Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Probability of extracting a ball after two balls were swapped We have $2$ boxes, the first one contains $10$ white balls and $11$ black balls. The second box contain $12$ white balls and $13$ black balls.
We swap two balls between the boxes then we extract a ball from the first box. What is the probability that the bal... | Here’s another approach. [Added: Also, see more on this approach here]
Equivalently, we can do the following.
Reach into the first box and write “S” on one ball (the one to swap).
Now choose a ball from the first box. If it is not the “S” ball, that’s your ball. The probability of this happening is $20\over21$, and th... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $6^n=6^{n+5} (mod 100)$ How can we prove that $6^n=6^{n+5} (\text{mod}~ 100)$? I tried by writing $6^{n+5}=7776 \cdot 6^n = 76 \cdot 6^n (\text{mod}~ 100)$ but this approach does not lead to the above result.
| $\color{#c00}{a\mid 1\!+\!b,\ n\geq 2}\,\Rightarrow\, (\color{#c00}a\,\color{#0a0}{b})^{\large 2}\!\mid (1\!+\!b)^{\large n+b}\!-{(1\!+\!b)^{\large n}} = \overbrace{(\color{#c00}{1\!+\!b})^{\large\color{#c00} n}}^{\Large \color{#c00}{a^{\Large 2}}\,(\cdots)}\,\underbrace{\overbrace{((1\!+\!b)^{\large b} - 1)}^{\Large ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3094035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate $\int x^x \ln x\, dx$ The integral $$\int x^x \ln x\, dx= ?$$
I know of the integral $\int x^x dx$ can be further simplified as $\int e^{x\ln x} dx$. And this requires identity to simplify. What about the product in the integral $\int x^x\ln x\,dx=\int e^{x\ln x}\ln x\, dx.$ Is there any identity to be used fo... | Define $g(x) = x^x$. Then $\ln g(x) = x\ln x$ and differentiating both sides
$$\frac{g'(x)}{g(x)}=\ln x+1,$$
which means $g'(x) = x^x(\ln x + 1)$. Now, up to a constant
$$x^x = \int g'(x)\,dx = \int x^x \ln x \,dx + \int x^x\,dx$$
thus
$$\int x^x \ln x\,dx=x^x-\int x^x\,dx. $$
| {
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Constraining the sum of Gaussian random variables? Suppose I have $n$ Gaussian random variables $X_i$ with $i=1,2,...,n$, each with zero mean $\mu=0$ and the same constant standard deviation $\sigma\neq 0$. I would like to constrain the elements collectively drawn from these distributions to satisfy
$$\sum_{i=1}^nX_i=0... | As pointed out by d.k.o. in a comment, $X_n=-\sum_{i=1}^{n-1}X_i$ represents one such solution. However, it is not homogeneous in the treatment of each $X_i$. Turns out, this can be easily fixed by considering all possible cases
$$X_j=-\sum_{{{i=1},{i\neq j}}}^nX_i$$
instead, and defining the new random variables $X'_j... | {
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General solution to linear system with general form I am trying to find the general solution of the system
$$X' = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}X$$
where $a+d \not= 0 $ and $ad-bc=0$
I find that the eigenvalues are 0 and $a+d$ with corresponding eigenvectors both $[0,0]$ which implies that the... | We are given
$$X' = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}X\\\text{where}~ ~~a+d \ne 0 , ~~ad-bc=0$$
We can find the eigenvalues using the characteristic polynomial by solving $|A - \lambda I| = 0$, yielding
$$\lambda_{1,2} = \frac{1}{2} \left(-\sqrt{a^2-2 a d+4 b c+d^2}+a+d\right),\frac{1}{2} \left(\s... | {
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Invertible Matrices within a Matrix Suppose A, B are invertible matrices of the same size. Show that
$$M = \begin{bmatrix} 0& A\\ B& 0\end{bmatrix}$$ is invertible.
I don't understand how I could show this. I have learned about linear combinations and spanning in my college class, but I don't know how that would help... | Since you mentioned linear combinations, you can also follow that path and prove that all the matrix rows are linearly independent, which is equivalent to invertibility.
For that, let $a_i$ and $b_i$ be the $i$-th rows of $A$ and $B$, respectively. Now assume that we have a linear combination of the rows of $M$. These... | {
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Show that f is non linear and g is linear I have been given the Basis B of V and C of W in Q with
$$B = \{v_1,v_2,v_3\}$$
$$C = \{w_1,w_2\}$$
And f and g with:
$$f:\mathbb{V} \to \mathbb{W}, f(k_1v_1, k_2v_2, k_3v_3)=(3k_1+k_2)w_1+k_3^7w_2$$
$$g:\mathbb{V} \to \mathbb{W}, g(k_1v_1, k_2v_2, k_3v_3)=5k_1w_1+(2k_2+7k_3)w_... | this just means that you have (for B) three linearly independent vectors that span your vector space V. It is a more general way of writing vecor spaces, you don't always have a standard basis (i.e. $\mathbb{R}^{2}=span{((1,0), (0,1))}$).
For your example you can simply handle this the exact way you would if you had be... | {
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About Definition 4.5, Theorem 4.6, Theorem 4.8 in "Principles of Mathematical Analysis" by Walter Rudin. Definition 4.5
Suppose $X$ and $Y$ are metric spaces, $E \subset X, p \in E$ and $f$ maps $E$ into $Y$. Then $f$ is said to be continuous at $p$ if for every $\varepsilon > 0$ there exists a $\delta >0$ such that... |
(Rudin's related explanation in the later chapter)
You are right here. He can simply discard the complement of $E$ in $X$, and view it as a function from $E$ to $Y$. As IEm points out, E is a metric space in its own right using the metric inherited from $X$.
| {
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Sum involving $\ln{(2)}$ I got this sum. How do can this sum be equal to $8\ln{(2)}?$
$$\sum_{n=2}^{\infty}\frac{(-1)^n}{n}\left[\frac{35n-37}{(2n-1)(n-1)^2}+\frac{35n+37}{(2n+1)(n+1)^2}\right]=8\ln{(2)}$$
I have try to expand out the sum but it is too messy. Dealing the sum in this form, I haven't got any idea.
Any he... | First note that
\begin{eqnarray*}
&&\frac{1}{n}\left[\frac{35n-37}{(2n-1)(n-1)^2}+\frac{35n+37}{(2n+1)(n+1)^2}\right]\\
&=&\frac{74}{n}+\frac2{(n+1)^2}-\frac2{(n-1)^2}+\frac{41}{n+1}+\frac{41}{n-1}-\frac{156}{2n+1}-\frac{156}{2n-1}
\end{eqnarray*}
and
\begin{eqnarray*}
&&\sum_{n=2}^{\infty}\frac{(-1)^n}{n-1}=\ln2,\sum... | {
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For which integer $n$ is $28 + 101 + 2^n$ a perfect square? This question
For which integer $n$ is $$28 + 101 + 2^n$$ a perfect square. Please also suggest an algorithm to solve similar problems. Thanks
Btw, this question has been taken from an Aryabhatta exam for 8th graders in India.
| One way is to notice that
$28+101+2^n=128+1+2^n=2^7+1+2^n=2^n+2*2^6+1$.
Therefore, if we let $n=12$ we have
$2^{12}+2*2^6+1=(2^6)^2+2*2^6+1=(2^6+1)^2$
Which is a perfect square, as desired.
This relies on noticing this specific pattern, so doesn't lend itself to a general method. One could always guess and check.
| {
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Can we find two non-congruent right triangles with whole-number lengths and congruent hypotenuses? I know some ways to find some Pythagorean triples.
And I understand that if $a^2 + b^2 = c^2$ then $(a-b)^2 + (a+b)^2 = 2c^2$.
I feel like that suggests a way forward, but I cannot find that way.
