Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What does $\text{dim}_K L$ mean when $K,L$ are fields? +Example. What does $\text{dim}_K L$ when $K,L$ are fields?
From the context in which I have seen it used I believe it means the dimension of the vector space we get from viewing $L$ as a vector space over $K$ and that $\text{dim}_K L=[L:K]$ but I would like some c... | All of what you said is correct except the last part. You wrote correctly that $\{1,\pi,\pi^2,\pi^3,...\}$ are linearly independent over $\mathbb Q$ since $\pi$ is trascendental. So $\mathbb R$ contains an infinite set of $\mathbb Q$-linearly independent vectors, therefore $\mathrm{dim}_{\mathbb Q}\mathbb R$ is infini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1816010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
When a loop with inverse property is commutative Question
How to prove that a loop $L$ with inverse property and $x^3=e$ for all $x$ is commutative iff $(x y)^2=x^2 y^2$ for all $x,y$?
Definitions:
A loop is a quasigroup with identity $e$.
$L$ has the inverse property if every element has a two sided inverse and $x... | Note that $x^3 = e$ means that $x^2 = x^{-1}$. Similarly, this means that $(xy)^2 = (xy)^{-1}, y^2 = y^{-1}$. We now write our condition as
$$
(xy)^{-1} = x^{-1}y^{-1}
$$
This implies that $(x^{-1}y^{-1})^{-1} = xy$.
We now apply the inverse property repeatedly:
$$
x^{-1} = (x^{-1}y^{-1})y\\
(x^{-1}y^{-1})^{-1}x^{-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1816162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convexity and Proof of one sided Derivative Working on some real analysis work, I've been able to show that for a function $f$, which is convex on $[a,b]$, for $a\leq x_1< x_2< x_3\leq b$:
$$\frac{f(x_2)-f(x_1)}{x_2-x_1} \leq \frac{f(x_3)-f(x_2)}{x_3-x_2}$$
and for $h>0$, for some $x_0 \in [a,b]$:
\begin{equation}
\fr... | Consider some $k > 0$ and any $h > 0$ such that
$$a < x_0-k < x_0 < x_0 + h < b.$$
Using your inequality for a convex function, we have
$$\frac{f(x_0) - f(x_0-k)}{k}= \frac{f(x_0) - f(x_0-k)}{x_0 - (x_0-k)} \leqslant \frac{f(x_0+h) - f(x_0)}{x_0+h - x_0} = \frac{f(x_0+h) - f(x_0)}{h}. $$
Thus, $[f(x_0 + h) - f(x_0)]/h$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1816383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the required group theory knowledge needed to understand Verhoeff's algorithm? The Wikipedia page tells me I need to understand permutation groups and dihedral groups. Can someone clearly outline what exactly the perquisites of understanding this is and how much time I'll take to understand this ?
I know some b... | Bit of a strange question. As (finite) groups go, I'd rate the family of Dihedral groups as the second easiest to get a handle on, after cyclic groups. Only 2 generators - basically a rotation and a reflection of a n-gons. In any book on group theory you are still in an early chapter when you reach this topic :-)
For... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1816726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that as $PP'$ varies,the circle generates the surface $(x^2+y^2+z^2)(\frac{x^2}{a^2}+\frac{y^2}{b^2})=x^2+y^2.$ $POP'$ is a variable diameter of the ellipse $z=0,\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ and a circle is described in the plane $PP'ZZ'$ on $PP'$ as diameter.Prove that as $PP'$ varies,the circle generates... | The circle has two endpoints $(-X,-Y)$ and $(X,Y)$ with $\dfrac{X^2}{a^2}+\dfrac{Y^2}{b^2}=1$ and is perpendicular to the $z$-plane.
Let $(x,y,z)$ be a point on this circle.
Then we have:
$x^2+y^2+z^2=X^2+Y^2$ and $y/x=Y/X$.
So let $Y/y=X/x=c$.
We get: $c^2\left (\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} \right)=1$
as well as:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1816806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all functions $f:\mathbb R \to \mathbb R$ satisfying $xf(y)-yf(x)=f\left( \frac yx\right)$
Find all functions $f:\mathbb R \to \mathbb R$ that satisfy the following equation:
$$xf(y)-yf(x)=f\left( \frac yx\right).$$
My work so far
*
*If $x=1$ then $f(1)=0$
*If $y=1$ then $f\left(\frac1x\right)=-f(x)$
*If $... | $$xf(y)-yf(x)=f\left( \frac yx\right) \\
\iff xf(y)-f(\frac{y}{x})-yf(x) = 0 \tag{1}$$
Substitute $x \to y/x$, $y \to y$:
$$\frac{y}{x}f(y) - yf( \frac{y}{x}) = f(x) \\
\iff \frac{y}{x}f(y) - yf( \frac{y}{x})-f(x) = 0 \tag{2}$$
Consider $(2)-(1)\times y$:
$$(\frac{y}{x}-xy)f(y)+(y^2-1)f(x)=0 \tag{3}$$
Let $y = 0$, As ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1816896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Given that $\tan 2x+\tan x=0$, show that $\tan x=0$ Given that $\tan 2x+\tan x=0$, show that $\tan x=0$
Using the Trigonometric Addition Formulae,
\begin{align}
\tan 2x & = \frac{2\tan x}{1-\tan ^2 x} \\
\Rightarrow \frac{2\tan x}{1-\tan ^2 x}+\tan x & = 0 \\
\ 2\tan x+\tan x(1-\tan ^2 x) & = 0 \\
2+1-\tan ^2 x & = 0 ... | By the double angle formula we get $$\tan(2x)+\tan(x)=\frac{2\tan(x)}{1-\tan^2(x)}+\tan(x)=\frac{3-\tan^2(x)}{1-\tan^2(x)}\tan(x),$$ so that $\tan(x)=0$ is certainly a solution.
But $\tan(x)=\pm\sqrt3$ as well, so that the initial claim is false.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Bijectivity and Lipschitz continuity of a function on a Banach space I don't really know how to solve the following exercise, I need a little help:
a) Let $(X,\|\cdot \|)$ be a Banach space and $F\colon X \to X$ Lipschitz-continuous (i.e. $|F(x) - F(y)| \le L|x-y|$) with Lipschitz constant $L < 1$. Show that $G\colon ... | *
*The reverse triangle inequality yields $|G(x) -G(y)|\ge (1-L)|x-y|$. This gives the Lipschitz continuity of the inverse map, as soon as $G$ is shown to be surjective. For surjectivity, you are right to use the Banach fixed point theorem: the equation $x+F(x)=y$ has a solution for $x$ because the map $x\mapsto y-F(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show that $n \to \infty, \sqrt{n}(Y_{n}-p) \rightarrow N(0,p(1-p)$
Can anyone show me the correct working out to find the variance for $Y_n$, My variance seems to be $\frac{p(1-p)}{n}$
| The variance is $\frac{p(1-p)}{n}$:
\begin{align*}
Var(Y_n) &= E\left((Y_n-E(Y_n))^2\right)\\
&=\frac{1}{n^2}E\left(\left[\sum_{i=1}^n\big(X_i-E(X_i)\big)\right]^2\right)\\
&= \frac{1}{n^2}\sum_{i=1}^n E\left((X_i-E(X_i))^2\right) \\
&= \frac{1}{n}\left[p(1-p)^2+p^2(1-p) \right]\\
&=\frac{p(1-p)}{n}.
\end{align*}
Then,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why does every number of shape ababab is divisible by $13$? Why does it seems like every number $ababab$, where $a$ and $b$ are integers $[0, 9]$ is divisible by $13$?
Ex: $747474$, $101010$, $777777$, $989898$, etc...
| Note that
$$
[ababab] = a\times 101010 + b \times 10101 =
13 (7770a + 777b)
$$
Also noteworthy:
$$
10101 = 1 + 10^2 + 10^4 \equiv \\
1 + 3^2 + 3^4 = 1 + 9 + 9^2 \equiv\\
1 +(-4) + (-4)^2 = 1 - 4 + 16 = 13 \equiv 0
$$
where $\equiv$ indicates equivalence modulo $13$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 9,
"answer_id": 4
} |
about series and integral - solve $\int_1^2 (x^2-x)dx$ by limit of definite integral So I have to compute $$\int_1^2 (x^2-x)dx$$ by using the Riemann sums.
