Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Borsuk - Ulam Theorem for $n=2$ Show that Borsuk -Ulam Theorem for $n=2$ is equivalent to the following statement : For any cover $A_1, A_2, $ and $A_3$ of $S^2$ with each $A_i$ closed, there is at least one $A_i$ containing a pair of antipodal points.
For one direction, the function $f:S^2\rightarrow \mathbb R^2$ wher... | For any continuous function $ f: S^2 \to \mathbb R^2 $ consider the function $ g (x) = f (-x) -f (x) $. Note that $ g (x) $ is odd for reflection through the origin.
Now cover $\mathbb R^2$ by three closed sets $ B_i $ such the only pair of points given by reflection through the origin that is contained in each set is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that a string cant be outside a circle How can I prove that a chord can't be outside the circle itself.
Is there a way to prove that you can't draw a chord outside the circle.
| Let $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$ be points on the unit circle, and let $P=(x,y)$ be a point on the chord $P_1P_2$, not equal to $P_1$ or $P_2$. So we can write $P=tP_1 + (1-t)P_2$ for some $t \in (0,1)$.
So we have $x = tx_1 + (1-t)x_2$ and $y = ty_1 + (1-t)y_2$, giving
$$\begin{align}
x^2+y^2 & = t^2(x_1^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to multiply and reduce ideals in quadratic number ring. I am studying quadratic number rings and I have a problem with multiplying and reducing ideals, for example:
Let $w=\sqrt{-14}$. Let $a=(5+w,2+w)$, $b=(4+w,2-w)$ be ideals in $\mathbb Z[w]$. Now, allegedly, the product of ideals $a$ and $b$ in $\mathbb Z[w]$ ... | You reduce an ideal by adding a multiple of one generator to another with the goal of simplifying things (making the numbers smaller and removing appearances of $w$)
First, reduce $a$ and $b$ :
$a = (5+w,2+w) = (3,2+w) \\ b = (4+w,2-w) = (6,2-w)$
Then, multiply every generator of $a$ with veery generator of $b$, and re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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How to proof the following function is always constant which satisfies $f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $? Suppose that $f(x)$ is a bounded continuous function on $\mathbb{R}$,and that there exists a positive number $a$ such that
$$f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} ... | Below, we will see that the function $f$ is actually the restriction of an entire function to $\Bbb{R}$, i.e. the sume of a convergent power-series with infinite radius of convergence. Once this is shown, the (sadly rather downvoted) answer of @Leucippus actually becomes a valid argument.
For this, let $K := \Vert f \V... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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"answer_id": 2
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Need to prove that the equation has only 1 solution. I have been trying to solve the following equation:
$5^x+7^x=12^x.$
Obviously, x=1 is a solution but how do I prove that there are no other solutions.
| we'll show it has no solutions for $x > 1$, hope you can use the idea to deal with the other case. your observation (that $1$ is a solution) will be crucial. write $x = 1 + y$ with $y > 0$, then
$$ 5^x + 7^x = 5 \cdot 5^y + 7 \cdot 7^y < (5+7)\cdot 7^y < 12 \cdot 12^y = 12^x$$
inequalities are strict because $y > 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/951088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Convergence of $\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$ Does the series
$$\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$$
converges?
My attempt: Since the ratio test is inconclusive, my idea is to use the Stirling Approximation for n!
$$\frac{(2n)!}{n!n!4^n} \sim (\frac{1}{4^n} \frac{\sqrt{4\pi n}(\frac{2n}{e})^{2n}}{\sqrt{2 n \p... | A much simpler way:
$$
\frac{a_{n+1}}{a_n}=\frac{(2n+2)(2n+1)}{4(n+1)(n+1)}=\frac{2n+1}{2n+2}\ge
\sqrt{\frac{n}{n+1}},
$$
since
$$
\left(\frac{2n+1}{2n+2}\right)^2=\left(1-\frac{1}{2n+2}\right)^2\ge 1-\frac{2}{2n+2}
=\frac{n}{n+1},
$$
and hence
$$
a_n=a_1\prod_{k=2}^n\frac{a_{k}}{a_{k-1}}\ge a_1\prod_{k=2}^n\sqrt{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/951171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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help with strange Double Integral: $\iint_E {x\sin(y) \over y}\ \rm{dx\ dy}$ i'm having trouble with this double integral:
$$
\iint_E {x\sin(y) \over y}\ \rm{dx\ dy},\ \ \ \
E=\Big\{(x,y) \in \mathbb{R^2} \mid 0<y\le x\ \ \ \land\ \ \ x^2+y^2 \le \pi y\Big\}
$$
i've tried using polar coordinates, but after i made th... | $$
\iint_E {x\sin(y) \over y}\ \rm{dx\ dy},\ \ \ \
E=\Big\{(x,y) \in \mathbb{R^2} \mid 0<y\le x\ \ \ \land\ \ \ x^2+y^2 \le \pi y\Big\}\\
\\
\begin{cases}
x^2+y^2\le\pi y \iff x^2\le\pi y -y^2 \iff \vert x\vert \le \sqrt{\pi y -y^2}\\
0<y\le x
\end{cases} \implies \\
\implies
0<y\le x \le \sqrt{\pi y -y^2} \implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/951256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given 2 colors, how to calculate the mix amount between them? As an input I have two colors, let's say red (RGB = 1,0,0) and magenta (RGB = 1,0,1).
Now I have an image which includes additive mixes between these two colors, for example purple (RGB = 0.5,0,1).
I want to calculate the mix amount between these two colors ... | Things became much easier to me when I thought of the RGB components as XYZ coordinates of a 3-dimensional point, and that the lengths between those is the key to success!
To get the mix-amount between two colors, you require some basic vector maths:
*
*We have the background color A and foreground color B. C is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/951395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show independence of number of heads and tails I am independently studying Larry Wasserman's "All of Statistics"
Chapter 2 exercise 11 is this:
Let $N \sim \mathrm{Poisson}(\lambda)$ and suppose we toss a coin $N$ times. Let $X$ and $Y$ be the number of heads and tails. Show that $X$ and $Y$ are independent.
First, I ... | The problem can also be solved without the use of expectations, which at this point of the book (chapter 2 [1]) are yet to be introduced (they only appear in chapter 3 [1]).
As you correctly identified, we are going to be seeking $f_{X}(x)$, $f_{Y}(y)$ and $f_{X, Y}(x, y)$, to show that $$f_{X}(x)f_{Y}(y) = f_{X, Y}(x,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing that the $\text{ord}(a+b) = \text{min}(\text{ord}(a),\text{ord}(b))$ in a DVR This is Problem 2.29 from Fulton's Algebraic Curves. First a bit of background because I don't know how standard his terminology is.
For a discrete valuation ring $R$ with maximal ideal $\mathfrak{m} = \langle t \rangle$, any element... | $u + v t^{m-n}\notin\mathfrak m$, so it is invertible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/951619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $xH=yH$, then $xy^{-1} \in H$. Use a counterexample to disprove the following statements:
*
*If $xH=yH$, then $xy^{-1} \in H$.
*If $xy^{-1} \in H$, then $xH=yH$.
I was thinking for the first statement:
$$xH=yH \rightarrow y^{-1}x \in H$$
But we do know if H is commutative. Is this correct?
Would it be the sam... | I don't know any group theory. Here's how I figured out a counterexample to your first statement, just knowing the basic definitions, and following my nose.
First, I figured out that $xH=yH$ means $y^{-1}x\in H$. So I'm looking for an example where $y^{-1}x\in H$ while $xy^{-1}\notin H$. So $x$ and $y$ should not commu... | {
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"url": "https://math.stackexchange.com/questions/951672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Predicate logic proof problem Where the domain of the variables are Real Numbers, determine the truth value for the following:
$$ \forall x \exists y(y^2-x<200) $$
I don't understand how to formally prove this problem. Since $y^2\geq 0$ it would stand to reason that any $x< 0$ could disprove this statement. I tried to ... | One way to understand the statement $$ \forall x \exists y(y^2-x<200) $$ is to think of it as a game. One player takes the $\forall$ quantifiers and one takes the $\exists$ quantifiers. The $\forall$ player is trying to make the statement at the end (the $y^2-x<200$) false, and the $\exists$ player is trying to make ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Throw X dice, sum face $1,2,3,4$ minus Y, then add $ 5,6$ formula I am trying to program something to find out the probability of all sum possible when I throw $X$ dice, sum all the face showing $1,2,3,4$ then I minus $Y$ from that sum (with a minimum of $0$, no negative), then I add the sum of face showing $5,6$.
Exam... | The simpler formula is a generating function as I showed to you yesterday.