Is there an algorithm to ... | Let $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$ be arbitrary non-proportional pythagotrean triples, and let $d:={\rm gcd}(c_1,c_2)$. Then
$${c_2\over d}\left(a_1,b_1,c_1\right),\qquad{c_1\over d}\left(a_2,b_2,c_2\right)$$
are pythagorean triples to noncongruent right triangles with the same hypotenuse $c={c_1c_2\over d}$.
| {
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$\int_{0}^{1} t^2 \sqrt{(1+4t^2)}dt$ solve $$\int_{0}^{1} t^2 \sqrt{(1+4t^2)}dt$$
my attempt
$$t = \frac{1}{2}\tan(u)$$
$$dt = \frac{1}{2}\sec^2(u)du\\$$
$$\begin{align}
\int_{0}^{1} t^2 \sqrt{(1+4t^2)}dt&=\int_{0}^{1} \frac{\tan^2(u)}{4} \sqrt{1+\tan^2(u)}\frac{1}{2}\sec^2(u)du\\
&=\frac{1}{8}\int_{0}^{1} \tan^2(u)\se... | Hint. I would say
$\int \sec^5u\,du = \int\frac{\cos u\,du}{\cos^6u}=\int\frac{\cos u\,du}{(1-\sin^2u)^3}$ and after substitution
$\sin u = v,\quad \cos u \,du = dv$
it will be an integral part of the rational function.
Edit:
I will add: The given integral is the so-called binomial integral with a specific solution. S... | {
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Simplifying $\frac{ (2x+1) (x-3) }{ 2x^3 (3-x) }$ In simplifying
$$\frac{2x^2-5x-3}{6x^3-2x^4}$$
I got this far
$$\frac{ (2x+1) (x-3) }{ 2x^3 (3-x) }$$
but there aren't same brackets to cancel out.
| $$\dfrac{2x^2-5x-3}{6x^3-2x^4}=\dfrac{(2x+1)(x-3)}{2x^3(3-x)}=\dfrac{(2x+1)(x-3)}{2x^3[-(x-3)]}=\dfrac{(2x+1)(x-3)}{-2x^3(x-3)}=\dfrac{2x+1}{-2x^3}$$
| {
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A homeomorphism $T$ from extended complex plane to itself preserving cross ratio is a Mobius map. Cross ratio preserving means $(Ta,Tb,Tb,Td)=(a,b,c,d)$ where $(a,b,c,d)=\dfrac{(a-b)(c-d)}{(a-d)(c-b)}$.
If we assume $T$ fixes infinity, can we prove $T$ is affine?
| My answer needs two facts about Möbius transforms:
*
*They preserve cross ratios (this can be checked by a direct calculation).
*A Möbius transform can map three arbitrary points to any other three arbitrary points (this is not completely obvious, but at least very intuitive since Möbius transforms have 3 independe... | {
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How do i calculate $\lim_{h\to0}\frac{f(a+h^2)-f(a+h)}{h}$? All do i know about this problem is that f can be derived in "a".
What troubles me is the h squared,i just can't get rid of it or make it useful,no matter what i do, i always end up with it giving me an undefined limit, so it stays like that,any idea on how to... | Hint:$$\lim_{h\to0}\frac{f(a+h^2)-f(a)}h=\lim_{h\to0}h\frac{f(a+h^2)-f(a)}{h^2}=h\times f'(a)=0.$$
| {
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Logical equivalences for $P \implies Q$ and $\neg Q \implies \neg P$ This is probably quite a basic question, but it's something I'm having trouble wrapping my head around.
I came across a proof which employed the following strategy
The goal was to prove
$$P \implies Q$$
And the strategy used was to prove the followin... | In classical logic this boils down to the definition of $\implies$. You can then check this equivalence by testing all possible combinations of truth values for $P$ and $Q$. This principle is called contraposition.
In intuitionistic logic, you only get that $P\implies Q$ implies $\lnot Q\implies \lnot P$, but not the ... | {
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Linear transformations with the same kernel If two linear transformations between finite-dimensional vector spaces, we say
$$L,T : V \longrightarrow W$$
has the same kernel, and kernel is not the space zero. Is $T$ a multiple of $L$?
| Hint: Take square matrices of order $2$ like $A=[a_{ij}]$ where $a_{11}=1, a_{12}=a_{21}=a_{22}=0$ and $B=[b_{ij}]$ where $b_{21}=1, b_{11}=b_{12}=b_{22}=0$
| {
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Let $a,b,n$ be positive integers, if $n|a^n-b^n$ then $n|\frac{a^n-b^n}{a-b}$ Let $a,b,n$ be positive integers, if $n|a^n-b^n$ then prove that $n|\frac{a^n-b^n}{a-b}$.
My approach: If $n$ is a prime, write $n=p$, then since $$a^p\equiv b^p\pmod p$$
and by Fermat's little theorem, we have $$a^p\equiv a, b^p\equiv b\pmod... | Suppose $(n, a-b)=d$ is the highest common factor. Use the $a=b+kd$ and the binomial expansion to show that $d^2|a^n-b^n$.
| {
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Function such that $f^{(n)}(x) = \frac{x}{f(x)^n}$
Let $n$ be a fixed positive integer. Find all function $f:(0, \infty) \to \mathbb{R}$ that can be differentiated $n$ times such that $f^{(n)}(x) = \frac{x}{f(x)^n}$ if $f^{(n)}(x)$ is the $n$-th derivative of $f$.
I tried to differentiate the given identity and I wro... | Not an answer
(I just didn't have enough room to put all this in a comment.)
If you take $f^n(x)=(f\circ f \cdots \circ f) (x)$ as definition, may something nicer happen? For example, consider $n=2$ and restrict yourself to the case
$$f(x) = Kx^\alpha,$$
$\alpha,K \in \Bbb R$ and $x \in \Bbb R^+$, so that
$$f''(x)=K\al... | {
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Complex operator $i$ and Exponents I am trying to understand the complex numbers and exponents.
I came across this question. I wonder how to explain the difference between $${2\cdot i} \text{ and } 2^i$$ as $i=\sqrt{-1}$
edit:
Rather than explaining the meaning of above two numbers in yet another equally difficult math... | $$2^i=\mathrm e^{i\ln 2}=\cos(\ln 2)+i\sin(\ln 2)\ne 2i. $$
| {
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A variant of submodularity A function $f: \mathbb{R}^2 \to \mathbb{R}$ is said to be submodular if for all $x,y \in \mathbb{R}^2$ it holds
$$
f(x \vee y)+f(x \wedge y)\le f(x)+f(y).
$$
In particular, if $x_1 \ge y_1$ and $x_2 \le y_2$, this means that
$$
f(y_1,x_2)+f(x_2,y_1) \le f(x_1,y_1)+f(x_2,y_2),
$$
or, equivalen... | I don't know about the functions that you've described here, but a related concept is something termed "continuous submodlarity", which is a way to extend the notions of submodularity to $\mathbb{R}^n$. The idea is to use the lattice on $\mathbb{R}_n$ obtained by component-wise partial ordering.
For $i=1, \dots n$, le... | {
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Find sum $u_0 u_1 + u_1u_2+...+u_{n-2}u_{n-1} $ I have
$$ u_k = \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n}$$
And I should calculate:
$$ u_0 u_1 + u_1u_2+...+u_{n-2}u_{n-1}+u_{n-1}u_0 $$
But I have stucked:
Firstly I calculate
$$u_0 u_1 + u_1u_2+...+u_{n-2}u_{n-1} $$
and put
$$ \alpha_k = u_k \cdot u_{k+1} = ... = e... | Note that $$\alpha_k = u_{k}\cdot u_{k+1} = e^{i\frac{2\pi}{n}(2k+1)} = e^{i\frac{2\pi}{n}}\cdot e^{i\frac{4\pi}{n}k}.$$
Therefore,
$$\sum_{k=0}^{n-2}{\alpha_k} = e^{i\frac{2\pi}{n}}\sum_{k=0}^{n-2}{\left(e^{i\frac{4\pi}{n}}\right)^{k}} = e^{i\frac{2\pi}{n}} \frac{1-e^{i\frac{4\pi(n-1)}{n}}}{1-e^{i\frac{4\pi}{n}}} = e^... | {
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If $G$ is a group and $a,b\in G$ and $e$ neutral element why $ab = aeb$? The Problem I have if this would be true then one could also say $aaa^{-1}b=ab$.
Why do I know that there don't exist counterexamples for which the inequality holds?
| $aaa^{-1}b = a(aa^{-1})b = aeb = (ae)b = ab$
So yes, you CAN say that, and you'd be one hundred percent correct when you do say that.