I got stuck the point where I got to $$\lim_{n\to \infty} \frac{1}{n^3} \sum_{i=0}^n (i^2+in).$$
Can I split the sum? If I can then I know how to continue. so the q... | $$I=\int_{1}^{2}(x^2-x)\,dx = \int_{0}^{1}(z^2+z)\,dz\tag{1} $$
hence:
$$ I = \lim_{n\to +\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k^2+kn}{n^2}=\lim_{n\to+\infty}\frac{1}{n^3}\sum_{k=1}^{n}\left[2\binom{k}{2}+(n+1)\binom{k}{1}\right]\tag{2} $$
and:
$$ I = \lim_{n\to +\infty}\frac{1}{n^3}\left[2\binom{n+1}{3}+(n+1)\binom{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
pointwise and uniformly convergence of a series of functions $f_n(x)$ I have $\sum \frac{e^{nx}}{2+e^{nx}} arctan(n^{|x|})$. I have pointwise convergence in $(-\infty,0)$.But for the uniform convergence?
If I consider $\sum sup_{(-\infty,0)} |f_n(x)|\le \sum \frac{\pi}{2} \frac{e^{nx}}{2+e^{nx}}=\frac {\pi}{4}$ series... | HINT:
For any fixed $N$ with $x=1/N$, we have that
$$\begin{align}
\left|\sum_{n=0}^\infty \frac{e^{nx}}{2+e^{nx}}\arctan\left(n^{|x|}\right)-\sum_{n=0}^{N} \frac{e^{nx}}{2+e^{nx}}\arctan\left(n^{|x|}\right)\right|&=\left|\sum_{n=N+1}^\infty \frac{e^{nx}}{2+e^{nx}}\arctan\left(n^{|x|}\right)\right|\\\\
&=\sum_{n=N+1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding convergence zone/range for $\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$ $$\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$$
I used the ratio test and I end up with:
$$|x|*\frac{n}{n+2}$$
What steps do I need to take to continue?
Looking for hints or steps, not full solution/
| Did you notice the powers of $\;x\;$ are not consecutive? They are $\;1,4,9,16,...\;$ , so the actual coefficients are
$$a_n:=\begin{cases}\cfrac1{\sqrt{n(n+1)}},\,&n\;\text{is a square}\\{}\\0,\,&\text{otherwise}\end{cases}$$
and thus
$$\lim\sup_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}\frac1{\sqrt[n]{\sqrt{n(n+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is a Hurwitz matrix minus a positive-definite matrix still a Hurwitz matrix? In a certain multi-agent systems, the $i$th system can be described by the form
\begin{equation}
\begin{array}{cc}
\left\{ {\begin{array}{l}
{{\dot x}_i}=A_i{x_i} +B_iu_i\\
y_i = C_ix_i
\end{array}} \right.,& i=1,2,\cdots,n
\... | If you are asking whether the sum of two positive definite matrices is still positive definite, then the answer is yes. As when $xA_1x^T>0$ and $xA_2x^T>0$, one have $x(A_1+A_2)x^T = xA_1x^T + xA_2x^T > 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Generalization of inner product I was wondering if there was a widely accepted generalization of inner product spaces where the inner product look something like $\langle\bullet , \bullet\rangle:V\times V \to \mathbb{F}$, where $\mathbb{F}$ does not have to equal to $\mathbb{R}$ or $\mathbb{C}$. Could you define a mea... | Yes, you can do this. Choose a linearly independent basis for an as yet unconstructed vector space $V$. Define $V$ as the set of linear combinations of basis elements with coefficients drawn from any field $\mathbb{F}$ you like. (You'll need to make sure your basis elements can be multiplied by any of the fields' eleme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Three-Dimensional Random Walk A particle starts at an origin $O$ in three-space. Thinking of point $O$ as the center of a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight corners of the cube. That is to say, at every walk, the particle has a 50/50 chance of movi... | One mistake is that you have approximated the expected number of times of return to the origin, rather than the probability of at least one return. They are related by
$$
P=\frac{<n>}{<n>+1}
$$
so your approximation should be giving a probability of about $\frac13$. The real value of the sum you have done is
$$
\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Prove formally that $P(|X+Y|>\varepsilon)\leq P(|X|>\frac{\varepsilon}{2})+ P(|Y|>\frac{\varepsilon}{2})$ I'm stuck with a simple probability inequality. For arbitrary random variables
$$P(|X+Y|>\varepsilon)\leq P(|X|>\frac{\varepsilon}{2})+ P(|Y|>\frac{\varepsilon}{2}) $$
Using $P(A)+P(B) \geq P(A \cap B) = P(X,Y > \... | Hint: verify
$$ \{ |X+Y|\geq \varepsilon\} \subset \{ |X|\geq \varepsilon/2 \}\cup \{ |Y|\geq \varepsilon/2 \}$$
and use finite subadditivity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can I show that the $\lim_{R\to\infty}\int_{C} \frac{e^{iz}dz}{(1+z^2)^2}$ is zero, where $C$ is a semicircle? Where $C$ is the positive semicircle of radius $R$, that is to say, it is the semicircle covering the first and second quadrant, $0 \leq \theta < \pi$
I know I have to use the $ML$ inequality, and I unders... | Let $z = R(\cos\theta+i\sin\theta)$. Then
$$e^{iz} = e^{iR(\cos\theta+i\sin\theta)} = e^{iR\cos\theta+(-R)\sin\theta}=e^{-R\sin\theta}e^{iR\cos\theta}.$$
Hence
$$|e^{iz}| = |e^{iR\cos\theta+(-R)\sin\theta}|=|e^{-R\sin\theta}e^{iR\cos\theta}| = e^{-R\sin\theta}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
how to solve double summation $1/k^2$ How should I solve
$$\sum^{\infty}_{n=1}\sum^n_{k=1} \frac{1}{k^2} $$
I know that $\sum^\infty_{k=1} \frac{1}{k^2} = \frac{\pi^2}{6}$
| This is probably off-topic.
In answers, you have been explained that the infinite summation does not converge and why it does not.
Let us look at what happens $$\sum^n_{k=1} \frac{1}{k^2}=H_n^{(2)}$$ where appear the generalized harmonic numbers. So, using their properties, $$S_p=\sum^{p}_{n=1}\sum^n_{k=1} \frac{1}{k^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
2nd order Runge-Kutta method for linear ODE Could someone please help me with the next step of this 2nd order Runge-Kutta method?
I am solving the following initial value problem (IVP)
$$x' = - \frac12 x(t), \qquad x(0)=2$$
I wish to use the second order Runge-Kutta method
$$x(t+h)=x(t)+1/2(F_1+F_2),$$
where \begin{ali... | Integrating the ODE
$$\dot x = -\frac{1}{2} x$$
we get
$$x (t) = x_0 \cdot \exp\left(-\frac{t}{2}\right)$$
Hence,
$$\begin{array}{rl} x (t+h) &= x_0 \cdot \displaystyle\exp\left(-\frac{t+h}{2}\right)\\\\ &= x_0 \cdot \displaystyle\exp\left(-\frac{t}{2}\right) \cdot \exp\left(-\frac{h}{2}\right)\\\\ &= \displaystyle\exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluate $\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$ for fixed integers $n,k\geq 1$ My question is the following
Question. Can you compute some of the following $$c_{n,k}=\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$$
where $n\geq 1$ is a fixed integer and $k\geq 1$ is also a fixed integer? What of those? ... | This is not an answer since the result comes from a CAS.
$$\int_{0}^{\frac{\pi}2}\sin^n(x)\,dx=\frac{\sqrt{\pi } \Gamma \left(\frac{n+1}{2}\right)}{2 \Gamma
\left(\frac{n}{2}+1\right)}$$ $$\int_{0}^{\frac{\pi}2}\sin^n(x)e^{-\cos(x)}\,dx=\sqrt{\pi }\, 2^{\frac{n}{2}-1} \,\Gamma \left(\frac{n+1}{2}\right)
\left(I_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Creating nearest neighbors Laplacian matrix I want to create a general formula for an $N \times N$ nearest neighbors Laplacian matrix such that I can write an mfile in MATLAB to compute the matrix for given $N$.
The nearest neighbors Laplacian matrix is of the below form
\begin{equation}
L=\left[
\begin{array}{ccc}
-2 ... | I might be misreading what you have written as the $L$ matrix, but I believe you can use:
L = gallery('tridiag',ones(1,N-1),-2*ones(1,N),ones(1,N-1));
This will return the tridiaganol matrix in sparse form. You can wrap it in full() to convert it to a standard matrix. Then you can manually change the top and bottom cor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Norms That Define An Open Set How can a norm define a set in a vector space. I don't understand for example how 2 different norms can define a same open set.
It's not intuitive to me.
An open set doesn't need a norm to be open (or to even exist)
| Are you talking about open balls or open sets? If $|| \cdot ||$ is a norm on a vector space $V$, then an open ball in $V$ (with respect to the norm $|| \cdot ||$) is a set of the form $$ \{ v \in V : ||v - v_0|| < r \}$$ for a fixed $v_0 \in V, 0 < r \in \mathbb{R}$. An open set (with respect to $|| \cdot ||$) is a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
why is $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$ I'm having a hard time wrapping my head around why $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$
| Consider the domain of real numbers, where $P$ means "equals $\pi$" and $Q$ means "is rational".