A generating function is a polynomial that can represent many things but essentially the coefficients of the polynomial represent the frequency of the value of it exponent.
By example: to represent a simple dice that go from 1 to 6 the generatin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/951902",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Given $a+b+c=4$ find $\max(ab+ac+bc)$ $a+b+c = 4$.
What is the maximum value of $ab+ac+bc$?
Could this be solved by a simple application of Jensen's inequality? If so, I am unsure what to choose for $f(x)$. If $ab+ac+bc$ is treated as a function of $a$ there seems no easy way to express $bc$ in terms of $a$.
EDIT: The ... | Jensen's Inequality gives
$$
\left[\frac13(a+b+c)\right]^2\le\frac13(a^2+b^2+c^2)
$$
and we know that
$$
\begin{align}
ab+bc+ca
&=\frac12\left[(a+b+c)^2-(a^2+b^2+c^2)\right]\\
&\le\frac12\left[(a+b+c)^2-\frac13(a+b+c)^2\right]\\
&=\frac13(a+b+c)^2
\end{align}
$$
and equality can be achieved when $a=b=c$. Therefore, if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/951997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Differential equation multivariable solution I don't understand how I would solve the following problem:
Where does the $F(t,y) = -5$ come from? I tried solving it normally, do I create a multivariable function that satisfies $F(t,y) = -5$? Guidance would be appreciated, thanks.
| This DE is separable. Rewrite as
$$(9y^2-30e^{6y})\,dy=\frac{25t^4}{1+t^{10}}\,dt$$
and integrate.
On the left, we get $3y^3-5e^{6y}$. On the right, make the substitution $t^5=u$.
After a short while, we arrive at $5\arctan(t^5)+C$.
Use the initial condition to find $C$. Since $y=0$ when $t=0$, we get $C=-5$. At this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/952104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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eigenvalues of the sum of a diagonal matrix and a skew-symmetric matrix Suppose $A$ is a skew-symmetric matrix (i.e., $A+A^{\top}=0$) and $D$ is a diagonal matrix. Under what conditions, $A+D$ is a Hurwitz stable matrix?
| For any $x\in\mathbb{R}^n$, $x^T(A+D)x=x^TDx$. If $x^T(A+D)x<0$ for all nonzero $x$, all eigenvalues of $A+D$ have negative real parts. Consequently, if the diagonal of $D$ is negative, $A+D$ is Hurwitz stable.
To see that for a real matrix $B$, $x^TBx<0$ for all nonzero $x$ implies the negativity of the real part of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/952233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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High-School level question concerning circle and arcs This question somehow is unsolvable to me. Any idead/hints wil be much appreciated.
$AB$ is a chord which is cut ny the chords $CD$ and $EC$ in the circle.
Givens:
$\frown{AC} +\frown{BE}=\frown{AD}+\frown{BC}$
$S_{CFG}=S_{CGH}$
I need to show:
$AB \perp HG$
I rea... | Since $$\text{arc}AC+\text{arc}BE=\text{arc}AD+\text{arc}BC,$$
we have
$$\angle{CBA}+\angle{BCE}=\angle{ACD}+\angle{BAC},$$
i.e.
$$\angle{CBG}+\angle{BCG}=\angle{ACF}+\angle{FAC}.$$
Considering $\triangle CBG$ and $\triangle CAF$, we can say
$$\angle{CGF}=\angle{CFG}\Rightarrow CF=CG.$$
Then, as you wrote, since $CF=CH... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/952324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show the minimum value for v Really struggling with this question...
If $$v=\frac{(L\cdot V1-V1\cdot x+V2\cdot x)\cdot R}{2Lrx-2rx^2+LR}$$
Prove that the minimum values ($x>0$) for $v$ will occur at:
$$x=\frac{L}{V2-V1}\cdot [-V1 \pm \sqrt{V1\cdot V2-\frac{R}{2rL} \cdot (V1-V2)^2}]$$
How do I do this?
I have tried diff... | Suppose that $L,r,R,V_1,V_2$ are constants.
If we set
$$a=-2r,b=2Lr,c=LR,d=R(V_2-V_2),e=RLV_1,$$
we have
$$v=\frac{dx+e}{ax^2+bx+c}\Rightarrow \frac{dv}{dx}=\frac{-dax^2-2eax+dc-eb}{(ax^2+bx+c)^2}.$$
Hence, we have
$$-dax^2-2eax+dc-eb=0$$$$\iff -R(V_2-V_1)(-2r)x^2-2RLV_1(-2r)x+R(V_2-V_1)LR-RLV_1\times 2Lr=0$$
$$\iff (V... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/952414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A cute limit $\lim_{m\to\infty}\left(\left(\sum_{n=1}^{m}\frac{1}{n}\sum_{k=1}^{n-1}\frac{(-1)^k}{k}\right)+\log(2)H_m\right)$ I'm sure that for many of you this is a limit pretty easy to compute, but my concern here
is a bit different, and I'd like to know if I can nicely compute it without using special functions. Do... | $$
\begin{align}
\lim_{m\to\infty}\left(\sum_{n=1}^m\sum_{k=1}^{n-1}\frac{(-1)^k}{nk}+H_m\log(2)\right)
&=\lim_{m\to\infty}\sum_{n=1}^m\sum_{k=n}^\infty\frac{(-1)^{k-1}}{nk}\\
&=\sum_{n=1}^\infty\sum_{k=n}^\infty\frac{(-1)^{k-1}}{nk}\\
&=\sum_{k=1}^\infty\sum_{n=1}^k\frac{(-1)^{k-1}}{nk}\\
&=\sum_{k=1}^\infty(-1)^{k-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/952535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Show $\frac{1}{|x-y|^{p}}-\frac{1}{|x|^{p}}\in L^{2}(\mathbb{R})$ for fixed real y This is homework so no answers please
Show $\frac{1}{|x-y|^{p}}-\frac{1}{|x|^{p}}\in L^{2}(\mathbb{R})$ for fixed real y
where $|p|< \frac{1}{2}$.
Any strategies?
Attempt:
I will type as I go, I am just curious to see strategies suggest... | The result is false for $p=1/2$, since $1/|x|$ is not integrable at $0$.
If $p<1/2$, you only have to worry about integrability at $\infty$. Use the Mean Value Theorem. You will find the condition $p>-1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/952622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Olympiad problem: Erdos-Selfridge The following problem is a special case of Erdos-Selfridge theorem: http://projecteuclid.org/euclid.ijm/1256050816
Problem: Prove that for any positive integer $n$, the product $(n+1)(n+2)...(n+10)$ is not a perfect square.
It appeared as an olympiad problem, so there is a solution usi... | the product is equal 2^5 x (odd number) . 5 is not even, thus the product cannot be a square.
Best regards
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/952800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Showing that $-\ln{X} \sim \exp{\alpha}$ for $X \sim Beta(\alpha, 1)$ The CDF for $X \sim Beta(\alpha,1)$ is given by:
$$F(x) = \frac{\int_{0}^{x}t^{\alpha-1}dt}{\int_{0}^{1} t^{\alpha-1}dt}$$
I am given to understand that $-\ln{X} \sim \exp(\alpha)$ if $\alpha > 0$.
How do I go about showing this? In all my attempts, ... | Let $g(x) = -\log x$, hence $Y = g(X) = -\log X$, so that $X = g^{-1}(Y) = e^{-Y}$. Then $$F_Y(y) = \Pr[Y \le y] = \Pr[-\log X \le y] = \Pr[X \ge e^{-y}] = 1 - F_X(e^{-y}).$$ Then differentiation of the CDF gives $$f_Y(y) = f_X(e^{-y})e^{-y}$$ by the chain rule. Now if $X \sim \mathrm{Beta}(\alpha,1)$, then $f_X(x) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/952888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving an irrational number in the cantor set I'm trying to prove that $0.2020020002\ldots_3 \in \Bbb Q^c\cap C$ where $C$ denotes the Cantor set. I'm trying to get a contradiction assuming $0.2020020002\ldots_3 \in \Bbb Q$ (without using the fact that every rational number is periodic or terminating).
