboring numbers are different.
Look at example.
$a_0 = 0$
$a_1 = 0$
$a_2 = 2$ (because of $1212$ and $2121$)
And my idea is:... | For $a_3$, you'll add two $3$'s somewhere to each one of those so they're not next to each other. There are $5$ possible places: the two ends, and three in between two numbers, so there are ${5 \choose 2}$ ways of putting the two $3$'s in so they're not next to each other.
For case $a_n$: There are $2n+1$ possible pl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all solutions to the equation $x^2 + 3y^2 = z^2$ Find all positive integer solutions to the equation $x^2 + 3y^2 = z^2$
So here's what I've done thus far:
I know that if a solution exists, then there's a solution where (x,y,z) = 1, because if there is one where $(x,y,z) = d$, then $\frac{x}{d}, \frac{y}{d}, \frac{... | I don't think y is necessarily even, $13^2 + 3*3^2 = 14^2.$
There seem to be two sets of solutions
$2x = p^2 - 3q^2, y = 2pq, 2z = p^2 + 3q^2$ and $pq$ odd
$x = p^2 - 3q^2, y = pq, z = p^2 + 3q^2$ and $pq$ even
$gcd(p,q) = 1 = gcd(p,3)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/898608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Probability of drawing at least 1 red, 1 blue, 1 green, 1 white, 1 black, and 1 grey when drawing 8 balls from a pool of 30? Given a pool of 30 balls (5 of each color). When drawing 8 balls without replacement, what is the probability of getting at least one of each color?
Related: Probability of drawing at least one r... | The quickest approach is to count ways to get $6$ colors on $8$ balls. There are essentially two cases, $\langle 3,1,1,1,1,1\rangle$ and $\langle 2,2,1,1,1,1\rangle$. There are $\binom{6}{1}\binom{5}{3}\binom{5}{1}^5=187,500$ ways to get the first case. The $\binom{6}{1}$ counts the number of ways of choosing one colo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Upper and lower bounds for the smallest zero of a function The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function:
$$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, }x\ge1;n=1,2,3,\cdots,\tag{1}$$
$$F_m(x)=\sum_{n=1}^{m}f_n(x)\text{, }\tag{2... | While this is just a partial answer, I hope this serves at least as a step in the right direction for proving what you need to.
First, to work with something more concrete, I substituted the expressions for $f_n(x)$ into $F_m(x)$ and then that into $G_m(x)$ in order to get an explicit set of functions:
$$F_m(x) = \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
How to deal with linear recurrence that it's characteristic polynomial has multiple roots? example ,
$$
a_n=6a_{n-1}-9a_{n-2},a_0=0,a_1=1
$$
what is the $a_n$?
In fact, I want to know there are any way to deal with this situation.
| A bare-hands method would be as follows:
$$a_n - 3 a_{n-1} = 3 ( a_{n-1} - 3 a_{n-2} )$$
$$a_n - 3 a_{n-1} = 3^{n-1} ( a_1 - 3 a_0 )$$
$$\sum_{k=1}^n 3^{n-k} ( a_k - 3 a_{k-1} ) = \sum_{k=1}^n 3^{n-k} \left( 3^{k-1} ( a_1 - 3 a_0 ) \right)$$
$$\sum_{k=1}^n 3^{n-k} a_k - 3^{n-(k-1)} a_{k-1} = \sum_{k=1}^n 3^{n-1} ( a_1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
An infinite group $G$ and $\forall x\in G, x^n=e$
Let $G$ be an infinite group and $n\in \mathbb N$. If for any infinite subset $A$ of $G$ there is $a\in A$ such that $$a^n=e,~~~~(e=e_G)$$ then prove that for every element $x\in G$ we have $x^n=e$.
This question was asked me and honestly I could not find any way to b... | Notice that $G$ has no elements of infinite order. Suppose there exists $y \in G$ with $|y| \nmid n$. Replacing $y$ with a power of $y$, we may assume $\text{gcd}(|y|, n) = 1$. If $y^G = \{gyg^{-1} \mid g \in G\}$ is infinite, we are done (since $|y| = |gyg^{-1}|$ for any $g \in G$).
On the other hand, if $|y^G| = |G ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Finding all the possible values of an Integral in the Complex Plane I am studying Complex Analysis by Lars V Ahlfors. I am unable to solve one of his exercises. It is:
Find all possible values of $$\int \frac{dz}{\sqrt{1-z^2}}$$
over a closed curve.
I do not have any clue as to how to solve the above problem, if the de... | The exercise considers regions in the complex plane such that the two points $1$ and $-1$ belong to the same component of the complement. On such regions, an analytic branch of $\frac{1}{\sqrt{1-z^2}}$ exists, which is the first part of the exercise, and the second part asks for the possible values of
$$\int_\gamma \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/898985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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