Then $\forall x (P(x)\to Q(x))$ means "All real numbers that equal pi are rational", which is blatantly false.
Where as $\forall x~P(x) \to \forall x~Q(x)$ means "If every real number equals pi, then every real number is ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
find two matrices A and B such that $A^2 -BA-AB+B^2 = O_{2\times 2}$ Can someone please explain this question to me
Question : Construct a $2\times 2$ matrix A and B with A different from B and neither A = $O_{2\times 2}$ nor $B = O_{2\times 2}$ such that $A^2 - BA - AB + B^2 = O_{2\times 2}$
Attempt: I found a matrix ... | $A^2 - AB - BA + B^2 = (A-B)^2 = O$
Find a matrix $M$ such that $M^2 = O$
How about $M = \begin{pmatrix}0&1\\0&0\end{pmatrix}$
Now find chose $A, B$ such that $(A-B) = M$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A question about extending Lebesgue measure to all subsets of the real numbers. Is it known to be consistent with just ZF set theory-without the Axiom of Choice-that there exists an extension of Lebesgue measure to all subsets of real numbers, which is countably additive and isometrically invariant?.....I am confused a... | It is known that the consistency of "All sets of reals are Lebesgue measurable" over $ZF+DC$ has large cardinal strength; however, note the role of $DC$, Dependent Choice, in the base theory. With many related results, dropping the requirement of $DC$ also drops the consistency strength of the related regularity proper... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1818917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability of being dealt four-of-a-kind in a set of $5$ cards?
You are dealt a hand of five cards from a standard deck of playing cards. Find the probability of being dealt a hand consisting of four-of-a-kind.
If possible, please provide a hint first before the answer.
One of the first things that came to me was $... | The simplest explanation might be the following:
there are ${52}\choose{4}$ possible combinations of 4 cards in a deck of 52. Then, with 5 cards, you can have 13 * 5 possible four of a kind. Divide the latter by the former.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
How do I simplify this Log with a Fraction in it? So I have:
$$
\log_2(5x) + \log_2 3 + \frac{\log_2 10}{2}
$$
I understand that when there is addition, and the bases are the same, I can simply multiply what is in the parenthesis. So for the first part, I'd get $\log_2(15x)$. I'm stuck now, because I'm not sure what t... | You may write
$$
\log_2(15x)+\frac12 \log_2(10)=\log_2(15x)+\log_2(10^{1/2})=\log_2(15x \cdot10^{1/2}).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
Prove that for every $n \ge m \ge 1 , \sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
I've tried saying that the RHS represents the number of binary series with m "1" 's and n+1-m "0"'s, but I couldn't find out what k represents at the L... | $$
\begin{align}
\sum_{k=0}^m\binom{n-k}{m-k}
&=\sum_{k=0}^m\binom{n-k}{n-m}\tag{1}\\
&=\binom{n+1}{n-m+1}\tag{2}\\[3pt]
&=\binom{n+1}{m}\tag{3}
\end{align}
$$
Explanation:
$(1)$: $\binom{n}{k}=\binom{n}{n-k}$
$(2)$: Equation $(9)$ from this answer
$(3)$: $\binom{n}{k}=\binom{n}{n-k}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
If an event $A$ is independent from $B$ and $C$, is A also independent from $B \cap C$? $A$ and $B$ are independent. $A$ and $C$ are independent. Are $A$ and $B \cap C$ also independent?
|
Let $A$ and $B$ be independent events, and let $A$ and $C$ be independent events. How do I show that $A$ and $B\cap C$ are independent events as well?
You cannot show this result because it does not hold for all $A, B, C$
enjoying these properties. Consider the following counter-example.
Consider two independent toss... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
find the measure of $AMC$ if $M$ is the midpoint of $BC$ then find the measure of $AMC$.
I tried to use the angles to find $AMC$ but I don't know how to use that $M$ is the midpoint of $BC$.
| Let $X$ be the point on $BC$ exactly below $A$ and $MX=z$ for comfort.
$BX=\frac{BC}{2}+z$ and $XC=\frac{BC}{2}-z$
Now:
$$\tan(\alpha )= \frac{AX}{BX}\implies \tan(\alpha )= \frac{AX}{\frac{BC}{2}+z}$$
$$\tan(AMC )= \frac{AX}{Z}$$
$$\tan(\beta )= \frac{AX}{XC} \implies \tan(\beta )= \frac{AX}{\frac{BC}{2}-z}$$
Now you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Pareto optimum in game matrix I have to find Pareto optimum squares in game matrix. They are marked in following picture
What questions do I have to ask myself for every square to decide if it is Pareto optimum?
Why square E/A (5,4) is not optimal?
|
Pareto optimality is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off.
E&A with a value $(5,4)$ can be improved to F&D with a value of $(5,5)$ since the second player is better off and the first player is not worse off.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Integral $\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx$ I would like to evaluate this integral:
$$\mathcal F(a)=\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx,\quad a>0.\tag1$$
For all $a>0$ the integrand is a smooth oscillating function decaying for $x\... | Interesting question. By using the reflection formula, your integral can be written as a convolution integral:
$$ I(a) = \pi \int_{-\infty}^{+\infty}\frac{dx}{\Gamma(x+a)\Gamma(1-x)}.\tag{1}$$
We may also notice that when $n\in\mathbb{N}$ we have:
$$ \int_{-\infty}^{+\infty}\frac{\sin(\pi x)\,dx}{x(x+1)\cdot\ldots\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
} |
Angle bracket (quadratic variation) process for martingales Probability with Martingales:
What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales?
It seems that $$\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$$ is suppo... | Since $M^2-A=N$ is a martingale, the stopped process $(M^2-A)^{S(k)}$ is also a martingale. Also, $A^{S(k)}$ is previsible. However, the Doob decomposition of an $\mathcal{L}^1$ adapted process into a martingale and a previsible process is essentially unique. Hence, $\left<M^{S(k)}\right>=A^{S(k)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What should I do further? I came across simple question, The length of all sides of a $\triangle{ABC}$ are in integral units. If length of $AB=10$ and $AC= 15$ then the number of distinct possible values of $BC$ is finite. We can simply apply Triangle Inequality, I tried to show creativity as follows:
Taking $\angle A=... | I think the Triangle Inequality is the cleanest method, applied twice to find the (excluded) lower bound of $5$ and the (excluded) upper bound of $25$, giving the full set of solutions for the length of $BC$ as
$$\{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24\}$$
A more creative method would of course be inter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Sum $1+(1+x)^2+(1+x+x^2)^2+\cdots+(1+x+x^2+\cdots+x^{N-2})^2 $ Is there a way to find the sum the following series:
$$1+(1+x)^2+(1+x+x^2)^2+\cdots+(1+x+x^2+\cdots+x^{N-2})^2 \text{ ?}$$
Any ideas ? Perhaps someone knows already the result..
Thank you in advance for your time.
| $$
1+(1+x)^2+(1+x+x^2)^2+\cdots+(1+x+x^2+...+x^{N-2})^2 = \sum_{i=0}^{N-2}(1+x+\cdots + x^i)^2$$
Let $x<1$, though same can be repeated for $x>1$ - we do not consider $x=1$ since the answer is straightforward in this case
$$\sum_{i=0}^{N-2}(1+x+\cdots + x^i)^2 = \sum_{i=0}^{N-2}\left(\frac{1-x^{i+1}}{1-x}\right)^2 = \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Under what condition does $f(f^{-1}(f(A))) = f(A)$? Basic question regarding function.
Let $f: X \to Y$, then for what $f$ does $f(f^{-1}(f(A))) = f(A)$? hold?
Obviously this relationship holds when $f$ is a bijection.
This does not hold when $f$ is pure surjection because the inverse does not exist.
Does this also ho... | When $f$ is an injection function then $f^{-1}(f(A))=A$, as a result
$$f(f^{-1}(f(A))) = f(A)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$ Wolfram Alpha evaluates this integral numerically as
$$\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=0.379064 \dots$$
Its value is apparently
$$\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2}... | Hint:
Consider the parametric integral
\begin{equation}
I(a) = \int_0^\infty \frac{t^{a-1}}{\cosh^{2} t}\ dt=4 \int_{0}^{\infty} \frac{t^{a-1}}{(e^{t}+e^{-t})^{2}}\ dt = 4 \int_{0}^{\infty}\frac{t^{a-1} e^{-2t}}{(1+e^{-2t})^{2}}\ dt
\end{equation}
Hence, your integral is simply
\begin{equation}
\int_0^{\infty} \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Any Nilpotent Matrix is not Diagonalizable I'm trying to go about the proof that any matrix that is nilpotent
(i.e. $\exists N \in\Bbb N. A^N = \mathbf{0}$) cannot be diagonalizable.