Suppose $$\sum_... | Suppose $$ \sum_{k=1}^{\infty}\frac{2}{3^{\frac{k(k+1)}{2}}} \text{ is rational.}$$ Then there exists $ p,q\in \mathbb{Z}^{+} $ such that $\gcd(p,q)=1$ and
$$ \sum_{k=1}^{\infty}\frac{2}{3^{\frac{k(k+1)}{2}}}=\frac{p}{q} .$$
Then you need to choose $ n\in \mathbb{N} $ such that $ n=\frac{k_{0}(k_{0}+1)}{2} $ for some ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why do polynomial terms with an even exponent bounce off the x-axis? Like if it's $f(x) = (x-5)^2(x+6)$
Why, at $x=5$, does the graph reflect off the x-axis?
| Well... at $x=5$, it is clear that
$$f(5)=(5-5)^2(5+6)=0^2*11=0$$
So from here, you need to observe that if you pick some number greater than $5$ for $x$ (we can write this as $x=5+x_0$, where $x_0 >0$) then we get
$$f(5+x_0) = ((5+x_0)-5)^2((5+x_0)+6)=(x_0)^2(11+x_0)>0$$
The $>$ above is true because $(x_0)^2$ is al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/953128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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ker$(A^n) =$ ker$(A^m), \forall m > n$
If $A$ is a square matrix and ker$(A^n) =$ ker$(A^{n+1})$, then
ker$(A^n) =$ ker$(A^m), \forall m > n$.
I'm trying to prove that this is correct, but I'm having trouble figuring out what the relation between the ker$(A^n)$ and ker$(A^{n+1})$ is precisely. I know "ker" represen... | You can think of the problem in the following way:
If we know that $A^{n+1}x = A^n \cdot Ax = 0 \implies A^n x = 0$ (for all $x$), show that
$A^m x = 0 \implies A^n x = 0$ for all $m>n$ (for all $x$).
In fact, it's enough to show the following:
If we know that $A^{n+1}x = A^n \cdot Ax = 0 \implies A^n x = 0$ (for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/953228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that there exists a real function $f$ such that $x\to a, |f(x)|\to |L|$ but the limit of $f(x)$ does not exist. Show that there exists a real function $f$ such that $x\to a, |f(x)|\to |L|$ but the limit of $f(x)$ does not exist.
Just before this problem I was asked to prove the theorem:
If $L=\lim_{x\to a}f(x)$ ex... | The classic Sign Function is also Okay:
$$f(x) =\left\{\begin{matrix}
1 & x>0 \\
0 & x =0 \\
-1 & x<0
\end{matrix}\right..$$
$\lim_{x \to 0} f(x)$ does not exist, but $\lim_{x \to 0} | f(x) | = 1.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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If the dot product between two vectors is $0$, are the two linearly independent? If we have vectors $V$ and $W$ in $\mathbb{R^n}$ and their dot product is $0$, are the two vectors linearly independent?
I can expand $V_1 \cdot V_2 = 0 \Rightarrow v_1w_1+...+v_nw_n = 0$, but I don't understand how this relates to linear ... | More generally, Yes any orthogonal set of vectors are linearly independent.
So if we consider $R^n$ together with the inner product being the dot product then if the set of vectors have dot product of zero then they are linearly indepedent:
Consider the set A = {$x_1$,$x_2$,...,$x_n$}
Now we need to prove that the se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/953512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Calc III - Parameterization Given x(t) = (2t,t^2,t^3/3), I am asked to "find equations for the osculating planes at time t = 0 and t = 1, and find a parameterization of the line formed by the intersection of these planes."
I have already solved the vector-valued functions for x.
I know this is vague and I'm asking for a... | First find the unit tangent vector
$${\bf T} = \frac{{\bf x'}}{\|{\bf x'}\|}$$
Then the unit normal (or just normal for ease):
$${\bf N} = \frac{{\bf T'}}{\|{\bf T'}\|}$$
And you may need the binormal which is just
$${\bf B} = {\bf T} \times {\bf N}
$$
Then your point is ${\bf x}(0)$ and your vector is
${\bf B}(0)$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/953591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Graph Theory inequality I've been trying to prove the following inequality
If G is r-regular graph and $\kappa (G)=1 $, then $\lambda (G)\leq \left \lfloor \frac{r}{2} \right \rfloor$
I've tried manipulating the Whitney inequality, but it doesn't seem to help.
| Let $G=(V,E)$ be the $r$-regular graph. Since $\kappa(G)=1$, there exist three vertices $u$, $v$ and $x$ such that, after removing $x$ from $G$, $u$ and $v$ are not connected. Call
$G'$ the graph obtained by removing $x$ from $G$, and define
$$ G_u=\{ y \in V \;|\; y \mbox{ is connected to } u \mbox{ in } G' \}$$
$$ G_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Automorphism groups of vertex transitive graphs Does there exist a finite nonoriented graph whose automorphism group is transitive but not generously transitive (that is, it is not true that each pair $(x,y)$ of vertices can be interchanged by some automorphism)?
I know, for example, that Cayley graphs are always gener... | Cayley graphs for abelian groups have generously transitive automorphism groups. In general a Cayley graph for a non-abelian group will not be generously transitive.
In particular if $G$ is not abelian and $X$ is a so-called graphical regular representation (abbreviated GRR) for $G$, then its automorphism group is not... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/953856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Integration without substitution How to i integrate this with out substitutions or Partial fraction decomposition ?
($3x^2$+$2$)/[$x^6$($x^2$+1)]
I've got to : 2/x^6(x^2+1),but after this i haven't been able to eliminate the 2.
| You can use $\displaystyle\frac{3x^2+2}{x^6(x^2+1)}=\frac{2(x^2+1)}{x^6(x^2+1)}+\frac{x^2}{x^6(x^2+1)}=\frac{2}{x^6}+\frac{1}{x^4(x^2+1)}$
$\displaystyle=\frac{2}{x^6}+\left(\frac{x^2+1}{x^4(x^2+1)}-\frac{x^2}{x^4(x^2+1)}\right)=\frac{2}{x^6}+\frac{1}{x^4}-\frac{1}{x^2(x^2+1)}$
$\displaystyle=\frac{2}{x^6}+\frac{1}{x^4... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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In the theorem is it necessary for ring $R$ to be commutative? According to the statement of theorem that a commutative ring $R$ with prime characteristic $p$ satisfies
$$\begin{align}
(a+b)^{p^n} = a^{p^n} + b^{p^n}
\end{align}$$
$$\begin{align}
(a-b)^{p^n} = a^{p^n} - b^{p^n}
\end{align}$$
Where a,b $\in$ $R$, n $\... | The commutative property is key in proving (for example, by induction) the Binomial Formula for commutative rings $A$ with $1$:
$$ (a+b)^n = \sum_{k=0}^n {n \choose k} a^k b^{n-k} $$
for all $a,b\in A$.
In prime characteristic $p$, we can then use $p | {p^n \choose k}$, $k<p^n$ and $k>0$.
| {
"language": "en",
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"source": "stackexchange",
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Proof of the product rule for the divergence How can I prove that
$\nabla \cdot (fv) = \nabla f \cdot v + f\nabla \cdot v,$
where $v$ is a vector field and $f$ a scalar valued function?
Many thanks for the help!
| $$
\begin{align}
\nabla \cdot (fv) &= \sum \partial_i (f v_i) \\
&= \sum (f \partial_i v_i + v_i \partial_i f) \\
&= f \sum \partial_i v_i + \sum v_i \partial_i f \\
&= f \nabla \cdot v + v \cdot \nabla f
\end{align}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 1
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What does the notation for a group ${A_n}^*$ mean? I was on a website that catalouges lattices. I was looking through the alternating group lattices and the dihedral lattices and there were two kinds for each. For example, there was $A_2$ and ${A_2}^*$ as well as $D_3$ and ${D_3}^*$.
I can't seem to find any info on wh... | '*' indicates dual or weighted lattices. See this pdf. But, to fulfill this answer, I may assure you that $A_n\;^*$ means the lattice of vectors having integral inner
products with all vectors in $A_n:A_n\;^*=\{x\in R^n |(x,r)\in Z \;for\;all\;r\in A_n\}$. And, that,$A_n\;^*\;^*=A_n$.
| {
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$M^{-1}$ has at most $n^2-2n$ coefficients equal to zero Let $M\in GL_n(\mathbb{R})$ such that all its coefficients are non zero.
How can one show that $M^{-1}$ has at most $n^2-2n$ coefficients equal to zero ?
I have no idea how to tackle that problem, I've tried drawing some contradiction if $M^{-1}$ had $n^2+1-2n$ z... | Equivalently you need to prove that $M^{-1}$ has at least $2n$ nonzero entries. Let's in fact prove that every row of $M^{-1}$ had at least two nonzero entries. Suppose that the $i$th row of $M^{-1}$ has less than two nonzero entries; of course it cannot have a zero row (otherwise it would be singular), so it has exact... | {
"language": "en",
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Recurrence Relation from Old Exam I see this challenging recurrence relation that has a solution of $T^2(n)=\theta (n^2)$. anyone could solve it for me? how get it?
$$T(n) = \begin{cases} n,\quad &\text{ if n=1 or n=0 }\\
\sqrt{1/2[T^2(n-1)+T^2(n-2)]+}n ,\quad &\text{ if n>1} \end{cases}$$
| Hint: Let $a(n)=T^2(n)$. Then the recurrence relation becomes
$$a(n)={1\over2}a(n-1)+{1\over2}a(n-2)+n$$
with $a(0)=0$ and $a(n)=1$. Now try finding $c$ and $C$ for which there's an inductive proof that $cn^2\lt a(n)\lt Cn^2$ for sufficiently large $n$.