Why do I know that there don't exist counterexamples for which the inequality holds?
The same way you know that there aren't any counter examples to $k\times \frac 1k =1$ when $k \ne 0... | {
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Proof that for every $z \in \mathbb C$: $z=\frac{2}{n}\sum_{k=0}^{n-1}(Re(u_{k}\overline z)u_{k})$
Let $n \in \mathbb N$, $n>2$ and $u_{1},u_{2},...,u_{n-1}$ are all root of unity for n. Proof that for every $z \in \mathbb C$: $$z=\frac{2}{n}\sum_{k=0}^{n-1}(Re(u_{k}\overline z)u_{k})$$
I know that $Re(u_{0})=Re(u_{... | Note that $$\frac{2}{n}\sum_{k=0}^{n-1}(Re(u_{k}\overline z)u_{k}) = \frac{2}{n}\sum_{k=0}^{n-1}\frac{u_{k}\overline z+\overline{u_{k}}z}{2}u_{k} =\frac 1n (\overline z\sum_{k=0}^{n-1}u_k^2+z\sum_{k=0}^{n-1}|u_k|^2)$$
Using a geometric argument, a geometric sum, or Newton's identities it's easy to show $\sum_{k=0}^{n-1... | {
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Killing cohomology of surfaces in finite covers Let $S$ be a closed orientable surface and $R$ a commutative ring. Given a nonzero element $\alpha \in H^1(S;R)$ is there a finite cover $p : \tilde{S} \to S$ such that $p^*(\alpha) = 0 \in H^1(\tilde{S}; R)$?
| No. This has nothing to do with $S$ being a surface. In what follows all we assume is that $S$ can run the classification theory of covers (path-connected, locally path-connected, semilocally simply connected). It holds for any CW complex.
There is a natural isomorphism $H^1(S;G) \cong \text{Hom}(\pi_1 S, G)$ for any $... | {
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Integral $\int\frac{2x^2}{2x\cos(2x)+(x^2-1)\sin(2x)} \mathrm d x$
Integrate $\displaystyle\int\dfrac{2x^2}{2x\cos(2x)+(x^2-1)\sin(2x)} \mathrm d x$
I tried dividing by $\cos^2(x)$ and then substituting $\tan(x)=t$.
| $$2x\cos (2x)+(x^2-1)\sin 2x=(x^2+1)\bigg[\frac{2x}{x^2+1}\cos 2x+\frac{x^2-1}{x^2+1}\sin 2x\bigg]$$
$$=(x^2+1)\cos\bigg(2x-2\alpha\bigg)$$
$\displaystyle \sin(2\alpha)=\frac{x^2-1}{x^2+1}$ and $\displaystyle \cos(2\alpha)=\frac{2x}{x^2+1}$
integration $$=\int\sec\bigg(2x-\tan^{-1}\frac{x^2-1}{2x}\bigg)\frac{2x^2}{x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3097078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$
Calculate $$\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2}dx$$
I have tried to put $\displaystyle x=\frac{1}{t}$ and $\displaystyle dx=-\frac{1}{t^2}dt$
$$ \int^{\infty}_{0}\frac{t^2\ln^2(t)}{(t^2-1)^2}dt$$
$$\frac{1}{2}\int^{\infty}_{0}t\ln^2(t)\frac{2t}{(t^2-1)... | You're definetly on the right track with that substitution of $x=\frac1t$
Basically we have: $$I=\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2}dx=\int_0^\infty \frac{x^2\ln^2 x}{(1-x^2)^2}dx$$
Now what if we add them up?
$$2I=\int_0^\infty \ln^2 x \frac{1+x^2}{(1-x^2)^2}dx$$
If you don't know how to deal easily with the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3097187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
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Show that the projection moves you closer to the set, i.e. $|P_C(x) - y| \leq |x-y|$ for all $y\in C$ Let $H$ be a Hilbert space with induced norm $|\cdot|$, with $C\subset H$ a closed, convex subset of $H$ and $P_C(x)$ the projection of $x$ onto $C$, i.e.,
$$P_C(x) = \arg\min\limits_{y\in C}|x-y|^2$$
I want to show th... | First $z = P_C(x)$ if and only if
$$
\langle z - x, y-z\rangle \ge0 \quad \forall y\in C.
$$
This follows from
$$
\frac12|y-x|^2 - \frac12|z-x|^2 = \langle z-x, y-z\rangle + \frac12 |y-z|^2.
$$
Now take $x_1,x_2$, then
$$
\langle P_C(x_1) - x_1, P_C(x_2)-P_C(x_1)\rangle + \langle P_C(x_2) - x_2, P_C(x_1)-P_C(x_2)\rang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3097393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Complex Infinite Series Having trouble with this infinite series and deciding whether it converges or diverges.
The series:
$$\sum_{n=1}^\infty n(\frac{1}{2i})^n$$
My thoughts are that you take the modulus of the fraction and get $\frac{1}{2}$ to the exponent $n$ makes it go to $0$ and then multiplied by $n$ make it
$... | First let’s look if the series converges absolutely.
For this, we need to see if $\sum b_n = \sum \frac{n}{2^n}$ converges. And this is immediate using the ratio test
as $\lim\limits_{n\to \infty}\frac{b_{n+1}}{b_n} =1/2<1$.
Conclusion: the given series converges absolutely hence converges
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3097485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Angle bisector in triangle, quick question: $|AE| = \frac{bc}{a+c}$ Triangle $ABC$; $AB=c, BC=a, AC=b$; angle bisector of angle $(c, a)$ cuts $AC$ in point $E$.
Why is the following true?
$$|AE| = \frac{bc}{a+c}$$
Where does that come from?
| The angle bisector theorem:
$$\frac{CE}{AE}=\frac ac$$
Then:
$$\frac{bc}{a+c}=\frac{b}{\frac ac+1}=\frac{b}{\frac{CE}{AE}+1}=\frac{AE\cdot b}{CE+AE}=AE$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3097573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\displaystyle 1+ \frac{\log_2(2/3(n+1))}{\log_2(3/2)} = \frac{\log_2(n+1)}{\log_2(3/2)}$ Show $\displaystyle 1+ \frac{\log_2(2/3(n+1))}{\log_2(3/2)} = \frac{\log_2(n+1)}{\log_2(3/2)}$
from LS
$\displaystyle 1+ \frac{\log_2(2/3(n+1))}{\log_2(3/2)} = \frac{\log_2(3/2) + \log_2(2/3(n+1))}{\log_2(3/2)} = \frac{\log_2\big(... | I would write $$\log_{2}\frac{3}{2}+\log_{2}\frac{2}{3}+\log_{2}(n+1)=\log_{2}(n+1)$$ since $$\log_{2}{\frac{2}{3}}=-\log_{2}{\frac{3}{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3097695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Behaviour of positive function on compact sets Let $v$ be any strictly positive bounded function on $\Omega \subset \mathbb{R}^n$ and zero on $\partial \Omega.$
Can we say that for each compact subsets (w.r.t usual topology defined on $\mathbb{R}^n $) of $\Omega$ there exists a constant $c$ such that $v \geq c >0$?
If... | Let $n=1$, $\Omega =(-1,1)$, $u(\frac 1 n)=\frac 1 n$ for $n=1,2,...$ and $u(x)=1$ for all other $x \in \Omega$, $u(x)=0$ for $x =\pm 1$. Consider the compact set $[0,\frac 1 2]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3097828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Formal Deduction (logic) Question: $\lnot C, (B \to \lnot C) \to A \vdash (A \to C) \to F$ I've been stuck on this question for around two hours now.
I'm trying to prove that:
$\lnot C, \ (B \to \lnot C)\to A \vdash (A \to C)\to F $
I'm trying to get my second last step to be:
$\lnot C, \ (B \to \lnot C)\to A, \ A \to... | You can show that $((B\to \neg C)\to A) \iff ((B\land C)\lor A)$.
Now adding $A \to C$ to the hypotheses and using disjunction of cases:
\begin{array}{l}
A \\
A \to C \\
\hline
C
\end{array}
Which considering $\neg C$ is absurd. Also:
\begin{array}{l}
B\land C \\
\hline
C
\end{array}
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there 'Algebraic number' which cannot display with Arithmetic operation and root Let $a + bi$ be an algebraic number.