I believe that the best way to go about this is by showing
that a given eigenvalue's
geometric multiplicity is not the same as its algebr... | Not true (example: $0$ matrix). Though, the statement is true if $A \neq 0$.
If $A$ is diagonalizable, then there exists an invertible $P$ and a diagonal matrix $D$ such that $A = P^{-1} D P$. Then, $A^N = P^{-1} D^N P$, which gives $P^{-1} D^N P = 0$. Therefore $D^N = 0$ and so $(D_{ii})^N = 0$ for all $i$, hence $D_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 4
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How do I prove $\int^{\pi}_{0} \frac{\cos nx}{1+\cos\alpha \cos x} \mathrm{d}x = \frac{\pi}{\sin \alpha} (\tan \alpha - \sec \alpha)^n $? The result I wish to show is that for $n \in \mathbb{Z}$, $$\int^{\pi}_{0} \frac{\cos nx}{1+\cos\alpha \cos x} \mathrm{d}x = \frac{\pi}{\sin \alpha} (\tan \alpha - \sec \alpha)^n $$... | Hint:
Use the following relation
\begin{equation}
\frac{\sin\alpha}{1+\cos\alpha\cos x}=1+2\sum_{k=1}^\infty \left(\frac{\sin\alpha-1}{\cos\alpha}\right)^k\cos(kx)
\end{equation}
It can be obtained by writing cosine in the denominator of the integrand out in exponential form as the following
\begin{equation}
\frac{A^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Show that $x^n$ is a continuous function So the idea I thought of doing was by induction.
$\forall\; \epsilon > 0\;\; \exists \;\delta > 0$ such that $|x-x_0| < \delta \implies |x-x_0| < \epsilon $
By the definition of the limit choose $\delta=\epsilon$ and the statement holds for $n=1$. Thus $\lim_{x \rightarrow x_0} ... | Hint: if $f$ is continuous and $g$ is continuous then $fg$ is continuous. This is not that hard to show (ask if you need help). Then apply induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Prove that $\frac{dy}{dx} = -\frac1{(1+x)^2}$ for given that $x\sqrt{1+y} + y\sqrt{1+x} = 0$ $$x\sqrt{1+y} + y\sqrt{1+x} = 0$$
Please tell me where I went wrong. Why I am not getting correct answer ?
| There is nothing wrong. Put the value of $y$ to get your result.
However, a simpler approach:
$$x\sqrt{1+y} + y\sqrt{1+x} = 0$$
$$x\sqrt{1+y} = - y\sqrt{1+x}$$
Squarring both sides, we get
$$x^2(1+y) = y^2(1+x)$$
$$x^2(1+y) - y^2(1+x)=0$$
$$(x-y)(x+y+xy)=0$$
So either $x-y=0$ or $x+y+xy=0$.
Now if, $x-y=0$, then we h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to Solve a linear matrix equation of an array $M = BMC$ where $ B$ and $C$ are known Adding to the question's description :
I am doing Feature extraction from videos and i am trying to implement this one line of mathematical equation to matlab or even any algorithm .
let's say I have $ B $ and $ C $ which are $ 10... | If you multiply the equation $M = BMC$ by the inverse of $B$ on both sides and take everything on the same side of the equal sign, you get $$ B^{-1} M - MC = 0.$$
This is a Sylvester equation where the last term is the null matrix (all elements equal to zero). You should be able to solve your problem in Matlab by using... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to derive the gregory series for inverse tangent function? How to derive the Gregory series for the inverse tangent function?
Why is Gregory series applicable only to the set $ [-\pi/4,\pi/4] $ ?
| A fast way is by exploiting $\arctan'(x)=\dfrac1{1+x^2}$.
The Taylor series of that derivative is easily established to be the sum of a geometric series of common ratio $-x^2$,
$$\sum_{k=0}^\infty(-x^2)^k=\frac1{1-(-x^2)},$$ which only converges for $x^2<1$.
Then integrating term-wise,
$$\arctan(x)=\int_0^x\frac{dx}{1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Can all functions over $\mathbb{Z}/4\mathbb{Z}$ be "described" by a polynomial? Consider the set $T$ of functions from $\mathbb{Z}/4\mathbb{Z}$ to $\mathbb{Z}/4\mathbb{Z}$. I'm now asked to prove or disprove the statement that all functions in $T$ can be described by a polynomial over $\mathbb{Z}/4\mathbb{Z}$, i.e. for... | Consider a polynomial $p(X) = \sum_{k=0}^n a_k X^k$ with $a_k \in \mathbb{Z}/4\mathbb{Z}$. Then we have
$$ p(0) = a_0, \\
p(2) = a_0 + a_1 \cdot 2 + a_2 2^2 + \dots + a_n 2^n = a_0 + 2a_1. $$
Hence, we can't find a polynomial that satisfies for example $p(0) = 1, p(2) = 2$ as the equation $2a_1 = 1$ has no solution ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
The coefficient of $x^{100}$ in the expansion of $(1-x)^{-3}$. Please solve it and tell me the technique so that I can solve it in examination in multiple choice questions.
| This is an application of the generalized binomial theorem, that
$$(x+y)^s = \sum_{k=0}^\infty {s \choose k} x^{s-k}y^k$$
where
$${s \choose k} = \frac{s(s-1)(s-2)\cdots(s-k+1)}{k!}$$
is the generalized binomial coefficient. Hence in this case,
$$(1-x)^{-3} = \sum_{k=0}^\infty {-3 \choose k} (1)^{-3-k}(-x)^{k}$$
and so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does a separable metric space have a countable basis? I want to prove that if $X$ is a metric space and has a dense countable subset, then it has a countable basis. I know that every metric space is first countable, but I can't continue. Thanks for your help.
| Well, take a dense subset of $D\subset X$ and then the metric guarantees you a countable neighbourhood basis at every point of $D$. A countable union of countable sets is again countable. Check that this is enough.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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With $m\in\mathbb{Z^+}$ fixed, is $\sum_{m\ne n\ge1} (n^2-m^2)^{-1}$ evaluable really elementarily? I found this exercise at the begininning of the series section of a calculus workbook, so it shouldn't require machinery like integrals or special functions; merely telescopic summing or some other easy trick, but I can'... | Since $$\frac1{n^2-m^2}=\frac{1}{2m}\left(\frac{1}{n-m}-\frac{1}{n+m}\right)$$the sum does telescope "eventually"; for any specific $m$ you can see that it equals $\frac1{2m}$ times the sum of finitely many terms $$\frac{\mp1}{n\pm m};$$all the other terms cancel. (If that's not clear write out a large number of terms ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Where is the notion of anti-isomorphism useful Let $(S,\cdot)$ and $(T,\circ)$ be semigroups (or some algebraic structure with an operation), then they are anti-isomorphic if there exists some $\varphi : S \to T$ such that
$$
\varphi(xy) = \varphi(y) \circ \varphi(x).
$$
Now for what is this notion useful?
The noti... | Well, ... at the very least there are indeed some examples for that! I am going to mention some anti-automorphisms. (Post-) Composing with any isomorphism gives an anti-isomorphism.
*
*Taking inverses in a group: $(ab)^{-1} = b^{-1}a^{-1}$
*Taking the transpose in a semigroup of $n\times n$-matrices: $(AB)^T = B^T... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving that $\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$ $f:[a,b]\rightarrow R$ that is integrable on [a,b]
So we need to prove:
$$\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$$
1.) So we'll use a property of definite integrals: (homogeny I think it's called?)
$$\int_{-b}^{-a}f(-x)dx=-1\int_{-b}^{-a}f(x)dx$$
2.) Great, now ... | Let f(x) = x. then f(-x) = -x. Substituting -a and -b in the limits of the integration will lead to be f(a) = a then f(-a) = -(-a) = a.
it's simply this that you are multiplying the limits and the function by -1. if both are multiplied then they would get neutralized,
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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If $U^*DU=D=V^*DV$ for diagonal $D$, is $U^*DV$ diagonal too? All the matrices mentioned are complex $n\times n$ matrices. Let $U, V$ be unitary matrices such that $U^*DU=V^*DV=D$ for a diagonal matrix $D$ with nonnegative diagonal entries. Does this imply that $U^*DV$ is also diagonal? All I understand is that $U^*V$ ... | No. Consider $D=I$. Any unitary $U$ and $V$ satisfy the condition and yet it is easy to see $U^*V$ is not diagonal in general.