Added later (at OP's request): I'll do one inequality, which... | {
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"source": "stackexchange",
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2nd order odes with non constant coeff. I am trying to solve these two DE: $ y''+(2x)/(1+x^2)y'+1/(1+x^2)^2y=0 $ and $ xy''-y'-4x^3y=0 $ and I am looking for methods on how to find the solutions. Should I go with the series method or is there a simpler way?
| Hint:
(2) You have that $y_1 = \text {sinh}\ x^2 $ is a solution of your DE. Write $$\begin{align}y_2 &= v\ \text{sinh}\ x^2 \\ y'_2 &= v'\ \text{sinh}\ x^2 + \ v\ (2x\ \text{cosh}\ x^2) \\ y''_2 & = v''\text{sinh}\ x^2 + 2v' (2x\ \text{cosh}\ x^2) + v(2\ \text{cosh}\ x^2+ 4x^2\ \text{sinh}\ x^2) \ \end{align}$$
P... | {
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"question_score": "3",
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volume of unit ball in high dimension with maximum norm If $z=(x,y)$, $x\in X, y\in Y$, space $X, Y$ could have any distance $d_x, d_y$, and space $z\in Z$ has maximum distance: $$d(z,z')=\max\{d_x(x,x'),d_y(y,y')\}.$$
If the volume of unit ball in $X, Y$ are $c_x, c_y$, the volume of unit ball in $Z$ is $$c_z=c_xc_y.$... | The reason is that the unit ball in $Z$ is the Cartesian product of the unit balls of $X$ and $Y$. Indeed, if $z=(x,y)$, then
$$
d(z,0)\le 1 \iff d(x,0)\le 1 \text{ and } d(y,0)\le 1
$$
And the volume (I assume Lebesgue measure) of the product of two sets is the product of their volumes; this comes directly from the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/954665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Lang Fiber Products STATEMENT: Let $\mathcal{C}$ be a category.A product in $\mathcal{C}_z$ is called the fiber product of $f$ and $g$ in $\mathcal{C}$ and is denoted by $X\times_zY$, together with its natural morphisms on $X,Y$, over $Z$, which are sometimes not denoted by anything.
QUESTION: What is $X\times_zY$. Is ... | If $Z$ is a final object then the fibered product coincides with the usual product, however this is not true in general.
Note that the definition of the fibered product also includes ''projections'' $\pi_X: X\times_Z Y\to X$ and $\pi_X: X\times_Z Y\to Y$, as well as two morphisms $\alpha:X\to Z$ and $\beta:Y\to Z$ s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is $f(x)^{-1}$ used to denote the inverse of a function, and not its reciprocal? Function notation says that any operations applied to a variable inside the parenthesis are applied to the variable before it enters the function, and anything applied to the function as a whole is applied to the entire function or to ... | See, Inverse of $a$ i.e. $a^{-1}$ (where $a$ is any element of some group) means that $a*a^{-1}=e$ where $e$ is the identity w.r.t. binary operation $*$. I guess you don't understand Group theory and all i.e. abstract algebra. But only thing to notice here is that in Functions binary operation is not multiplication but... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Exact Differential Equation? I tried to solve this equation so far, since the partial derivative respect to $x$ and $y$ are not exact, I have to find the $u(x)$ to make them exact
$e^x \cos(y)dx+\sin(y)dy=0$
Partial derivative of $y = e^x\sin(y)$
partial derivative of $x = -\sin(y)$
Now i have to find the $u(x)$ to... | You don't need to make this equation exact, it's seperable. First subtract $sin(y)dy$ to get
$e^x\cos ydx=-\sin y dy$
Then divide both sides by $\cos y$ to get
$e^xdx=-tanydy$.
Proceed to integrate both sides.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Use the law of logarithms to expand an expression $$\log\sqrt [ 3 ]{ \frac { x+2 }{ x^{ 4 }(x^{ 2 }+4) } } $$
How is this answer incorrect?
$$\frac { 1 }{ 3 } [\log(x+2)-(4\log x+\log(x^ 2+4))]$$
| with this simplify you have to know that your domain will be changed and that is not exactly with your initial question.suppose you want to simplify
this you can't input negative integer in this,but if you simplify that:
that you can input negative numbers,so domain changed.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Find the value of $\sqrt{10\sqrt{10\sqrt{10...}}}$ I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it simplifies to $x=10\sqrt{x}$ which has solutions $x=0,100$. I don't unde... | Denote the given problem as $x$, then
\begin{align}
x&=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{\cdots}}}}}\\
&=10\cdot10^{\large\frac{1}{2}}\cdot10^{\large\frac{1}{4}}\cdot10^{\large\frac{1}{8}}\cdot10^{\large\frac{1}{16}}\cdots\\
&=10^{\large1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots}\\
&=10^{\large y}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/955190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
"answer_count": 7,
"answer_id": 4
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Prove that $ \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x $ In a related question the following integral was evaluated
$$
\int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x
=\int_0^{\pi} \frac{\mathrm{d}x/2}{1 +... | $$
\frac{\cos(\pi/2-x)^2}{1+\cos(\pi/2-x)\sin(\pi/2-x)} =
\frac{\sin(x)^2}{1+\sin(x)\cos(x)}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 4
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Spaces where all compact subsets are closed All compact subsets of a Hausdorff space are closed and there are T$_1$ spaces (also T$_1$ sober spaces) with non-closed compact subspaces. So I looking for something in between.
Is there a characterization of the class of spaces where all compact subsets are closed? Or at le... | According my acknowledge, it hasn't a characterization of this class of spaces, I believe that the reason is that this class of spaces is not Hausdorff and many people don't care it since we always study the classes of spaces at least Hausdorff.
The following paragraphes may be useful for you, which is copied from Page... | {
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"timestamp": "2023-03-29T00:00:00",
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Any shorter way to solve trigonometric problem? If $10 \sin^4\theta + 15 \cos^4 \theta=6$, then find value of $27 \csc^2 \theta + 8\sec^2 \theta$
I know the normal method o solve this problem in which we need to multiply L.H.S. of $10 \sin^4\theta + 15 \cos^4 \theta=6$ by $(\sin^2\theta + \cos \theta^2)^2$ and then sim... | How can $20\sin^2\theta + 15\cos^2\theta = 6$? $20\sin^2\theta + 15 \cos^2\theta = 15 + 5\sin^2\theta$, implying that $5 \sin^2\theta = -9$, but $\sin^2\theta$ should be nonnegative.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Mean and Variance of X with possible outcomes I roll a six-sided die until I get a 6. Then I roll it some more until I get an
even number. Let X be the total number of rolls. So here are some possible
outcomes with the resulting value of X:
24 1 2 6 1 5 4 : X = 8
3 6 4 : X = 3
3 4 6 3 1 1 2 : X = 7
1 5 4 6 6 : X = 5
Fi... | Expressing $X$ as a sum of two simple random variables is indeed the best approach.
Let $U$ be the number of tosses until we get a $6$, and $V$ the number of additional tosses until we get an even number. Then $X=U+V$.
Note that $U$ and $V$ each have geometric distribution (with different parameters), and are independ... | {
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Why do we set $u_n=r^n$ to solve recurrence relations? This is something I have never found a convincing answer to; maybe I haven't looked in the right places. When solving a linear difference equation we usually put $u_n=r^n$, solve the resulting polynomial, and then use these to form a complete solution. This is how ... | *
*if $u_n=f(n)$ satisfies a linear recurrence relation (without looking at the starting condition(s)) then $u_n=cf(n)$ also satisfies the recurrence relation.
*if $u_n=f_1(n)$ and $u_n=f_2(n)$ both satisfy a linear recurrence relation (again: without the starting conditions(s)) then $u_n=f_1(n)+f_2(n)$ also satisfie... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that the limit, $\displaystyle\lim_{x \to 1^+} \frac{x-3}{3 - x - 2x^2}$, exists. For each of the following, use definitions (rather than limit theorems) to prove that the limit exists. Identify the limit in each case.
d)$\displaystyle\lim_{x \to 1^+} \frac{x-3}{3 - x - 2x^2}$
Proof: We need to prove the limit ... | To prove that the limit doesn't exist, it suffices to show that for any $M > 0$, there exists a $\delta$ such that for $0 < x - 1 < \delta$, $|f(x)| > M$. (Then, $f$ is unbounded as $x \rightarrow 1^{+}$.) One way to reverse engineer a proof is find an expression in terms of $\delta$ that will bound $|f(x)|$ as $\delta... | {
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If two groups are generated by the same number of generators, and the generators have the same orders, are the groups isomorphic? If, say, for two groups G and H, G = < a1, a2, ... , a_n > and H = < b1, b2, ... , b_n >,
such that |a_i| = |b_i| for all i from 1 to n,
is G isomorphic to H?