Then there is polynomial which coefficients are rational number and one of root is $a+bi$.
I think..
$$x = a + bi$$
we can subtract $c_1$ (which is rational number) from both sides.
$$x-c_1=a-c_1+bi$... | The algebraic numbers are divided into the explicit algebraic numbers and the implicit algebraic numbers.
The explicit algebraic numbers can be presented explicitly from the rational complex numbers by the arithmetic operations (addition, subtraction, multiplication, division, raising to integer powers, and $n$-th root... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Behavior of $f(x)=(-1)^x$
For what $x$ is $f(x)=(-1)^x$ a real number, and when is it a complex number?
When I graph it online, the graph glitches out and has points all over the place.
| When $-1$ is raised to power of $x$ the complex number so formed has a real part $\cos \pi x $ and imaginary part $\sin \pi x.$ This can be found using Euler formula $ e^{i \pi}=-1$. The two parts are graphed below .. with each part having a period or wave length $\lambda=2, $ the roots are at $x=..,-2,-1,0,1,2,...$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Geometrical proof for length of chord passing through vertex of parabola In a parabola $y^2=4ax$ , the length of focal chord making an angle $\theta$ with the x - axis is $4acosec^2\theta$ . If a chord is drawn parallel to that focal chord which passes through vertex of parabola at (0,0) , it's length comes out to be $... | Changing the notation a bit, let $\overline{AB}$ be a chord through the parabola's focus, $F$, and let $\overline{UV}$ be a parallel chord through the vertex, $V$. Let $\overleftrightarrow{A^\prime B^\prime}$ be the parabola's directrix, and let $C$ be the fourth vertex of rectangle $\square AA^\prime B^\prime C$.
Writ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Relation of complete homogeneous symmetric polynomials and the elementary symmetric polynomials I was reading about the symmetric polynomials and saw the following relation:
$$\sum _{{i=0}}^{m}(-1)^{i}e_{i}(X_{1},\ldots ,X_{n})h_{{m-i}}(X_{1},\ldots ,X_{n})=0\text{ for } m>0$$
The proof is constructed by using a gen... | We have
$$h_p(X_1,\ldots,X_n)=\sum h_r(X_1,\ldots,X_k)h_{p-r}(X_{k+1},\ldots,X_n)$$
$$e_p(X_1,\ldots,X_n)=\sum e_r(X_1,\ldots,X_k)e_{p-r}(X_{k+1},\ldots,X_n)$$
$$\sum_{i=0}^m\sum_{p=0}^i\sum_{q=0}^{m-i} (-1)^ie_{i-p}(X_1,\ldots,X_k)h_{(m-p-q) -(i-p) }(X_1,\ldots,X_k)e_p(X_{k+1},\ldots,X_n)h_q(X_{k+1},\ldots,X_n)$$
If w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Need help to decide if $\{x \in \mathbb R^n : 1 \leq x_1^2 + x_2^2 + \cdots + x_n^2 \leq 2 \}$ is convex
Is the following set convex? $$\{x \in \mathbb R^n : 1 \leq x_1^2 + x_2^2 + \cdots + x_n^2 \leq 2 \}$$
I did the following.
Assume $1≤x_1^2+x_2^2+...+x_n^2≤2$ and $1≤y_1^2+y_2^2+...+y_n^2≤2$
Assume $z=αx+(1-α)y,... | The set is not convex because $(\pm1,0,0,\ldots,0)$ belong to it, but $(0,0,0,\ldots,0)$ doesn't. Note that$$(0,0,0,\ldots,0)=\frac12(1,0,0,\ldots,0)+\frac12(-1,0,0,\ldots,0).$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\lim_{(x,y)\to(0,0)}(xy+y^{3})=0$.
Prove that $\lim_{(x,y)\to(0,0)}(xy+y^{3})=0$.
I am trying to determine how to set the $\delta$. Here is my rough work, which isn't much:
$|f(x,y)-0|=|xy+y^{3}|\leq |y||x+y^{2}|$
I am not sure whether I should separate $y$ or separate $xy$ and $y^{3}$ to make it $|xy| + ... | Hint
For $\vert x \vert, \vert y \vert \le 1$, you have
$$\vert xy+y^3 \vert \le \vert x \vert \vert y \vert + \vert y \vert^3 \le \vert x \vert \vert y \vert + \vert y \vert\le 2 \vert y \vert$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Proving Symmetric Difference of A and B Let A and B be sets. Define the symmetric difference of A and B as A∆B= (A ∪ B) − (A ∩ B).
(a) Prove that A∆B = (A − B) ∪ (B − A)
I tried to start this but am getting really lost. if someone could try to help that would be great
| Using that $A - B = A \cap B^C$:
$A \Delta B = $
$(A \cup B) - (A \cap B) = $
$(A \cup B) \cap (A \cap B)^C =$
$ (A \cup B) \cap (A^C \cup B^C) =$
$ (A \cap (A^C \cup B^C)) \cup (B \cap (A^C \cup B^C)) = $
$(A \cap B^C) \cup (B \cap A^C) = $
$(A - B) \cup (B - A)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How to calculate $\int_{0}^{\pi/3} \sqrt{\sec ^2 (x)} dx$? I'm trying to calculate the following integral:
$\int_{0}^{\pi/3} \sqrt{\sec^2 (x)} dx$
But I have no idea where to start. Can you give me some advice?
| As on the integration interval, $\sec x>0$, we may simplify to
$$\int_0^\tfrac\pi3\sec x\,\mathrm dx=\int_0^\tfrac\pi3\frac{\mathrm dx}{\cos x}=\int_0^\tfrac\pi3\frac{\cos x\,\mathrm dx}{\cos^2 x}.$$
Can you continue?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Solving $f'(x)=f(x+1)$ I was wondering if it was possible to find functions $f$ such that
$$
f'(x)=f(x+1)
$$
for all $x \in \mathbb{R}$. The only thing i've found so far is that it implies
$$
f^{\left(n\right)}\left(x\right)=f\left(x+n\right)
$$
is there any way to solve this ?
| By the Fourier transform,
$$2i\pi\xi F(\xi)=e^{2i\pi\xi}F(\xi).$$
Then $F(\xi)$ is only nonzero for the roots of $2i\pi\xi=e^{2i\pi\xi}$ and the spectrum is discrete. But the equation has no real roots in $\xi$ !
| {
"language": "en",
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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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showing a relation on $\mathbb Z$ \ $0$ is an equivalence relation We define a relation on $\mathbb Z \setminus {0}$ where a ~ b iff $0< ab$. How would you show this is an equivalence relation and describe the equivalence classes?
| I'm using the notation $a \simeq b$ for the relation since I can't get the $\LaTeX$ for the "tilde" sign to work right at the moment.
$ab > 0 \tag 1$
if and only if the signs of $a$ and $b$ are the same; thus
$a \simeq a, \tag 1$
since $a$ has the same sign as itself;
$a \simeq b \Longrightarrow b \simeq a, \tag 2$
sin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the density of $Y = a/(1 + X^{2})$, where $X$ has the Cauchy distribution. Find the density of $Y = a/(1 + X^{2})$, where $X$ has the Cauchy distribution.
MY SOLUTION
To begin with, let us remember that the Cauchy probability density function is given by
\begin{align*}
f_{X}(x) = \frac{1}{\pi(1+x^{2})}\quad\text{f... | A different way to approach this is to represent a Cauchy r.v. $X$ as the ratio of two independent $N(0,1)$ r.v.s. Once you write $X=S/T$, with $S,T\sim N(0,1)$ the representation $Y=aT^2/(S^2+T^2)$ drops out. And so on...
This method relies on "pattern matching" (you have to `know the ratio of Gaussians fact, you ha... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that every operator G such that GT = TG is a polynomial in T Let be $T:V \rightarrow V$ ($\dim V \lt \infty$) a diagonalizable operator with the algebraic multiplicity of eigenvalues $1$. Then every operator $G$ such that $GT = TG$ is a polynomial in $T.$
My attempt:
I Know that if $GT = TG$ then the eigenspac... | $T$ is diagonalizable. So it is easiest if just change basis to one where $T$ is diagonal.