If the diagonal $D$ has all distinct diagonal entries, then $U$ and $V$ are both diagonal. Any kind of multiplication involving any of the three matrices is diagonal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What assumptions do we need for Fourier transform of derivative formula Suppose $f: \mathbb R \to\mathbb C$ is differentiable and $f$ and $f'$ are in $L^1(\mathbb R)$. Do we need further assumptions to have the formula:
$$\widehat{f'}(t) = (2\pi it)\hat f(t) $$
My textbook also assumes $f$ is continuously differentiabl... | The minimum set of conditions needed can be expressed using the Lebesgue integral. Essentially you need $f$ to be absolutely continuous with derivative $f' \in L^1$. Absolutely continuous means $f$ is the integral of its derivative:
$$
f(y)-f(x)=\int_{x}^{y}f'(t)dt.
$$
This all works out very nicely using... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solutions to $[x^2]+2[x]=3x\text{ where } 0\le x\le 2$
Find all solutions to $$[x^2]+2[x]=3x\text{ where } 0\le x\le 2$$
and $[x]=\lfloor x\rfloor$
$$$$
I managed to simplify this to $[x^2]-[x]=3\{x\}$. Thus, $$[x^2]-[x]=\{0,1,2\}$$ However I got stuck here and was unable to proceed further. Somehow I have a feelin... | If $0\leq x<1$, $LHD=0$.
$1\leq x<\sqrt2$, $LHD=3$
$\sqrt2\leq x<\sqrt3,LHD=4$
$\sqrt3\leq x<2,LHD=5$
So x is a rational number by feature of equation, $x=0,1,2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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number of roots on SO(3) Suppose we have a smooth map$ f:SO(3) → SO(3)$ of manifolds s.t.$ f(X)=X^2$.
$I$ though since I is a regular value of this map and f is orientation preserving, to calculate degree of it, it is enough to check the number of roots of $X^2-I$ in SO(3).
If X is a root of it, then i have that $X=VD... | The equation $X^2=I$ simply means that $X$ is an involution, i.e. order 2 rotation about the origin. Such rotation is determined by its axis and, thus you get continuum of solutions. The point is that $I$ is not a regular value of $f$. Hint: Try to solve the equation $f(X)=R$, where $R$ is a fixed rotation by any angl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proving that $\int_{-b}^{b}f(x)dx=0 $ if $f$ is an odd function So I split this integral up into two:
$$ \int_{-b}^{0}f(x)dx+\int_{0}^{b}f(x)dx= \int_{-b}^{b}f(x)dx$$
Then I can do the following:
$$ \int_{-b}^{0}f(x)dx+\int_{0}^{b}f(x)dx= \int_{0}^{b}f(-x)dx+\int_{0}^{b}f(x)dx$$
and since $f$ is odd $(f(-x)=-f(x))$ we ... | Another approach. Let's assume $f$ is sufficiently regular on a given set. By the fondamental theorem of calculus and the chain rule, we get that
$$
\left(\int_{-b}^{b}f(x)dx\right)'=f(b)-(-f(-b))=0
$$ giving that
$$
\int_{-b}^{b}f(x)\:dx=\text{constant}
$$ by putting $b:=0$, the constant is seen to be $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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On every other finite field at least one of −1, 2 and −2 is a square, because the product of two non squares is a square
[Except on field extensions of $\mathbb{F}_2$] On every other finite field at least one of $−1$, $2$ and $−2$ is a square, because the product of two non squares is a square.
I don't see why this i... | Looking at the Wikipedia article, I think you meant to say "On every finite field of characteristic other than $2$" rather than "On every other finite field apart from $\mathbf{F}_2$". Also, the Wikipedia article doesn't mean to imply that the statement isn't true in finite fields of characteristic $2$—just that the ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Graphical explanation of the difference between $C^1$ and $C^2$ function? We are all aware of the intuitive (graphical) explanation of the concepts of continuous and differentiable function. Whenever these two concepts are formally defined, the following elementary explanations are given:
A continuous function is a fu... | If $f$ is everywhere differentiable, but $f'$ is not continuous at some point, then $f'$ has to be very discontinuous there, because otherwise $f'$ could not satisfy the intermediate value requirement. As an example consider the function $f(x):=x^2\sin{1\over x}$ $(x\ne0)$, $f(0):=0$.
On the other hand there are beaut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
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Question about proof of the product of a finite number of T1 spaces is a T1 space. Show that the product of a finite number of T1 spaces is a T1 space.
Since each $X_i$ is a $T_1$ space, $$\prod_{i \in I} X_i \backslash \{x_i\} = \{(a_1, a_2, ..., a_n):a_i \in X_i \land (a_1, a_2, ..., a_n) \neq (x_1, x_2, ..., x_n)\} ... | To make things more simple you can focus on closed sets instead of open sets.
If $X$ denotes the product of $X_1,\dots,X_n$ and $\pi_i:X\to X_i$ the projections then a singleton $\{x_1,\dots,x_n\}\in X$ equals $\bigcap_{i=1}^n\pi_i^{-1}(\{x_i\})$. The singletons in every $X_i$ are closed (since $X_i$ is $T_1$) and the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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General matrix solution for a variant orthogonality condition An $n\times n$ complex matrix $X$ satisfies a constraint looking like a rank deficient version of the orthogonality condition
$$ X^TX = \text{diag}\left(1,\dots,1,0\right), $$
where $X^T$ is the transpose of $X$.
How can I describe the most general form of ... | Let $x_{mn}$ denote the entries of $X$, i.e.
$$
X= \begin{bmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1} & x_{n2} & \cdots & x_{nn}
\end{bmatrix}.
$$
We may define vectors
$$
X_j=\begin{bmatrix}
x_{1j}\\
\vdots \\
x_{nj}
\end{bmatrix},
$$
i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Picking cards sequentially vs consecutively We have a pack of 6 cards over the table. Cards are: {A, A}, {B, B}, {C}, {D}.
There are 3 players (Papa, Pepe, Popo) sat around the table. Cards are all upside down, so the players cannot see which card they are picking.
Now we are presented with two scenarios:
*
*Case 1:... | The probabilities that Papa picks two A's are actually the same.
*
*Case 1
Probability of Papa picks A: $\frac{2}{6}$
Probability of Pepe picks cards other than A: $\frac{4}{5}$
Probability of Popo picks cards other than A: $\frac{3}{4}$
Probability of Papa picks A again: $\frac{1}{3}$
So, probability of Papa pick... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find the inverse of $f(x) = 1 + \frac{1}{x}, x \gt 0$ I'm tasked to find the inverse of the function $$f(x) = 1 + \frac{1}{x}, x \gt 0$$
The book offers a solution, simply to set $$1 + \frac{1}{x} = s$$
and solve $$x = \frac{1}{s-1}$$
and I think I understand why that works. But why won't this work?
$f_i$ is the func... | You need to replace $x$ with $1+\frac1x$ in both parts of the formula. So
$$f_i\left(1+\frac1x\right)=\left(1+\frac1x-1\right)\left(1+\frac1x\right)^2
=\frac1x\left(1+\frac1x\right)^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Proving the ratio of curvature and torsion is constant. This question has been asked slightly differently in a few different forums, but I wanted to discuss my approach and see if I was on the right track:
Prove that if the tangent lines of a curve make a constant angle with a fixed direction, then the ratio of its cur... | To begin with, we should name the fixed direction some other way, say $v$. You already have that $T\cdot v$ is constant, so its derivative is zero. As $\dot v= 0$, we get $\kappa N \perp v$, and $v$ belongs to the vector space generated by $T$ and $B$. Since these are orthonormal, we get
$$
v= \langle v,T\rangle T+ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Spectrum of difference of two projections Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ?
The exercise is to show that $\lVert p-q \rVert \leq 1$.
Any hints are appreciated.
| You have, since $0\leq q\leq I$ and $0\leq p\leq I$,
$$
-I\leq -q\leq p-q\leq I-q\leq I.
$$
So, as you mentioned, it follows that $\sigma(p-q)\subset[-1,1]$.
Note also that the argument does not use that $p,q$ are projections, only that they are positive elements of the unit ball.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Finding the second derivative of $f(x) = \frac{4x}{x^2-4}$. What is the second derivative of
$$f(x) = \frac{4x}{x^2-4}?$$ I have tried to use the quotient rule but I can't seem to get the answer.
| Avoiding the quotient rule, just for an option:
$$\begin{align}
\ln(f(x))
&=\ln(4)+\ln(x)-\ln(x+2)-\ln(x-2)\\
\frac{f'(x)}{f(x)}
&=x^{-1}-(x+2)^{-1}-(x-2)^{-1}\\
f'(x)
&=f(x)\left(x^{-1}-(x+2)^{-1}-(x-2)^{-1}\right)\\
f''(x)
&=f(x)\left(-x^{-2}+(x+2)^{-2}+(x-2)^{-2}\right)+f'(x)\left(x^{-1}-(x+2)^{-1}-(x-2)^{-1}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
at what price he should sell his product? Can anyone help me with this problem?