If so, what is the proof of ... | No; for n=2, we get two distinct groups, where $1$ is the identity
$$G=<a,b | a^2 = 1, aba^{-1}b = 1, b^2=1>$$
$$H=<a,b | a^2 = 1, b^2 = 1>$$
Notice that $G$ is finite - it is $\mathbb{Z_2\times Z_2}$. However, $H$ can be described as the set of words which alternate $a$ and $b$ (with the appropriate multiplication ope... | {
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Are there other accumulation functions that holds $a(n-t)={a(n) \over a(t)}$? This might be a beginner's question regarding accumulation methods and their functions, but so far I have learned that compound interest satisfy
$$a(n-t)={a(n) \over a(t)}$$
Which allows nice results such as
$$s_{\bar{n}\rceil} = (1+i)^n a_... | You can rearrange your equation by multiplying through to get $a(n-t) a(t) = a(n)$. Let $n = s+t$ and you get $a(s) a(t) = a(s+t)$. So you want to find a function satisfying $a(s) a(t) = a(s+t)$ for all choices of $s$ and $t$.
Now, you can rewrite this to get $a(s) a(s) = a(2s)$, and then $a(s) a(2s) = a(3s)$ so $a(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Sum the series (real analysis) $$\sum_{n=1}^\infty {1 \over n(n+1)(n+2)(n+3)(n+4)}$$
I tried to sum the above term as they way I can solve the term $\sum_{n=1}^\infty {1 \over (n+3)}$ by transforming into ${3\over n(n+3)} ={1\over n}-{1\over(n+3)}$ but I got stuck while trying to transform $12\over n(n+1)(n+2)(n+3)(n+4... | By working it out, by various methods, it is seen that
\begin{align}
\frac{1}{n(n+1)(n+2)(n+3)(n+4)} = \frac{1}{4!} \, \sum_{r=0}^{4} (-1)^{r} \binom{4}{r} \, \frac{1}{n+r}
\end{align}
Now,
\begin{align}
S &= \sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)(n+3)(n+4)} \\
&= \frac{1}{4!} \, \sum_{r=0}^{4} (-r)^{r} \binom{4}{r} ... | {
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"answer_id": 1
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What do limits of functions of the form $te^t$ have to do with l'Hopital's rule? I have an improper function that I have to integrate from some number to infinity. Once integration is done, the function is of the form $te^t$.
What I'm wondering is what does this have to do with l'Hopital's rule?
From reading my book, I... | The invocation of l'Hôpital's rule is
$$ \lim_{t\to-\infty} \frac{t}{e^{-t}} = \lim_{t\to-\infty} \frac{1}{-e^{-t}}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/956315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to evaluate the limit $\lim_{x\to 0} ((1+2x)^{1/3 }-1)/ x$ without using the l'Hospital's rule? $$\lim_{x\to 0} \frac{(1+2x)^{\frac{1}{3}}-1}{x}.$$
Please do not use the l'hospital's rule as I am trying to solve this limit without using that rule... to no avail...
| Hint: Put $2x+1 = y^{3}$. And note that as $x \to 0 \implies y \to 1$. Then the limit becomes
\begin{align*}
\lim_{x \to 0} \frac{(2x+1)^{1/3}-1}{x} &= \lim_{y \to 1} \frac{y-1}{\frac{y^{3}-1}{2}}\\
&= \lim_{y \to 1} \frac{2}{y^{3}-1} \times (y-1)\\
&= \lim_{y \to 1} \frac{2}{y^{2}+y+1} = \frac{2}{3}
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/956371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Strong induction on a summation of recursive functions (Catalan numbers) I've been stuck on how to proceed with this problem. All that's left is to prove this with strong induction:
$$\forall n \in \mathbb{N}, S(n) = \sum_{i=0}^{n-1} S(i)*S(n - 1 - i)$$
Some cases: S(0) = 1, S(1) = 1, S(2) = 2, S(3) = 5, S(4) = 14.
As ... | You neglect to mention that $S(n)$ is the $n$th Catalan number. You're probably trying to prove some formula, say $S(n) = \varphi(n)$. For given $n$, you already know (this is the strong induction hypothesis) that $S(k) = \varphi(k)$ for $k < n$. Therefore
$$ S(n) = \sum_{i=0}^{n-1} S(i) S(n-1-i) = \sum_{i=0}^{n-1} \va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/956446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Conjunctive Normal Form Is a statement of the form $\phi \vee \psi \vee \xi$ considered to be in its conjuntive normal form (CNF), given that $\phi \vee \psi$ is considered to be in CNF?
Example: While converting $\phi \wedge \psi \rightarrow \xi$ to its CNF, we get $\neg(\phi \wedge \psi) \vee \xi$ which gives $(\neg... | Here's a definition of a clause adapted from Merrie Bergmann's An Introduction to Many-Valued and Fuzzy Logic p. 20:
*
*A literal (a letter or negation of a letter) is a clause.
*If P and Q are clauses, then (P $\lor$ Q) is a clause.
Definition of conjunctive normal form.
*
*Every clause is in conjunctive norm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/956522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Task "Inversion" (geometry with many circles) Incircle $\omega$ of triangle $ABC$ with center in point $I$ touches $AB, BC, CA$ in points $C_{1}, A_{1}, B_{1}$. Сircumcircle of triangle $AB_{1}C_{1}$ intersects second time circumcircle of $ABC$ in point $K$. Point $M$ is midpoint of $BC$, $L$ midpoint of $B_{1}C_{1}$.... | HINT for a less tour-de-force approach (impressive though MvG, or his software, may be)
The obvious idea is to invert (as the question hints). The first thing to try is clearly the incircle. Denote the inverse point of $X$ as $X'$. So the line $B_1C_1$ becomes the circle $AB_1C_1$ and hence $L'=A$. Like $A'=L$, $K'$ li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/956600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
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PDF of an unknown probability after a successful test. Let $X$ be a random variable having the possible outcomes $\{0, 1\}$ representing failure or success, with unknown probability. We test $X$ just one time and the outcome is success. Let $X(1) = p \in [0, 1]$, ie, $p$ is the probability that another test of $X$ will... | Okay, I think I have an idea here.
Let P(1) denote the probability of the first outcome being a success. If $p$ is given then $P(1)=p$, but since $p$ is unknown and different values of $p$ are mutually exclusive events with probabilities $f(p)dp$:
$$
P(1) = \int p{f(p) dp} = \int {pd(F(p))}\\ = pF(p) - \int{F(p)dp}
$$
... | {
"language": "en",
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"source": "stackexchange",
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How did we find the solution? In my lecture notes, I read that "We know that $$x^2 \equiv 2 \pmod {7^3}$$ has as solution $$x \equiv 108 \pmod {7^3}$$"
How did we find this solution?
Any help would be appreciated!
| Check this proof of Hensel's Lemma, which is basically the justification/proof of why what André did works, to go over the following:
$$x^2=2\pmod 7\iff f(x):=x^2-2=0\pmod 7\iff x=\pm 3\pmod 7$$
Choose now one of the roots, say $\;r=3\;$ , and let us "lift" it to a solution $\;\pmod{7^2}\;$ (check this is possible sinc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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How to compute the induced matrix norm $\| \cdot \|_{2,\infty}$ The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^n, \| \cdot \|_q)$ is given by
$$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p}.$$
I would like to compute this norm for $p,q\... | You have, for $x$ with $\|x\|_2=1$ and using Cauchy-Schwarz,
$$
\|Ax\|_\infty=\max_k\,\left|\sum_{j=1}^nA_{kj}x_j\right|\leq\max_k\left(\sum_{j=1}^n|A_{kj}|^2\right)^{1/2}=\max_k \|R_k(A)\|_2,
$$
where $R_k(A)$ is the $k^{\rm th}$ row of $A$. Now suppose that we fix $k_0$ such that $\|R_{k_0}(A)\|_2$ is maximum among t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Cantor's Theorem (surjection vs bijection) Let me state Cantor's Theorem first: Given any set $A$, these does not exist a function $f:A \rightarrow P(A)$ that is surjective.