All eigenvalues have multiplicity $1$, so all diagonal entries are distinct. This means that any matrix which commutes with $T$ is also diagonal in this basis (so the subspace of endomorphisms $V\to V$ which commutes with $T$ has... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Analytically determine if $f(x) = f'(x)$ is possible? I was taking a test and two true/false type questions were asked.
In one of them, I had to say if there is a function $f(x)$ such that $f(x) = f'(x)$. Of course, $e^x$ is such a function and almost everyone who has taken a calculus course knows this fact well.
In th... |
My question is: is there an analytical way to determine if such a function exists?
There's a theorem for that. Specifically, the existence-uniqueness theorem for differential equations.
Wikipedia link
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3099641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Equation in the complex field $(z+2)^6=z^6$ I'm wondering why if $z$ is a solution of the equation
$$
(z+2)^6=z^6
$$
then we must have $\Re(z)=-1$. I've tried to take the real part of both sides, noticing that
$$
\Re((z+2)^6)=\Re([(z+2)^3]^2)=|(z+2)^3|^2
$$
but it doesn't seem to work. Thank you in advance.
| Doing $z = w - 1$:
$$0 = (z + 2)^6 - z^6 = 4w(3w^4 + 10w^2 + 3),$$
$w = 0$ ($z = -1$) is obviously solution and the biquadratic factor $3w^4 + 10w^2 + 3$ has only purely imaginary solutions because $w^2 = -3,-1/3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3099836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Find rational numbers $\alpha $ and $\beta$ in $\sqrt[3]{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$ How should we find two rational numbers $\alpha$ and $\beta$ such that
$$\sqrt[3]{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$$
The answer I got $\alpha = 1 $ and $\beta = 1$. If I'm wrong, please correct me. Thank you
| By Gauss' lemma, if $\alpha$ and $\beta$ are rational, then they are integers. Simply cube both sides to get
$$7+5\sqrt{2}=(\alpha+\beta\sqrt{2})^3.$$
Expanding the right hand side and comparing coefficients shows that
\begin{eqnarray*}
7&=&\alpha^3+6\alpha\beta^2&=&\alpha(\alpha^2+6\beta^2),\\
5&=&3\alpha^2\beta+2\bet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3099920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Homeomorphism between sets Let $$ Y = \left\{ \frac{1}{2} + \frac{1}{n} \mid n \in \mathbb{N}\right\}\cup\left\{ \frac{1}{2} - \frac{1}{n} \mid n \in \mathbb{N}\right\}$$ and
$$ X = \left\{ \frac{1}{n} \mid n \in \mathbb{N}\right\} $$ be subspaces of the Euclidean space $ \mathbb{R} $.
Are $X$ and $Y$ homeomorphic (i.e... | Your topologies are discrete, so every map is continuous. Therefore every bijection between $\mathbb{Z}^*$ and $\mathbb{N}$ will do.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3100003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Find Basis and Dimension of specific subspace of $M_{2 \times 2}$. Let $V=M_{2 \times 2}(\mathbb{R})$ be the set of all $2 \times 2$ real-valued matrices, and let the field $K = \mathbb{R}$. Then $V$ is a vector space under matrix addition and Scaler multiplication.
Let $$A = \begin{bmatrix}1&0\\1&2\end{bmatrix}$$
If ... | To show that $C(A)$ is a subspace, we need to show it closed under both addition and scalar multiplication; so let
$B_1, B_2 \in C(A); \tag 1$
then
$B_1A = AB_1, \; B_2A = AB_2; \tag 2$
thus,
$(B_1 + B_2)A = B_1A + B_2A = AB_1 + AB_2 = A(B_1 + B_2), \tag 3$
which of course implies
$B_1 + B_2 \in C(A); \tag 4$
likewise ... | {
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"url": "https://math.stackexchange.com/questions/3100113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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notation for this summation I have a summation of the product of two variable like this $v_{j}f_i$ where
$j=1,2,\dots,d,d+1,\dots, 2d,\dots, nd$ and after each $d$ step $i$ changes from $1$ to $n$, is my notation following correct in this situation?
$$\sum_{j=1}^{nd}\sum_{i=1}^{n} v_j f^{(i)}?$$
what I mean is I want $... | So you have
$$
\begin{split}
S &= (v_1+\dots+v_d)f^{(1)}+(v_{d+1}+\dots+v_{2d})f^{(2)}+\dots \\
&= f^{(1)}\sum_{k=1}^d v_k + f^{(2)}\sum_{k=d+1}^{2d} v_k + \ldots \\
&= \sum_{i=0}^n f^{(i+1)}\sum_{k=di+1}^{d(i+1)} v_k
\end{split}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3100212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Eigenvalues of a matrix whose square is zero Let $A$ be a nonzero $3 \times 3$ matrix such that $A^2=0$. Then what is the number of non-zero eigenvalues of the matrix?
I am unable to figure out the eigenvalues of the above matrix.
P.S.: how would the answer change if it were given that $A^3=0$?
| Another approach is this one:
Since $A^2 = 0$, the polynomial $g(x) = x^2$ annihilates A (and this means that the Linear operator defined by $g(T)$ is the null operator). However, the minimal polynomial of A must divide every polynomial that annihilates $A$, so if $m(x)$ is such polynomial, $m$ must divide $g$.
Hence,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3100338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Ideas on calculating the volume of $K:=\{(x,y,z)\in \mathbb R^{3}: x^2+y^2+z^2\leq 1, x^2+y^2\leq a\}$ Calculate the volume of $K:=\{(x,y,z)\in \mathbb R^{3}: x^2+y^2+z^2\leq 1, x^2+y^2\leq a\}$ and note that $0 < a < 1$
My ideas:
depending on the size $a$, I have a cylinder whose height is restricted by the radius of... | This is a solid of revolution so can be found using the cylindrical shell method
$$2\int_0^{\sqrt{a}}2\pi rh\,dx$$
where $r=x$ and $h=\sqrt{1-x^2}$
Note that this is an elementary integral and you will get the result
$$ V=\frac{4\pi}{3}\left[1-(1-a)^{3/2}\right] $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3100424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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S is a dense subset of $L^{p'}$, $\int_{E}fg = 0$ for all $f \in S$, then $g= 0$ Problem:
$E$ is a measurable set and $1 \leq p < \infty$. Let $p′$ be the conjugate of $p$, and $S$ is a dense subset of $L^{p′}(E)$. Show that if $g \in L^p(E)$ and $\int_{E}fg = 0$ for all $f \in S$, then $g= 0$.
Definition of Density:
... | This problem was probably designed for you to use duality, namely the fact that if $1 \leq p < \infty$ and $g \in L^{p}(E)$, then
$$
\| g \|_{L^p(E)} = \sup_{\| h\|_{L^{p’}(E)}=1} \left| \int_E gh \right|.
\tag{1}
$$
(See, e.g. Grafakos Volume I, Chapter 1.)
We will henceforth use $\| \cdot \|_p$ and $\| \cdot \|_{p’}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3100678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Proving $\frac{n}{2n+1} < \sqrt{n^2+n} -n < \frac{1}{2}$
I would like to prove using Mean Value theorem for $n \ge 1$
$$\frac{n}{2n+1} < \sqrt{n^2+n} -n < \frac{1}{2}$$
RHS can be proved by rationalizing the square root term, not sure about the LHS.
| Why MVT? For the LHS you are asking about you have
\begin{eqnarray*} \sqrt{n^2+n} -n
& = & \frac{n^2 + n - n^2 }{\sqrt{n^2+n} + n} \\
& \color{blue}{>} & \frac{n}{\sqrt{n^2+2n+1}+n} \\
& = & \frac{n}{\sqrt{(n+1)^2}+n} \\
& = & \frac{n}{2n+1} \\
\end{eqnarray*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3100809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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How can we find this limit $\lim_{n\to\infty \\x\to\infty}f^n(x)$? Let $f:\mathbb{R}\rightarrow \mathbb{R}$ , $f(x)=\frac{ax+b}{cx+d}$ and $a,b,c,d>0$ then $f^1(x)=f(x), f^2(x)=f(f(x)), f^3(x)=f(f(f(x)))$ and $f^n(x)={f(f(f\cdots f(x)\cdots )))}$, where $ f^n(x)$ is the $ n $ composition of the function $f(x).$
It's ... | One approach is using linear algebra: For a matrix $ M = \begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}$, put $f_M(x)=\frac{\alpha x+\beta}{\gamma x+\delta}$. Then direct computation shows that
$f_M(f_N) = f_{MN}$.