A merchant has determined that if the selling price of peaches is 15 cent each, he will sell 400 of them each day. He has also determined that for each cent the price is increased he will sell 16 fewer peaches each day. At what price in ce... | As you have correctly deduced, we have $P(n) = (15 + n)(400 - 16n) = -16n^2 + 160n + 6000$
At this point, you want to complete the square so that we have
$$P(n) = -16n^2 + 160n + 6000 = -16(n^2 - 10n) + 6000 = -16(n^2 - 10n + 25) + 6400 = -16(n-5)^2 + 6400$$
Is the expression on the right-most side of the above equati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Calculate $\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$ How can we evaluate the following integral:
$$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$
I know basically how to calculate by using the substitution $x=\tan{\theta}$ :
$$\int_0^1 \frac{dx}{\sqrt{1+x^2}}$$
But I... | There is a better way.
We shall prove that:
$$\int_0^s f(x+ks)dx=\int_{ks}^{(k+1)s}f(x)dx \tag1$$
And hence:
$$\int_0^s[f(x)+f(x+s)+...+f(x+(n-1)s)]dx=\int_0^{ns}f(x)dx \tag2$$
Proof:
Using substitution $t\mapsto x+ks$,
$$\int_0^s f(x+ks)dx=\int_{ks}^{(k+1)s}f(t)dt$$
\begin{align}
& \int_0^s[f(x)+f(x+s)+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Existence of non obtuse angle of n+2 vectors in n-dimensional euclidean space. There are n+2 distinct vectors $v_1,v_2,v_3,\cdots ,v_{n+2}$ in n-dimensional euclidean space.
Prove that there must be a integer pair of $(i,j)$ which satisfies $1\leq i<j\leq n+2$, and $dot(v_i,v_j)\geq 0$.
i.e. There are at most n+1 vecto... | W.l.o.g assume that the vectors are normal and suppose for contradiction that all these $n+2$ vectors have pairwise dot product less than zero.
In the following I further assume that the dimension of the span of these vectors is $n$ but I think for a smaller rank we can reduce it to the full-rank case.
Let $w_1=v_1$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Geometry-Triangle Let $ABC$ be a triangle with $DAE$, a straight line parallel to BC such that $DA=AE$. If $CD$ meets $AB$ at X and $BE$ meets $AC$ at $Y$, prove that $XY$ is parallel to $BC$
I tried to use the angle approach by couldn't work out the problem as I couldn't effectively prove the thing using alternate ang... |
We have that $\triangle{AXD}$ and $\triangle{BXC}$ are similar and that $\triangle{EYA}$ and $\triangle{BYC}$ are similar, and so
$$AX:BX=AD:BC=AE:BC=AY:CY$$
from which the claim follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that there exists $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$? Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions.
We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$ are linearly in... | Suppose, this is not true. Then, we would have $\mu(A\cap\{\lambda_i\ne0\})=0$ for all $i\in\{1,2,\dotsc,l\}$. Hence,
$$1_A X_1 = \sum_{i=1}^l \lambda_i1_AY_i$$
implies that $1_AX_1=0$. This, the vector $(1_A,0,\dotsc,0)$ satisfies
$$1_AX_1 + 0X_2 + \dotsb + 0X_k = 0$$
which contradicts the linear independence of $(X_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Is $\int_0^a\frac{f(x)^2}{\int_0^x f(t)\,dt}\,dx\geq\int_0^a \frac{g(x)^2}{\int_{0}^{x}{g(t)}\,dt}\,dx.$? Suppose $f(x)\geq g(x)>0$ and that $f,g$ are suitably defined so that the integrals below make sense. I want to know if
$$\int_0^a\frac{f(x)^2}{\int_0^x f(t)\,dt}\,dx\geq\int_0^a \frac{g(x)^2}{\int_{0}^{x}{g(t)}\... | This is equivalent to write
\begin{align}
\int_0^a\frac{f(x)^2}{\int_0^x f(t)\,dt}\,dx-\int_0^a \frac{g(x)^2}{\int_{0}^{x}{g(t)}\,dt}\,dx \ge0
\end{align}
So,
\begin{align}
0 &\le\int_0^a\left[{\frac{f(x)^2}{\int_0^x f(t)\,dt} -\frac{g(x)^2}{\int_{0}^{x}{g(t)}\,dt} }\right]dx
\\
&=\int_0^a\left[{\frac{f(x)^2\int_{0}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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How to calculate this integral in one variable I want calculate this integral
$$ \int_0^{2\pi} - \frac{\cos t \; ( 2 (\sin t)^2 + (\cos t)^2)}{(\cos t)^4 + (\sin t)^2} \, dt $$
Can I use an opportunity substitution?
| \begin{align}
& I=\int_{0}^{2\pi }{-}\frac{\cos t\ (2{{\sin }^{2}}t+{{\cos }^{2}}t)}{{{\cos }^{4}}t+{{\sin }^{2}}t}dt=-2\int_{0}^{\pi }{\frac{\cos t\ (1+{{\sin }^{2}}t)}{{{(1-{{\sin }^{2}}t)}^{2}}+{{\sin }^{2}}t}dt} \\
& I=-2\int_{0}^{\frac{\pi }{2}}{\frac{\cos t\ (1+{{\sin }^{2}}t)}{{{(1-{{\sin }^{2}}t)}^{2}}+{{(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Is A+B digonalizable if they share the same basis of eigenvectors I was given the following statement:
I know that the sum of two diagonalizable matrices is not allways diagonalizable,
but i'm not sure how the added element of the shared base contributes..
I would very much appreciate an explanation or some guidance..... | the same basis is also a basis of $A+B$, because if $v$ is an element of this basis we have $(A+B)v=Av+Bv=\lambda v+\gamma v=(\lambda +\gamma)v$, so $v$ is an eigenvectors of $A+B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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How is the function $f: \mathbb{Z} \to \mathbb{R}$ continuous? Where $\mathbb{Z}$ is the set of integers and $\mathbb{R}$ the set of real numbers.
In a question in a problem sheet, it said this statement was correct, however I do not understand how.
You clearly cannot even begin to draw this function without a lot of ... | $f:\Bbb Z \to \Bbb R$ is continuous since it cannot be noncontinuous.
Let me remind you, a function, $f$, is noncontinuous if there is an $x_0$ in its domain in which $f$ is noncontinuous in $x_0$.
By definition${}^\dagger$:
*
*If a function $f$ is noncontinuous in $x_0$, then (there exists a small distance $\epsilon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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find the the greatest value of $m$ such that $\text{lcm}(1,2,3,..,n)=\text{lcm}(m,m+1,..,n).$ I am stuck and unable to proceed. Value of n can be very large. For eg: if $n=6,\ \text{lcm}(1,2,...,6)=60$, so answer will be $4$ in this case.
Since $\text{lcm}(2,3,4,5,6)=60,\ \text{lcm}(3,4,5,6)=60,\ \text{lcm}(4,5,6)=60... | Let $L_n$ be the left-hand side, and let $R_{m,n}$ be the right-hand side. Note that $L_n$ is divisible by all of the greatest prime powers $p^r \leq n$. In fact, $$L_n= \prod_{p \leq n} p^r$$ where $p$ is prime and $r$ is the greatest exponent such that $p^r \leq n$.
Clearly $R_{m,n} \mid L_n$. Therefore, $$L_n=R... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A function taking values in a countable product of metric spaces is uniformly continuous iff its coordinate functions are. Let $\lbrace (M_n,d_n) \rbrace _{n \in I}$ be a countable family of metric spaces and define a metric in the product $\prod _{n \in I}M_n$ as follows:
$$d(\vec{x},\vec{y})=\sum _{n \in I} \min \lbr... | HINT: You’re not taking full advantage of the definition of $d$. Let $\epsilon>0$. Then there is an $m\in\Bbb N$ such that
$$\sum_{n\ge m}\frac1{2^n}<\frac{\epsilon}2\;.$$
Then for any $\vec x,\vec y$ we have
$$d(\vec x,\vec y)<\frac{\epsilon}2+\sum_{n<m}d_n(x_n,y_n)\;,$$
and we’re essentially dealing with only finite... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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What is the meaning of subtracting from the identity matrix? If I subtract the matrix $A$ from the identity matrix $I$, $I - A$, is there a meaning to the resulting matrix perhaps given some conditions like invertibility or symmetry? For example,
$$
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
- \begin{bmatrix} a & b \... | The difference $I - A$ is sometimes used with respect to projection matrices, where $A$ is the projection onto a subspace and $I - A$ is the projection onto the orthogonal complement of that subspace.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1823891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Using limits to determine if $a_n=\frac{(\ln n)^2}{\sqrt{n}}$ (without L'Hospital) is convergent $a_n=\frac{(\ln n)^2}{\sqrt{n}}$ $(n\ge1)$
Would I use the squeeze theorem or can I just take the limit of top and bottom?
using L'Hopital, I get $a_n=\frac{(4\ln n)}{\sqrt{n}}$ $(n\ge1)$
| $$\ln{n}=4\ln{n^{\frac{1}{4}}}$$ and $$\sqrt{n}=(n^{\frac{1}{4}})^2$$ and you know $\lim_{x \rightarrow \infty} \frac{\ln{x}}{x}=0$ thus $$\lim_{n \rightarrow \infty}a_{n}=\frac{16(\ln{n^{1/4})^2}}{(n^{1/4})^2}=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Notation for enumerating a set Is there a common notation for enumerating a set? For example if $A=\{2,4,6,\ldots,n \}$ is the set of even numbers, I would like to know the notation that enumerates ordered pairs $(e,i) \in \operatorname{enumerate}(A)$, where $e$ is the $i$:th element of $A$. So the sequence would be $(... | Normally, you would just write $A = \{a_i : i \in \omega\}$ or $A = \{a_i : i \leq n\}$ where $a_i = 2i$.