I understand the proof of this theorem, but I'm wondering why it's enough to show that the function $f$ is not surjective, but rather not bijectiv... | A surjective map from $A \rightarrow B$ maps each element of $A$ to one element of $B$, such that every element of $B$ is mapped to by at least one element of $A$.
So the Cantor proof does show that there is no surjective mapping, because surjective does not allow one element of $A$ to be mapped to multiple elements o... | {
"language": "en",
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Finding the integer solutions an equation $
3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14
$ .
I already solved this using the Cauchy–Schwarz inequality and got $x=4$ and $y=5$. But I'm sure there is a prettier, simpler solution to this and I was wondering if anyone could suggest one.
| Using the Cauchy-Schwarz inequality is a very clever move: Letting
$${\bf a}:=(3,2,1), \quad{\bf b}:=\bigl(
\sqrt{x+y},\>\sqrt{8-x},\>\sqrt{6-y}\bigr)\tag{1}$$
we get
$$3\sqrt{x+y}+2\sqrt{8-x}+\sqrt{6-y}={\bf a}\cdot{\bf b}\leq|{\bf a}|\>|{\bf b}|=\sqrt{14}\>\sqrt{14}\ ,$$
with equality iff ${\bf b}=\lambda{\bf a}$ for... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prime ideals in $k[x,y]/(xy-1)$. Let $k$ a field. Let $f$ be the ring injective homomorphism
$$ f:k[x] \rightarrow k[x,y]/(xy-1)$$
obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ f:k[x,y] \rightarrow k[x,y]/(xy-1)$.
Prove that there isn't any prime ideal in the ring ... | HINT:
$k[x,y]/(xy-1)$ is naturally isomorphic to the ring of fractions $k[x][\frac{1}{x}] = S^{-1}\ k[x]$, where $S= \{1,x,x^2, \ldots\}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integrate cos (lnx) dx Integrating by parts:
I'm having a hard time choosing the $u$, $du$, $v$ and $dv$...
I gave it a shot.
$u = \ln x \implies du = 1/x \ dx$
$v= \ ?$
$dv = \cos \ dx$
| I think you're better off using $u$-substitution here. Setting $\ln x = u$, you get $du = \frac{dx}{x} = \frac{dx}{e^u}$, and so, $dx = e^u\,du$. Then, your integral becomes
$$\int \cos\left( \ln x \right) \, dx = \int e^u\cos u \, du,$$
which you can evaluate by using integration by parts twice.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integrate $(1+7x)^{1/3}$ from $0$ to $1$
Integrate $(1+7x)^{1/3}$ from $0$ to $1$
So using substitution, I'm able to get to $$\frac17 \frac34 u ^{4/3}$$ Pretty sure that part is right, but I'm getting stuck after that.
| It looks like you did the integration correctly. Now you just have to evaluate your answer at the upper and lower limits.
The problem is that the upper and lower limits 1 and 0 are for the variable $x$. You did a variable change so that your integral became in terms of $u$. This means you have to evaluate your answe... | {
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Amateur Math and a Linear Recurrence Relation I haven't received a formal education on this topic but a little googling told me this is what I am trying to find. I would like to put
$$ a_n = 6 a_{n-1} - a_{n-2} $$
$$ a_1 =1, a_2 = 6 $$
into its explicit form. So far I have only confused myself with billions of 6's.
I w... | As user164587 answered, the closed form is $$a_n = A\alpha_1^n + B\alpha_2^n$$ where $\alpha_1$ and $\alpha_2$ are the roots of the characteristic equation $u^2 - 6u + 1 = 0$ that is to say $\alpha_1=3-2 \sqrt{2}$, $\alpha_2=3-2 \sqrt{2}$.
Solving for $A$ and $B$, you would find that $$A=-\frac{{a_2}-{a_1} {\alpha_2}}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sums with squares of binomial coefficients multiplied by a polynomial It has long been known that
\begin{align}
\sum_{n=0}^{m} \binom{m}{n}^{2} = \binom{2m}{m}.
\end{align}
What is being asked here are the closed forms for the binomial series
\begin{align}
S_{1} &= \sum_{n=0}^{m} \left( n^{2} - \frac{m \, n}{2} - \frac... | HINT:
As $\displaystyle n\binom mn=m\binom{m-1}{n-1},$
$\displaystyle\sum_{n=0}^mn\binom mn^2=m\sum_{n=0}^m\binom mn\binom{m-1}{n-1}=m\binom{2m-1}m$
comparing the coefficient of $x^{2m-1}$ in $(1+x)^m(1+x)^{m-1}=(1+x)^{2m-1}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Chain Rule for Second Partial Derivatives I am trying to understand this:
Let $g:\mathbb{R}\rightarrow\mathbb{R}$ and $f(r,s) = g(r^2s)$, where
$r=r(x,y) = x^2 + y^2$ and $s = s(x) = 3/x$.
What is (with Chain Rule) $$ \frac{\partial^2f}{\partial y\partial x}$$
g is a function of one variable so is the first part just:
... | $f$ is a function of two variables ($r$ and $s$), so the chain rule would give
$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial f}{\partial s} \frac{\partial s}{\partial x}$$
To compute for example $\partial f / \partial r$, remember that g is a function of ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Relative distance in Codes I'm studying coding theory. In my lecture we saw that Hadamard codes have an optimal relative distance of $1/2$. The relative distance of a code $C$ with minimum distance $d(C)$ and block length $n$ is defined by:
$$\dfrac{d(C)}{n}.$$
According to this, the minimum distance for a Hadamard cod... | It depends what kind of code you want. In its simplest form, a $q$-ary block code of length $n$ is just a subset of $\mathbf{F}_q^n$. Hence it is trivial to construct a block code of length $n$ with minimal distance $n$: $\{0^n,1^n\}$. This code is also linear, and even cylic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/957792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Prove by contradiction that $(x-y)^3+(y-z)^3+(z-x)^3 = 30$ has no integer solutions By factorizing it I found that
$(x-y)(y-z)(z-x) = 10$
| Just as Karvens comments:
Let $a=x-y$ and $b=y-z$. Then $-(a+b)=z-x$. Clearly, $a,b,c \in \Bbb Z$. So
$$30=a^3+b^3-(a+b)^3=(a+b)(a^2-ab+b^2)-(a+b)^3=-3ab(a+b)$$
And hence
$$10=-ab(a+b).$$
Therefore $a=\pm 1$ or $a=\pm 2$ or $a=\pm 5$ or $a=\pm 10$. Claerly they cannot be the solution.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Choice Problem: choose 5 days in a month, consecutive days are forbidden I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand.
Example 3.19. A medical student has to work in a hospital for five
days in January. However, he is not allowed to work two consecutive... | The prohibition on consecutive days means the constraints are really $$1 \le a_1$$ $$a_1+1 \lt a_2$$ $$a_2+1 \lt a_3$$ $$a_3+1 \lt a_4$$ $$a_4+1 \lt a_5$$ $$a_5 \le 31.$$ Rewrite these so that the right hand side of each line is the same as the left hand side of the next line as $$1 \le a_1$$ $$a_1 \lt a_2-1$$ $$a_2-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/957940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Series Expansion $\frac{1}{1-x}\log\frac{1}{1-x}=\sum\limits_{n=1}^{\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)x^n$ How do I prove that, if $|x|<1$, then $$\frac{1}{1-x}\log\frac{1}{1-x}=\sum_{n=1}^{\infty}\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)x^n$$
| $$\frac1{1-x}\log\frac1{1-x}=\sum_{i\geqslant0}x^i\cdot\sum_{k\geqslant1}\frac{x^k}k=\sum_{i\geqslant0,k\geqslant1}\frac{x^{k+i}}k=\sum_{n\geqslant1}\ldots x^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/958003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$ I need to prove the following:
If a function $\,f$ satisfies
$$f(x+T)=k\;f(x), \forall x \in \mathbb R$$
for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as $f(x)=a^xg(x)$ where $g$ is a periodical function with p... | If you do the reverse statement, then it is suggestive that $a=k^{1/T}$.
So let us set $a=k^{1/T}$ and consider $g(x)\equiv f(x)/a^x$. To check the periodicity of $g$:
$$
g(x+T)=\frac{f(x+T)}{a^{x+T}}=\frac{kf(x)}{a^{x+T}}=\frac{a^Tf(x)}{a^{x+T}}=\frac{f(x)}{a^x}=g(x).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/958210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Distance between powers of 2 and 3 As we know $3^1-2^1 = 1$ and of course $3^2-2^3 = 1$. The question is that whether set $$ \{\ (m,n)\in \mathbb{N}\quad |\quad |3^m-2^n| = 1 \} $$
is finite or infinite.
| Note first that if $3^{2n}-1=2^r$ then $(3^n+1)(3^n-1)=2^r$. The two factors in brackets differ by $2$ so one must be an odd multiple of $2$, and this is only possible if $n=1$ (the only odd number we can allow in the factorisation is $1$)
Now suppose that $3^n-1=2^r$ and $n$ is odd. Now $3^n\equiv -1$ mod $4$ so $3^n-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find the number of squares formed by given lattice points? Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points?