In your case, $f = f_A$, and $f^n=f_{A^n}$, with $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Fro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3101080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Do the eigenvalues of a matrix need to be in the same field as the elements of the matrix? Do the eigenvalues of a matrix need to be in the same field as the elements of the matrix? I believe they do not since, for instance, I was taught that the trace is the sum of the eigenvalues and this would mean that this is not ... | That depends upon the context. If, say, you have a $n\times n$ real matrix and and someone asks you what are its eigenvalues, then it is implicit that you should provide the real eigenvalues. And perhaps that there is none. On the other hand, it may be implicit that, in fact, you are after the complex eigenvalues.
On t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3101280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Exchanging limits when functions may not converge uniformly
Let $X$ be a compact metric space and for each $i \geq 1$, let $f_i \colon X \to \mathbb{N}$ be continuous functions satisfying:
*
*$f_{i+1}(x) \geq f_i(x)$ and;
*for each $n \in \mathbb{N}$ there exists an $x \in X$ and $j\geq 1$ such that $f_j(... | I don't think it is true this general:
Consider $X=\{0\}\cup\{\frac {1}{n}:n\in\mathbb {N\}}$ and define $f_n(x)=\frac {1}{x}$ for $x\geq \frac {1}{n}$ and zero otherwise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3101370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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If the ground field has characteristic $p$ then every line through the origin is a tangent line to the curve $y = x^{p+1}$ I tried simple example $F_{2}$ and $y=x^3$. The there are only 2 points (0,0) and (1,1). Then how to prove that every line through the origin is a tangent line?
| Suppose $L$ is a line through the origin, so $L$ is given by the equation $y=mx$ for some $m$. The intersection of $L$ with the curve $y=x^{p+1}$ is $mx=x^{p+1}$ which is equivalent to $x(x^p - m)=0$, or $x(x-\sqrt[p]{m})^p=0$ because we are in characteristic p. By looking at the zeros of this equation, the intersectio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3101525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove that spaces $C^k(\mathbb{R}^n)$ and $C^\infty(\mathbb{R}^n)$ are infinite dimensional.
Prove that spaces $C^k(\mathbb{R}^n)$ and $C^\infty(\mathbb{R}^n)$ are infinite dimensional.
So in order to prove that they're infinite dimensional spaces, I need to form a linearly independent sequence which is infinite. W... | Yes and no. Yes, because they form an infinite linearly independent family, and that is enough to prove what you want to prove. And no because they do not form a basis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3101603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is there a "greatest function" that converges? We just hit convergence tests in calculus, and learned that $\sum_{n=1}^{\infty} \frac{1}{n^p}$
converges for all $p \gt 1$. I thought that this was sort of a "barrier" between what converges and what diverges. Specifically, that setting $a_n=\frac{1}{n^{1+\epsilon}}$ is ... | 1) $$f(n) = \frac{1}{n \log(n)^2}$$
2) No. Given any $g > 0$ such that $\sum_n g(n)$ converges, there is an increasing sequence $M_k$ such that $$\sum_{n \ge M_k} g(n) < 2^{-k}$$
Then
$ \sum_n g(n) h(n)$ converges, where $h(n) = k$ for $M_k \le n < M_{k+1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3101694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 2,
"answer_id": 0
} |
Weak solution for Burgers' equation I have the following IVP:
$$
u_t + u u_x = 0,\qquad
u(x,0) = \left\lbrace
\begin{aligned}
&0 && \text{if } x<-3 \\
&0.5 && \text{if } {-3}<x<-2 \\
&1 && \text{if } {-2}<x<0 \\
&1-x && \text{if } 0\leq x<1 \\
&0 && \text{if } x\geq 1
\end{aligned}\right.
$$
Questio... | Since your initial data are discontinuous and also decreasing on part of the domain, your solution contains rarefaction waves and shock waves. The rarefaction waves appear at $x=-3$ and $x=-2$ at time $t=0$ and the shock appears at some positive time when the characteristics collide. To extend the solution beyond the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3101906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding limit when values of derivatives at a point are given
Let $f:\Bbb R\to\Bbb R$ be such that $f''$ is continuous on $\Bbb R$ and $f(0)=1, f'(0)=0, f''(0)=-1$. The $\displaystyle{\lim_{x\to\infty}\left(f\left(\frac{\sqrt2}{x}\right)\right)^x}$ is .....
I did this using particular function $f(x)=1-\frac{x^2}2$ a... | Hint: Exponentiate the expression and express it in a form wherein you can use L'Hopital's rule. $$ \exp \ln\Biggl(f\biggl( \dfrac{\sqrt{2}}{x} \biggr)\Biggr)^x=\exp x\ln \Biggl(f\biggl( \dfrac{\sqrt{2}}{x} \biggr)\Biggr)=\exp \dfrac{\ln\Biggl(f\biggl( \dfrac{\sqrt{2}}{x} \biggr) \Biggr)}{1/x}$$Note that now you can si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that the logarithm family of sums diverges? Define $$a_k(n) =\frac{1}{n\log(n)\log(\log(n)) \cdots \log^{k}(n)}$$
Do all of the sums (for a fixed $k \in \mathbb{N}$):
$$A_k = \sum_{n} a_k(n)$$
Diverge?
| We can use integral test. Since $\alpha_k(n)$ is decreasing in $n$, we have
$$
\sum_n \alpha_k(n) =\infty \ \ \ \Longleftrightarrow \ \ \ \int_{A}^\infty \frac{\mathrm dx}{x\cdot \log (x) \cdot\log(\log x)\cdots \log^k(x)}=\infty.
$$ Let $t_k = \log^k (x)$, $t_0 = x$. We find that $t_k = \log(t_{k-1})$ and
$$
\mathrm d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $G: Z[t] → \mathbb{C}$ defined by $G(f)=f(\sqrt{-1})$ is a ring homomorphism with kernel $(t^2 + 1)$ and image the Gaussian integers. I'm clear about the ring homomorphism and image part. But I'm not sure how to formulate my language precisely about the statement of the kernel.
| First show that $\langle t^2+1 \rangle\subseteq \ker(G)$. This is easy as $G(t^2+1)=0$. So $t^2+1 \in \ker(G)$, hence the ideal generated by this is a subset of $\ker(G)$.
Now you need to show that $\ker(G) \subseteq \langle t^2+1 \rangle $. Let $f \in \ker(G)$, then $f(i)=0$. Since $f(t) \in \mathbb{Z}[t]$, thus $f(-i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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$c_{00}$ is not complete I try to show that the space $c_{00}=\{(x_n):x_n=0 \text{ all but finitely many }n\}$ is not complete with respect to the norm $\|x\|_\infty=\max |x_n|$.
My attempt:
Let $(z_n)=\left(1,\frac{1}{2},\dots,\frac{1}{n},0,0,\dots\right)$ be a sequence. Clearly $(z_n)\in c_{00}$. We have convergence ... | What you did is fine. Now, given $\varepsilon>0$, taket $N\in\mathbb N$ such that $\frac1N\leqslant\varepsilon$. Then$$m>n\geqslant N\implies\lVert z_n-z_m\rVert=\frac1{n+1}\leqslant\frac1{N+1}<\frac1N\leqslant\varepsilon.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Prove that $\sum_{i=1}^{n} i \times i! = (n+1)! - 1$ by induction \begin{align*}
\sum_{i = 1}^{k + 1} i(i!) & = \sum_{i = 1}^{k} i(i!) + (k + 1)(k + 1)!\\
& = (k + 1)! - 1 + (k + 1)(k + 1)! & \text{by the induction hypothesis}\\
& = (1 + k + 1)(k + 1)! - 1\\
... | It is because, by definition, $n! = n(n-1)!$ and $0! = 1$. Just take $n=k+2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Building a function out of a given domain and range How does one build a function while given domain and range ? For example, the domain $(0,5]$ and range $[0, \infty)$
| something like
$f(x)= \frac 1x - \frac 15$ seems like it would fit the bill.
or $g(x) = -\ln x + \ln 5$
I am looking to map the open end in the domain with the open end in the range, and the closed end of the domain with the closed end of the range. It is not entirely necessary, but if we don't it seems to me that we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Ways to arrange $n\geq2$ people around a circular table, given two permanent seats. How many ways to arrange $n\geq2$ people around a circular table, given two specific people who cannot stand next to each other?