Of course a sequence such as $(a_n)$ is defined to be a function; the sequence of pairs you have described is the graph of the function $a$. We could write $a(i)$ instead of $a_i$. And, in the usual set theory way... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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How would you find the roots of $x^3-3x-1 = 0$ I'm not too sure how to tackle this problem. Supposedly, the roots of the equation are $2\cos\left(\frac {\pi}{9}\right),-2\cos\left(\frac {2\pi}{9}\right)$ and $-2\cos\left(\frac {4\pi}{9}\right)$
How do I start? The Cosines seem especially scary...
| HINT:
Let $x=b\cos A$
$$\implies b^3\cos^3A-3(b\cos A)=1$$
As $\cos3A=4\cos^3A-3\cos A,$
$$\dfrac43=\dfrac{b^3}{3b}\implies b^2=4\text{ as }b\ne0$$
Let $b=2$
Consequently, $$2\cos3A=1\iff\cos3A=\dfrac12=\cos\dfrac\pi3$$
$$3A=2n\pi\pm\dfrac\pi3=\dfrac\pi3(6n\pm1)$$ where $n$ is any integer
$$A=\dfrac\pi9(6n\pm1)$$ wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Proving Bijections i'm working on this question here but I am having some trouble:
$f: ℝ ⇒ ℝ$, $f(x) = x^3 - 6x$.
A) Is $f(x)$ injective?
B) Is $f(x)$ surjective?
C) Is $f(x)$ bijective?
My attempted solution:
A) $f(x)$ is injective if and only if $f(x) = f(y)$ $⇒$ $x = y$
$⇔$ $x^3 - 6x = y^3 - 6y$
$⇔$ $x(x^2 - 6) = y(... | For proving surjectivity, note that $\lim\limits_{x\to\infty}f(x)=\infty$ and $\lim\limits_{x\to -\infty}f(x)=-\infty$. Further note that $f(x)$ is a polynomial and it is known that polynomials are continuous. An application of the Intermediate-Value-Theorem will show that for each possible value of $y$ it can be ach... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find a subspace $W$ of $\mathbb{F}^4$ such that... - checking my answer! The question is the same as one previously asked, but I can't comment so I had to ask my own. Find a subspace $W$ of $\mathbb{F}^4$ such that $\mathbb{F}^4 = U \oplus W$
Suppose $U=\{ (u_1,u_1,u_2,u_2)\in \mathbb{F}^4:u_1,u_2\in \mathbb{F}\}$.
Fin... | The example you provide doesn't work. The vector $(1,1,1,1)\in \mathbb{F}^4$ but can't be written as $(x,2x,y,2y)$. But you can still solve your problem by taking the orthogonal complement $W^{\bot}$ of $W$. With $W$ as defined, its orthogonal complement is given by:$$W^{\bot}=\left\lbrace X=(x,y,z,t)\in\mathbb{F}^4\mi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why is $\gamma(t)=(0,t)$ a geodesic in the hyperbolic plane? I'm having trouble to understand a very simple fact of the book of DoCarmo "Riemannian Geometry". In the page 73 he calculates the geodesics of the hyperbolic plane:
$$ \mathbb{R}^+_2 = \{ (x,y) \in \mathbb{R}^² :y>0 \} \qquad g_{ij}= y^{-2} \delta_{ij} $$
To... | As he says, it's the image of a geodesic. It's not parametrized by arclength.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Finding the pdf of the difference of minimum and maximum of a finite set of random variables. Let $X_i$ $ (1\leq i\leq n)$ be identically distributed uniformly on $(0,1)$. Let $U = \min_i(X_i)$, $V = \max_i(X_i)$. Find the pdf of $V-U$
This is what I did. I found the cdf and differentiated.
$$F_{V-U}(x)= \mathbb{P}(V... | Your argument for constructing the order statistic's density function is almost correct. You want the count of ways to select two of the samples to be required values, times the density functions for each of those values, times the probability that the remaining $n-2$ samples lie between these values.
Next, it will b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the generating function to split $n$ into odd parts I have been recently working with generating functions in my discrete
mathematics course, and I was interested in one particular generating function.
I want to find the generating function for the number of ways one can split
$n$ into odd parts. I can see that... | The generating function for this sequence is
$f(x)=\prod\limits_{n=1}^{\infty} \cfrac{1}{1-x^{2n-1}}$
since
$f(x)=(1+x+x^{1+1}+\cdots)(1+x^3+x^{3+3}+\cdots)\cdots (1+x^{2n-1}+x^{(2n-1)+(2n-1)})\cdots = (1+x+x^2+\cdots)(1+x^3+x^{2\cdot3}+\cdots)\cdots (1+x^{2n-1}+x^{2(2n-1)})\cdots=\cfrac{1}{1-x}\cfrac{1}{1-x^3}\cdots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the matrix of this particular quadratic form I have been working on problems related to bilinear and quadratic forms,
and I came across an introductory problem that I have been having issues with.
Take
$$Q(x) = x_1^2 + 2x_1x_2 - 3x_1x_3 - 9x_2^2 + 6x_2x_3 + 13x_3^2$$
I want to find a matrix $A$ such that $Q(x) ... | The matrix associated with your quadratic form is
$$Q=\begin{pmatrix}1 & 1 & -\frac{3}{2} \\ 1 & -9 & 3 \\ -\frac{3}{2} & 3 & 13 \end{pmatrix}$$
and the characteristic polynomial of $Q$ is
$$ q(x) = x^3-5x^2-\frac{501}{4}x+\frac{511}{4}$$
so the eigenvalues are
$$\approx -9.5372,\qquad \approx 0.9886,\qquad \approx 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Is this correct definition of $T_1$ space? I found this in a handwritten note:
Defination : A topological space $X$ is $T_1$ if $\forall x \neq y \in X$
there exist a neighborhood of $y$ such that s.t. $x \not\in V$.
I was almost certain that this was incorrect, because it looked like the definition for $T_0$ space... | The definition is correct, albeit a little sloppy. Here’s a slightly more careful version of it:
$X$ is $T_1$ if for each $x\in X$ and each $y\in X\setminus\{x\}$ there is a nbhd $V$ of $y$ such that $x\notin V$.
Let $p$ and $q$ be any two distinct points of $X$. If we set $x=p$ and $y=q$, we see that the definition ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove inequality for $xyz=1$ I am having trouble to prove an intermediate step $(xy+xz+yz)^2 \geq 3xyz(x+y+z)$ for $x,y,z\geq0$.
| Using that $xyz=1$ we see that the inequality we want to prove is equivalent
to
$$(1/x + 1/y+1/z)^2 \geq 3xyz(x+y+z) \fbox {1} $$
Set $x=1/x,y=1/y,z=1/z$, and notice that the restraint for the new variables stays the same i.e. $xyz=1$ again.
Now we substitube to $\fbox{1} $ and get$$ ( x+y+z )^2\geq 3xyz(1/x+1/y+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Proof if $f(x)$ is continuous on $[a,b]$,strictly increasing on $(a,b)$,then $f(x)$ is strictly increasing on $[a,b]$ My attempt:
Because $f(x)$ is continuous at $a$,then $\lim_{x\rightarrow a^+}f(x)=f(a)$
When $x\rightarrow a^+$,$f(x)$ is strictly decreasing,so for any $x \in (a,b)$, $f(x)>f(a)$. It can also show tha... | The idea is very simple. Since $f$ is strictly increasing on $(a, b)$ it follows that if we have $a < x < b$ then we can choose $y, z$ such that $a < y < z < x < b$ and then $f(y) < f(z) < f(x)$. Letting $y \to a^{+}$ we get $f(a) \leq f(z) < f(x)$ and thus $f(a) < f(x)$. Similarly it can be proved that $f(x) < f(b)$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Eigenvector of polynomial
Suppose that $T: V \rightarrow V$ is an endomorphism of the linear space V (about $\mathbb{K}$) and that $p(X)$ is a polynomial with coefficients in $\mathbb{K}$. Show that if $x$ is an eigenvector of $T$ than it is also an eigenvector of $p(T)$.