Additionally, if we were to figure out how many, minimally, more points should we ... | I dont have the reputation to comment so I am writing it in answer. This problem seems to be taken from http://www.codechef.com/OCT14/problems/CHEFSQUA live contest! please do remove it. Please respect the code of honor.
And more over you are complaining of no help.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Expected value of a non-negative random variable How do I prove that $\int_0^\infty Pr(Y\geq y) dy = E[Y]$ if $Y$ is a non-negative random variable?
| Assuming we have a continuous random variable with an existant probability density function $f_Y$.
$\begin{align}
\int_0^\infty \Pr(Y \geqslant y) \operatorname d y
& =
\int_0^\infty \int_y^\infty f_Y(z)\operatorname d z\operatorname d y
\\[1ex] & =
\int_0^\infty \int_0^z f_Y(z)\operatorname d y\operatorname d z
\\[1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/958472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Showing equivalence of two binomial expressions I wish to show that $\sum_{k=0}^n {n\choose k}(\alpha + k)^k (\beta + n - k)^{(n-k)} = \sum_{k=0}^n {n\choose k}(\gamma + k)^k (\delta + n - k)^{(n-k)}$ given that $\alpha + \beta = \gamma + \delta$.
Both sides appear to be the n-th coefficient in the product of some expo... | We can prove this using a close relative of the labelled tree
function that is known from combinatorics.
This will provide a closed form of the exponential
generating function of the four terms that are involved.
The species of labelled trees has the specification
$$\mathcal{T} =
\mathcal{Z} \times \mathfr... | {
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"timestamp": "2023-03-29T00:00:00",
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How many nonnegative integer solutions are there to the pair of equations $x_1+x_2+…+x_6=20$ and $x_1+x_2+x_3=7$? How many nonnegative integer solutions are there to the pair of equations
\begin{align}x_1+x_2+\dots +x_6&=20 \\ x_1+x_2+x_3&=7\end{align}
How do you find non-negative integer solutions?
| You are correct. You can also think of it in terms of permutations.
The number of non-negative integer solutions of $x_1+x_2+x_3=7$ is the number of permutations of a multiset with seven $1$'s, and two $+$'s. This is $$\frac{9!}{7!\ 2!}.$$
Similarly, the number of non-negative integer solutions of $x_4+x_5+x_6=13$ is t... | {
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Calculation of $\int\frac1{\tan \frac{x}{2}+1}dx$ Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$
$\bf{My\; Try}::$ Let $\displaystyle I = \displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$, Now let $\displaystyle \tan \frac{x}{2}=t\;,$ Then $\displaystyle dx=\frac{2}{1+t^2}dt$
So $\displaystyle I = 2\... | Hint:
Multiply the denominator and nominator of the fraction by $\cos x/2(\sin x/2-\cos x/2)$. At the bottom you get $-\cos x$, on the top you get $-\cos^2 x/2+1/2\sin x$.
What can you do with $\cos^2 x/2$?
| {
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"answer_id": 1
} |
How to derive the closed form of this recurrence? For the recurrence, $T(n) = 3T(n-1)-2$, where $T(0)= 5$, I found the closed form to be $4\cdot 3^n +1$(with help of Wolfram Alpha). Now I am trying to figure it out for myself. So far, I have worked out: $T(n-1) = 3T(n-2)-2, T(n-2) = 3T(n-3)-2, T(n-3) = 3T(n-4)-2$ leadi... | By making use of the generating function method the difference equation $T_{n} = 3 T_{n-1} - 2$, $T_{0} = 5$, can be seen as follows
\begin{align}
T(x) &= \sum_{n=0}^{\infty} T_{n} x^{n} = 3 \sum_{n=0}^{\infty} T_{n-1} x^{n} - 2 \sum_{n=0}^{\infty} x^{n} \\
T(x) &= 3 \left( T_{-1} + \sum_{n=1}^{\infty} T_{n-1} x^{n} \r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/958781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
If $\,f^{7} $ is holomorphic, then $f$ is also holomorphic. I need some help with this problem:
Let $ \Omega $ be a complex domain, i.e., a connected and open non-empty subset of $ \mathbb{C} $. If $ f: \Omega \to \mathbb{C} $ is a continuous function and $ f^{7} $ is holomorphic on $ \Omega $, then $ f $ is also holo... | This is a topological variant of Christian’s elegant solution.
Let $ g \stackrel{\text{df}}{=} f^{7} $. As $ g $ is holomorphic on $ \Omega $, either (i) it is $ 0 $ everywhere on $ \Omega $ or (ii) its set of roots is an isolated subset of $ \Omega $. If Case (i) occurs, then we are done. Hence, suppose that we are i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/958866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 4
} |
Spectral radius of a normal element in a Banach algebra I know that if $A$ is a C*-algebra, then $||x||=||x||_{sp}$ for every normal element $x\in A$. I like to find an example of a normal element $x$ in a involutive Banach algebra with $||x||\neq ||x||_{sp}$. Thanks for your regard.
| Take $\ell_1(\mathbb{Z})$ with the $*$-operatorion given by $\delta_n^* = \delta_{-n}$. Consider the element $\tfrac{1}{2}\delta_1 + \tfrac{1}{2}\delta_2$. It has norm one but its spectral radius is 1/2.
Instead of $\mathbb{Z}$ you can take your favourite finite, non-trivial group to have a finite-dimensional example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/958947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
I dropped mathematics and want to self-teach it to myself starting from the basics I will make this question as objective based as possible. My curriculum teaches maths in a way that does not work well with me. Further, I also do not like the textbooks used (It was voted the worst in the state). The only way to succeed... | Calculus and Analytic Geometry by George B. Thomas and Ross L. Finney. This is a great book for self learning, it balances clear explanations with enough rigor and will help you gain some intuition from the basics all the way to the multivariate integral theorems.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/959106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
unreduced suspension Is the definition $SX=\frac{(X\times [a,b])}{(X\times\{a\}\cup X\times \{b\})}$ of the unreduced suspension the standard defininition? If I consider $X=$ point, the suspension of $X$ is a circle. But I saw an other definition of the unreduced suspension such that the suspension of a point should b... | I would have said that the suspension was $X \times [-1, 1]$ modulo the relation that
$$
(x, a) \sim (x', a')
$$
if and only iff
*
*$a = a' = 1$ or
*$a = a' = -1$, or
*$a = a'$ and $x = x'$.
Wikipedia seems to agree with me. It looks as if your author was a little glib, and failed to mention that the "botto... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/959183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
$G$-representations, $W \otimes V^* \to \text{Hom}(V,W)$ Let $V$ and $W$ be finite-dimensional vector spaces. I know how to construct an explicit isomorphism of vector spaces $W \otimes V^* \to \text{Hom}(V,W)$ and show that it's an isomorphism. But if I supposed that $V$ and $W$ are $G$-representations of some group $... | Let $\rho_1: G \to GL(V)$ be a $G$-representation and $\rho_2: G \to GL(W)$ as well. We can make $V^*$ a $G$-representation $\rho_1^*: G \to GL(V^*)$ given by$$(\rho_1^*(g)(\phi))(v) = \phi(g^{-1}v).$$Then $W \otimes V^*$ is a $G$-representation $\rho_T: G \to GL(W \otimes V^*)$ by $$\rho_T(g)(w \otimes \phi) = (\rho_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/959301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
stuck on logarithm of derivative of sum $\frac{\partial\mathrm{log}(a+b)}{\partial a}$ I need to evaluate an expression similar to the following:
$\frac{\partial\mathrm{log}(a+b)}{\partial a}$
At this point I don't know how to proceed. $b$ is a constant so there should be some way to eliminate it.
How would you proceed... | Suppose you know $\frac{d}{dx} e^x=e^x$ then define $f(x)=e^x$ to have inverse function $g(x) = \ln (x)$ to be the inverse function of the exponential function. In particular this means we assume $e^{\ln(x)}=x$ for all $x \in (0, \infty)$ and $\ln(e^x) = x$ for all $x \in (-\infty, \infty)$. The existence of $g$ is no ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/959410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing $\cos{\pi \frac{mx}{R}}\sin{ \pi\frac{nx}{R}}$ are orthogonal in $L^{2}([0,R])$? I'm trying to show that $\sin \left(\frac{n\pi}{R}x\right)$ and $\cos \left(\frac{m\pi}{R}x\right)$ are orthogonal using the trigonometric identity that $2\cos{mx}\sin{nx} = \sin((m+n)x) + \sin((m-n)x)$ on $L^{2}([0, R])$.