I've observed that when $n=2$ and $n=3$ there exists no way to arrange them so that the two specific people... | Well there are $n$ seats. Assuming the seats are not numbered, so the first mandatory person can be placed anywhere.
Then, place the second mandatory person, there are $n-3$ choices (why?)
Place the rest of $n-2$ people into $n-2$ seats, so $(n-2)!$ ways
Total: $(n-3)(n-2)!$ ways
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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find the sum of$\sum_{n=1}^{\infty } \frac{4^{n}}{n!\left ( n+2 \right )!}$ $\sum_{n=1}^{\infty } \frac{4^{n}}{n!\left ( n+2 \right )!} = \sum_{n=1}^{\infty } \frac{4^{n}}{ \left ( 2n+1 \right )!B\left ( n+1,n+1 \right ) \left ( n+1 \right ) \left ( n+2 \right )}$
Given, $F(n) = \frac{4^{n}}{ \left ( 2n+1 \right )!B\l... | The sum is $$-\frac{1}{2}+\frac{{I}_{2}(4)}{4}$$
where $I_2$ is a modified Bessel function of order $2$. More generally,
$$ \sum_{n=0}^\infty \frac{x^n}{n!(n+k)!} = \frac{I_k(2 \sqrt{x})}{x^{k/2}}$$
for nonnegative integers $k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3102947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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When is the sum and difference of two projection matrices $P_1$ and $P_2$ a projection matrix? Let $P_1$ and $P_2$ be two projection matrices for orthogonal projections onto $S_1 \in \mathcal{R}^m$ and $S_2 \in \mathcal{R}^m$, respectively. When does $P_1+P_2$ and $P_1-P_2$ result in a projection matrix? Prove it.
I am... | If $P_1+P_2$ is a projection, then
$$
P_1+P_2=(P_1+P_2)^2=P_1+P_2+P_1P_2+P_2P_1.
$$
So $P_1P_2+P_2P_1=0$. Multiply on the left by $I-P_1$ to get $(I-P_1)P_2P_1=0$. So $P_2P_1=P_1P_2P_1$, selfadjoint, which then gives $P_1P_2=P_2P_1$. So $2P_1P_2=0$, and $P_1P_2=0$.
If $P_1-P_2$ is a projection, then
$$
P_1-P_2=(P_1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How can I simplify $\sqrt{\frac{5+\sqrt{5}}{2}}$? I've tried to see the root as
$\sqrt{\frac{5+\sqrt{5}}{2}} = \sqrt{a}+\sqrt{b},$
but this method doesn't give me something good.
| We can write it so:
$$\sqrt{\frac{5+\sqrt{5}}{2}}=\sqrt{\frac{5+\sqrt{5}}{2}}+\sqrt0.$$
Let there be rationals $a$ and $b$ for which $\sqrt{\frac{5+\sqrt{5}}{2}}=\sqrt{a}+\sqrt{b}.$
Thus, $$\frac{5+\sqrt5}{2}=a+b+2\sqrt{ab},$$ which gives
$$a+b=\frac{5}{2}$$ and
$$ab=\frac{5}{16},$$ which says that $a$ and $b$ are root... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Why is the Change of Basis map unique? I've been looking all over, but I haven't found anything satisfactory.
We've been shown in class by a commutative diagram that, given an $n$-dimensional vector space $V$ over a field, $\mathbb{F}$, and bases, $\mathcal{B}=\{v_1,...,v_n\}$ and $\mathcal{C}=\{u_1,...,u_n\}$, the coo... | Remember that any linear map on any linear space $\;V\;$ is uniquely and completely determined once we know its action on any basis of $\;V\;$ ...and that's all.
If you want to do this proof, suppose there's another map $\;Q:V\to V\;$ s.t. it coincides on "the old basis" $\;\mathcal B\;$ with $\;P:\;\; Qv_i=Pv_i\;\;\fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Problem solving a 4 by 4 linear system with infinite solutions I have a question here. The following is a 4 by 4 system of linear equations which has infinitely many solutions.
$$
\left\{
\begin{array}{c}
x+y+3z+t=0 \\
x-y-z-t=0 \\
3x+y+5z+3t=0 \\
x+5y+11z+8t=0
\end{array}
\right.
$$
An online calculator gave me t... | Substitute $x=y+z+t$ in the first equation and you have
$2y+4z+2t=0 \\ \Rightarrow y+2z=-t$
Substitute $x=y+z+t$ in the third equation and you have
$4y+8z+6t=0$
Then substitute $y+2z=-t$ and you get $t=0$. So $y=-2z$ and $x=y+z$.
Substitute $x=y+z$, $y=-2z$ and $t=0$ in the fourth equation and you have
$0=0$
So the fou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How do I solve this equation with 3 constants? (Setting one constant to an arbitrary case value?) $$m+re^{rx}=3x+2(mx+b+e^{rx}) +1 $$
Solution says:
if (r != 0) then m=2b+3; r=0; 0=3+2m ...and then solves for m, r, and b.
if (r == 0) then m=2b+1; r=0; 0=3+2m ...and then solves for m, r, and b.
I'm not sure how to fol... | You have two questions here:
*
*What's it called when you split into multiple cases like this?
*How am I supposed to know to do this?
For the first, I've heard it called "case analysis".
For the second: Suppose I tell you that the difference of my son's and my daughter's ages is twice my son's age, and my daugh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Chinese remainder theorem, can't figure it out! x mod 5 = 3
x mod 7 = 5
x mod 11 = 7
How to determine x? I've been searching on YouTube, but they're giving examples in different ways, for example
x ≡ 1(mod 3)
I don't understand it, is it the same as x mod 3 = 1?
Thanks, Im still new and quite slow at math, I'm tryin... | Chinese Reminder Theorem says that there exist one solution $\mod M= m_1 \cdot \dots m_n$ of the system
$$\cases{x \equiv b_1 \mod m_1 \\ x \equiv b_2 \mod m_2 \\ \dots \\ x \equiv b_n \mod m_n}$$
and the solution is of the form $$x=b_1M_1M_1'+ \dots + b_nM_nM_n'$$ where $$ M_i= \frac{m_1 \cdot \dots m_n}{m_i}$$ and $M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Use the Fundamental Theorem of Calculus to find the derivative of $h(x) = \int_{1}^{e^x} \ln (t) \,dt$ The fundamental theorem of calculus states:
If $f$ is continuous on $[a,b]$, then if $g(x) = \int_{a}^{x}f(t)\,dt,~ \textrm{then}~g'(x) = f(x)$.
In Example 4, the chain rule is used because the upper bound, $x^4$ ne... | See the proofs of the Leibniz integral rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Find all polynomials $f (x)$ such that $f (x^2+x+1)$ divides $f (x^3-1)$ I had come across a question in which involved finding polynomials (with real coefficients) satisfying the division criteria stated above. By inspection, it was easy to see that polynomials like $x$, $x^2$, $x^3$, etc. satisfied. So I went on to t... | We will prove that all solutions are of a form $ax^n$, where $n\in \mathbb{N}_0$ and $a\in \mathbb{R}$.
Say exsist $a_1\ne 0$ such that $f(a_1)=0$. Then there exsist $x_1\in \mathbb{C}$ such that $x_1^2+x_1+1=a_1$ and $|x_1-1|>1$.
Such $x_1$ exsist since the equation $x^2+x+1-a=0$ has two solution $x_1,x_2$ for whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3104079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
} |
Find the limit of $\frac {x^2+x} {x^2-x-2}$ where $x \to -1$? I need to find the limit of $\frac {x^2+x} {x^2-x-2}$ where $x \to -1$. Right now I am getting $\frac{0}{0}$ if I don't factor first, or $\frac{2}{0}$ if I do.