My attempt:
So if $x$ is an eigenvector of $T... | Here is an outline.
Step 1: Prove by induction that if $Tx=\lambda x$, then $T^n(x)=\lambda^n x$.
Step 2: Show that if $Tx=\lambda x$ and $Sx=\mu(x)$, then $(T+S)(x)=(\lambda+\mu)x$.
Step 3: Combine the first two steps to show that if $Tx=\lambda x$, then $P(T)(x)=P(\lambda)x.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Orthogonal Projection onto the $ {L}_{\infty} $ Unit Ball What is the Orthogonal Projection onto the $ {\ell}_{\infty} $ Unit Ball?
Namely, given $ x \in {\mathbb{R}}^{n} $ what would be:
$$ {\mathcal{P}}_{ { \left\| \cdot \right\| }_{\infty} \leq 1 } \left( x \right) = \arg \min_{{ \left\| y \right\| }_{\infty} \leq 1... | Just an addition which is helpful to program this projector efficiently for example with Numpy:
The awnser
$$
({\mathcal{P}}_{ { \left\| \cdot \right\| }_{\infty} \leq 1 } \left( x \right))_i = \begin{cases}
1, & \text{if} & {x}_{i} \geq 1 \\
{x}_{i}, & \text{if} & \left | {x}_{i} \right | < 1 \\
-1, & \tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Prove that $ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $ Prove:
$$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$
Hypothesis:
$$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$
Proof:
$$ P1 | F(x) = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} + (x+1)q^x = \frac{1-(x... | $$\begin{align}
S&=1+2q+3q^2+\qquad\cdots\qquad \qquad+nq^{n-1}\\
qS&=\qquad q+2q^2+3q^3+\cdots +\quad(n-1)q^{n-1}+nq^n \\
\text{Subtracting,}&\\
(1-q)S&=1+\;\ q \ +\ q^2 +\ q^3+\cdots \qquad \qquad +q^{n-1}-nq^n\\
&=\frac {\;\ 1-q^n}{1-q}-nq^n\\
S&=\frac{1-q^n-nq^n(1-q)}{(1-q)^2}\\
&=\frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
Compute $\int_0^{\infty}\frac{x}{x^4+1}dx$ I should compute the integral $$\displaystyle \int_0^{\infty}\frac{x}{x^4+1}dx.$$
This problem appears in complex analysis in the residue theorem and its consequences of real integrals.
I know the following: Let $p,q$ be polynomials with $\deg p\leq\deg q -2$ so that $q$ has n... | An alternative way, just for fun. Through the substitution $x=\frac{1}{z}$ we have:
$$\int_{1}^{+\infty}\frac{x\,dx}{1+x^4}=\int_{0}^{1}\frac{z\,dz}{1+z^4}$$
hence:
$$\begin{eqnarray*}I=\int_{0}^{+\infty}\frac{x\,dx}{1+x^4}&=&\int_{0}^{1}\frac{2x\,dx}{1+x^4}\\&=&\int_{0}^{1}2\sum_{n\geq 0}(-1)^{n}x^{4n+1}\,dx\\&=&\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does associativity imply commutativity? I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea:
$$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ Here using associativity of addition operation, we've shown commutativity.
In gene... | Consider an $\operatorname L$ operator defined as 'take the left': $$a \operatorname L b = a$$
For any three $a,b,c$:
$$(a \operatorname L b) \operatorname L c = a \operatorname L c = a = a \operatorname L b = a \operatorname L (b \operatorname L c)$$
but
$$a \operatorname L b = a \ne b = b \operatorname L a$$
unless $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "79",
"answer_count": 7,
"answer_id": 3
} |
What is the sum of the first $17$ terms of an arithmetic sequence if $a_9=35$? What is the sum of the first $17$ terms of an arithmetic sequence if $a_9=35$?
This is what I did: $a_9=a_1+8d=35$
$S_{17}=\frac{17}{2}(a_1+a_{17})=\frac{17}{2}(a_1+a_1+16d)=\frac{17}{2}(2a_1+16d)=\frac{17}{2}\cdot 70= 595$
This solution i... | Dang, @gt6989b is right:
\begin{align}
a_1 + a_{17} &= a_1 + (a_1 + 16 d) = 2 a_1 + 16 d = 2(a_1 + 8 d) = 2 a_9 \\
a_2 + a_{16} &= (a_1 + d) + (a_1 + 15 d) = 2 a_1 + 16 d = 2 a_9 \\
& \vdots \\
a_8 + a_{10} &= (a_1 + 7d) + (a_1 + 9 d) = 2 a_1 + 16 d = 2 a_9
\end{align}
so
$$
\sum_{i=1}^{17} a_i = 8 \cdot 2 a_9 + a_9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Can spaces where all singletons are closed and all singletons are open be homeomorphic? Suppose $(X, \mathfrak{T})$ is a space where all singletons are closed, and $(Y, \mathfrak{J})$ is a space where all singletons are open.
Can these two spaces be homeomorphic? My thought is that they cannot be, but I am having a di... | The preimage of a open set is still open. And the preimage of a singleton under a bijection is still a singleton. So the singletons is open in both spaces. A space with all singletons open can only have discrete topology.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Functions validity. Why does writing a function differently make it valid for a originally invalid input?
$e.g:$
$$f(x) = \frac{1} {(\frac1x+2)(\frac1x-3)} \implies x≠0$$
Which may alternatively be written as:
$$f(x) =\frac{x^2}{(1+2x)(1-3x)}$$
Which is valid for $x=0$?
Both graphically represent the same function.
... | let $$g(x)=\frac{x^2}{(1+2x)(1-3x)}$$ and $$f(x) = \frac{1} {(\frac1x+2)(\frac1x-3)}$$
we have
$D_f=R-\{0,\frac{1}{3},\frac{-1}{2}\}$ and $D_g=R-\{\frac{1}{3},\frac{-1}{2}\}$, thus $g\ne f$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
What is the probability that a person wearing a blue t-shirt will sit next to one wearing red? 9 people sit in a row linearly. 2 dressed in Red, 3 blue and 4 in yellow. What is the probability that a person in blue will sit next to a person in red? Why?
RRBBBYYYY this sequence from what I gather can be arranged in 9! w... | Comment: (If I understand the problem properly.)
In case it is of assistance checking analytical results, here is
a simulation in R that approximates the distribution of $X,$ the number of red-blue adjacencies, for people randomly
seated in a row. The program uses a trick in assigning numbers
to colors so that the abso... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Find the Minimum of $ F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R)$. Let $F: BV(\mathbb R) \to \mathbb R$ be a functional defined as:
\[
F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R).
\]
Show that there is no minimum on $W^{1,1}$, but the infimum is exactly the mini... | First of all, $\chi_{(0,2)}$ isn't optimal; the minimum (on $BV$) is assumed for $u=\chi_{(0,\infty)}$, and $F(u)=1$.
Now let me show that $F(u)\ge 2$ for all $u\in W^{1,1}$ (so the assertion about the infimum is incorrect). Consider $F_-(u)=\int_{-2}^0 |u(x)|\, dx + |Du|(-\infty,0)$. Since the derivative exists and is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx $ $$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx$$ where $a>b>0$
What shall I do?
I have diffucty when I meet multi value function.
| Let us assume $A^3=a, B^3=b$, for simplicity. Now make a substitution $x=t^3$ which will transform the integral like this $$I=\int_{0}^\infty \frac {3t^3dt}{(t^3+A^3)(t^3+B^3)}$$ Now break this into partial fractions like this
$$I=3\int_{0}^\infty [\frac {1}{(t^3+B^3)}-\frac {A^3}{B^3-A^3}(\frac {1}{t^3+A^3}-\frac {1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Does this pattern have anything to do with derivatives? In 6th grade I was first introduced to the idea of a function in the form of tables. The input would be "n" and the output "$f_n$" would be some modification of the input. I remember finding a pattern in the function "f(n)=n^2". Here is what the table looked like:... | Yes it is related to finite calculus, but others have answered why and how while no one mentioned it is also related to sum of arithmetic sequences so I will.
Let's have arithmetic sequence where $a_1 = 1$ and $d = 2$ (so basically it's sequence of odd numbers).
Sum of first $n$ elements of that sequence (denoted as $S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 8,
"answer_id": 7
} |
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