I could ... | Indeed, this is not true. For instance, taking $R=\pi$ for simplicity, consider $n=1$ and $m=0$. Then the integral is $\int_0^\pi \sin( x)\,dx$, which is the integral of a positive integrand and is definitely not 0 (in fact it is 2).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/959547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $ I need to prove or disprove the following inequality:
$$
(xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4}
$$
For $x,y,z \in \mathbb R^+$. I found no counter examples, ... | Pursuing a link from a comment above, here's the Iran 1996 solution attributed to Ji Chen.
Reproduced here to save click- & scrolling ...
The difference
$$4(xy+yz+zx)\cdot\left[\sum_\text{cyc}{(x+y)^2(y+z)^2}\right]-9\prod_\text{cyc}{(x+y)^2}$$
which is equivalent to the given inequality, is
presented as the sum of sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/959688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Expression for the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2}$ I'm having troubles with the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2} \in L^2(\mathbb{R}^{n})$. For the case $n=1$ I got $\hat{f}(\xi) = \pi e^{-2\pi |\xi|}$ using residues. Does the general case have a nice expression? How is that expressio... | It's not in $L^2$ for $n \ge 4$. For $n \le 3$, assume wlog $\xi$ is in the direction of one of the coordinate axes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/959791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a geometrical definition of a tangent line? Calculus books often give the "secant through two points coming closer together" description to give some intuition for tangent lines. They then say that the tangent line is what the curve "looks like" at that point, or that it's the "best approximation" to the curve... | The best geometrical definition of the tangent line to a curve is the one given by Leibniz in his 1784 article providing the foundations of his infinitesimal calculus. The definition of the tangent line is the line through two infinitely close points on the curve.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/959885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 9,
"answer_id": 8
} |
Integral of $\int \frac{\cos \left(x\right)}{\sin ^2\left(x\right)+\sin \left(x\right)}dx$ What is the integral of $\int \frac{\cos \left(x\right)}{\sin ^2\left(x\right)+\sin \left(x\right)}dx$ ?
I understand one can substitute $u=\tan \left(\frac{x}{2}\right)$
and one can get (1) $\int \frac{\frac{1-u^2}{1+u^2}}{\left... | Consider the integral
\begin{align}
I = \int \frac{\cos(x) \, dx}{\sin^{2}(x) + \sin(x) } .
\end{align}
Method 1
Make the substitution $u = \tan\left(\frac{x}{2}\right)$ for which
\begin{align}
\cos(x) &= \frac{1-u^{2}}{1+u^{2}} \\
\sin(x) &= \frac{2u}{1+u^{2}} \\
dx &= \frac{2 \, du}{1+u^{2}}
\end{align}
and the int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/959992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Standard Uniform Distibution with Random Variable Could someone help explain how to solve the following problem:
From my understanding, this problem states that we have a function, Uniform(0, 1), that will generate a random value from 0 to 1 with uniform distribution. What I don't understand is how this translates int... | One approach is to consider the cumulative distribution function $F(x)=P(X \le x)$
So taking a cumulative sum of the probabilities in your table, it might look like
x 3 4 5
P(X=x) 0.40 0.15 0.45
P(X<=x) 0.40 0.55 1.00
Then look at your standard uniform random variable $U$ and
*
*if $0 \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/960123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why not use two vectors to define a plane instead of a point and a normal vector? In Multivariable calculus, it seems planes are defined by a point and a vector normal to the plane.
That makes sense, but whereas a single vector clearly isn't enough to define a plane, aren't two non-colinear vectors enough?
What I'm thi... | Remember, vectors don't have starting or ending positions, just directions. So take the vectors <1,0,0> and <0,1,0>. These vectors will define a plane that only goes in the x-y direction, but the problem is, they will work for any z-coordinate. So you need some starting point to anchor your plane.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/960261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
The condition that a ring is a principal ideal domain If $R$ is a nonzero commutative ring with identity and every submodule of every free $R$-module is free, then $R$ is a principal ideal domain.
What I don't know is how to show that every ideal is free. Once an ideal is free, for nonzero $u,v\in I$, $uv-vu=0$ shows t... | Consider $R$ as $R$-module, then this is free, with basis $\lbrace 1 \rbrace $.
Suppose $I \subseteq R $ is an ideal, $I \neq 0 $; then $I$ is a submodule, whence it is free. If $u, v \in I $ they are linearly dependent over $R$, because $vu -uv = 0 $.
This implies that if $B= \lbrace u_1, u_2, \ldots \rbrace $ is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/960336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integration and Laplace-Stieltjes of a multiplied Weibull and Exponential distribution Function I have a trouble for integrating a multiplied weibull and exponential distribution. The expression is as follows:
$$
Y(t) = \int_0^t e^{-\lambda T}e^{-(T/\mu)^z}dT\,.
$$
Then, I need to take Laplace-Stieltjes from Y as follo... | A method of solving is shown below :
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/960455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that for any $n$, we have $\sum_{i=1}^n \frac{a_i}i \ge a_n $. Consider the sequence $(a_n)_{n \ge1}$ of real nos such that $$a_{m+n}\le a_m+a_n,\ \forall m,n \ge 1$$
Prove that for any $n$, we have $$\sum_{i=1}^n \frac{a_i}i \ge a_n .$$
| This is Problem 2 from the Asian Pacific Mathematical Olympiad (APMO) 1999.
I took the following proof essentially from AoPS:
We will prove this by strong induction. Note that the inequality holds for $ n=1$. Assume that the inequality holds for $ n=1,2,\ldots,k$, that is,
$$ a_1\ge a_1,\quad a_1+\frac{a_2}{2}\ge a_2,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/960602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Why does this loop yield $\pi/8$? Was doing some benchmarks for the mpmath python library on my pc with randomly generated tests.
I found that one of those tests was returning a multiple of $\pi$ consistently. I report it here:
from mpmath import *
mp.dps=50
print "correct pi: ", pi
den=mpf("3")
a=0
res=0
for n in x... | From your program one can infer that we must have
$$
den_n=f(n)=32 T(n)+3
$$
Where $T(n)=1+2+...+n=\frac{n(n+1)}{2}$ is a so-called Triangular Number. Then you form the sum
$$
\begin{align}
res_n&=\sum_{i=0}^n\frac{1}{f(i)}\\
&=\sum_{i=0}^n\frac{1}{32 T(i)+3}\\
&=\sum_{i=0}^n\frac{1}{16i(i+1)+3}\\
\end{align}
$$
So the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/960681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Embedding of linear order into $\mathcal{P}(\omega)/\mathrm{fin}$ I try to prove following problem (in Kunen):
Assume $\mathrm{MA}(\kappa)$ and $(X,<)$ be a total order with $|X|\le\kappa$, then there are $a_x\subset \omega$ such that if $x<y$ then $x\subset^* y$.
($x\subset^* y$ if $|x-y|<\omega$ and $|y-x|=\omega$... | If $x<y$, $(q,m)\in P$, $x,y\in Dom(q)$, and $N<\omega$ define $p$ by $Dom(p)=Dom(q)$, $q(z)=p(z)$ if $z<y$, and $q(z)=p(z)\cup\{m+N\}$ otherwise, then $(p,m+N+1)\leq (q,m)$, and there is an element $\geq N$ in $q(y)\setminus q(x)$.
As all elements of $G$ are pairwise compatible, the genericity of this set and what we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/960767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Spivak Sine confusion (possible error) quote from Spivak:
"Let us consider the function $f(x) = \sin(1/x)$." The goal is to show it is false that as $x \to 0$ that $f(x)\to 0$
He says we have to show "we simple have to find one $a > 0$ for, which the condition $f(x) < a$ cannot be guaranteed, no matter how small we req... | He is saying that there are arbitrarily small values of $x$ such that the $|f(x)<1/2|$ is not satisfied.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/960860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
What does X-2 mean given continous probability distribution X? I have the continous probability distribution X: $f(x) = 2x e^{-x^2} \, x \geq 0$ and zero everywhere else. One of my homework problems is to find the probability distribution of X-2, -2X, and X^2 but intuitively it doesnt make much sense to me.
For example... | We approach the problem through the cumulative distribution functions, even though it is more inefficient than the method of transformations.
1) Let $Y=X-2$. We want the density function of $Y$. First we find an expression for the cumulative distribution function of $Y$, that is, $\Pr(Y\le y)$.
We have
$$F_Y(y)=\Pr(Y\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/961035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
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