Here are my factoring steps:
$\frac {x^2+x} {x^2-x-2}$
$=\frac{x(x+1)}{(x-2)(x+1)}$
replace $x$ wi... | $$\frac{x^2+x}{x^2-x-2}=\frac{x(x+1)}{(x-2)(x+1)}=\frac x{x-2}\xrightarrow[x\to-1]{}\frac{-1}{-3}=\frac13$$
The above is justified by the fact that taking the limit when $\;x\to-1\;$ means $\;x\;$ gets closer and closer to $\;-1\;$ but never equals it in this limit process.
To calculate a limit as $\;x\to a\;$ is the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3104145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
} |
Probability of getting at least one job offer based on interview odds I have recently interviewed for a number of jobs, and am wondering what the odds are of getting accepted for one based on the odds of each interview and the total number of interviews.
*
*I had $7$ interviews which had a $1$ in $10$ chance of sec... | This is a problem in which the complementary approach will be the most fruitful - let's instead consider how likely you are to not get a job. We know that, for an event $A$, then
$$P(A) = 1 - P(\text{not} \; A)$$
That is to say, more relevant to your case,
$$P(\text{getting at least one job offer}) = 1 - P(\text{gettin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3104236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
What's wrong with the following proof of the theorem? I am reading How to prove it.
In Section 3.2, it asked us what's wrong with the following proof of the theorem?
Suppose $x$ and $y$ are real numbers and $x+y = 10$. Then $x \neq 3$ and $y \neq 8$.
Proof:
Suppose the conclusion of the theorem is false. Then $x=3$ and... | The theorem is false. 3 + 7 = 10, so it can in fact be the case that "x + y = 10" is true while "x is not 3 AND y is not 8" is false. So "x + y = 10" does not imply "x is not 3 AND y is not 8".
"x + y = 10" does imply "x is not 3 OR y is not 8". And this is what the proof in the text is a proof for: the author assumes ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3104403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to compute $I(n)=\int_{0}^1x_1^a\int_{x_1}^1x_2^a\cdots\int_{x_{n-1}}^{1}x_n^adx_n\cdots dx_2dx_1$? Let $a$ be a positive integer. For $n\ge 1$, how to compute
$$I(n):=\int_{0}^1x_1^a\int_{x_1}^1x_2^a\cdots\int_{x_{n-1}}^{1}x_n^adx_n\cdots dx_2dx_1?$$
I tried first two terms, and found $I(1)=\frac{1}{a+1}$, $I(2)=... | Method 1. We have
$$ I(n) = \int_{0 \leq x_1 \leq \cdots \leq x_n \leq 1} x_1^a \cdots x_n^a \, \mathrm{d}x_1\cdots\mathrm{d}x_n. $$
Now by symmetry, this is simply
$$ I(n) = \frac{1}{n!} \left( \int_{0}^{1} x^a \, \mathrm{d}x \right)^n = \frac{1}{n!(a+1)^n}. $$
*
*Addendum. More specifically, for each permutation $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3104498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find the particular solution for Augmented Thin Plate Splines in the context of the Dual Reciprocity Boundary Element Method In the dual reciprocity boundary element method (DRBEM) the non-homogeneous terms are expanded in terms of radial basis functions. This expansion involves approximating the solution to the... | Your particular solution isn't correct. Once you add in angular dependency, the Laplacian becomes
$$ \nabla^2 \Psi = \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho\frac{\partial \Psi}{\partial \rho}\right) + \frac{1}{\rho^2}\frac{\partial^2 \Psi}{\partial \theta^2} $$
So $\theta$ can't be treated as a constant
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3104849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
integral of $x - x_0$ How to evaluate:
$$\int (x-x_0)dx$$
One way is direct:
$$\int x dx - \int x_0 dx = \frac{x^2}{2} - x x_0 +C$$
The other way is assuming $x - x_0 = \tau$, $dx=d\tau$, we get
$$\int \tau d\tau = \frac{\tau^2}{2} = \frac{(x - x_0)^2}{2} + C$$
Which one is the right one? If both are correct when to ch... | Both are correct and equivalent.
$$\int x dx - \int x_0 dx = \frac{x^2}{2} - x x_0 +C_1$$
$$\int \tau d\tau = \frac{\tau^2}{2} = \frac{(x - x_0)^2}{2} + C_2=\frac{x^2}{2} - x x_0+\frac{x_0^2}{2}+C_2$$
Now you can see that $\frac{x_0^2}{2} + C_2$ is just another constant and can be thought of the same as $C_1$, i.e. $C_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3104940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the solution of $x^a+(1+x)^b=0 $? I am not a mathematician, but a theoretical physicist. I am faced with this equation coming from some plasma phenomenon and I am unable to 'recognise' it.
Mathematica software cannot solve it. I have searched the web and it might have relation with the Lambert function, but I... | Assuming (as per comments) that $a$ and $b$ are actually positive integers then we have four cases:
1) $a$ even, $b$ even - no real solution.
2) $a$ even, $b$ odd - one real solution in $(-\infty, -2)$ if $a<b$, otherwise no real solutions.
3) $a$ odd, $b$ even - one real solution in $(-1,0)$.
4) $a$ odd, $b$ odd - one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3105037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Series power function over exponential function A typical exercise from calculus is to show that any exponential function eventually grows faster than any power function, i.e.
$$ \lim_{k \to \infty} \frac{k^a}{b^k} = 0 \qquad \text{ for } a,b>1.$$
In fact, by the ratio test, we can show for $x=a=b$ the even stronger re... | I think this is a special case of the Lerch transcendent, defined as
$$
\Phi(z,s,\alpha)=\sum_{n=0}^\infty\frac{z^n}{(n+\alpha)^s}.
$$
Specifically, your proposed function $F$ is given by
$$
F(x)=\Phi\big(\tfrac{1}{x},-x,0\big).
$$
Plotting this function on WolframAlpha confirms that the global minimum you computed is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3105125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Prove that a primitive $q$-th root of unity is in the algebraic closure of $\Bbb F_p$
Let $p$ and $q$ be odd primes. Let $\Omega$ be the algebraic closure of $\Bbb F_p$. Let $\omega$ be a primitive $q$-th root of unity. Show that $\omega \in \Omega$.
How do I show that? Please help me in this regard.
Thank you very m... | By definition $\omega$ is a root of $X^q-1\in\Bbb{F}_p[X]$, and by definition every polynomial in $\Bbb{F}_p[X]$ splits into linear factors in $\Omega[X]$. Hence $X-\omega\in\Omega[X]$ and so $\omega\in\Omega$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3105275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A question about integer BMO 1984 Question 4 My question is derived from BMO 1984 question 4.
Given an integer n, how many r (with r bigger than 0 less than 1) can make 2nr an integer?
I tried some values of n and from 1 to 9 there are 1,2,3,3,3,5,3,4,5 r that meet the requirement.
I find it hard to generalise it to n... | First observe that the condition $0<r<1$ implies $0<2nr<2n$. Next observe that for any integer $m, 0 < m< 2n$, there is precisely one $r\in(0,1)$ with $m=2nr$ (namely, $r=m/2n$). Therefore the question is equivalent to the following question: for an integer $n$, how many integers $m$ satisfy $0 < m < 2n$. Clearly this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3105378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Series of complex numbers over an uncountable set
Question: Let I be an arbitary index set and $(a_i)_{i\in I} \subset \Bbb C$ such that $\sum_{i \in I}a_i$ converges. Show that there exists a countable set $I_0 \subset I$ such that $a_i=0$ for all $i \in I\setminus I_0$.
I think I am able to prove this if all the ... | Note that positivity isn't really the assumption you need for the easier argument you mention - rather, you just want all the real parts to have the same sign and all the imaginary parts to have the same sign. As long as this happens, you're still happy since all the nonzero terms still "point away from zero in the sam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3105535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to tell if a matrix of a certain rank contains a certain kernel and image? I'm struggling with the following question
True or False: There exists a 3x4 matrix A of rank 2 such that ker(A) contains the vector v = $$
\begin{bmatrix}
1\\
1\\
1\\
1\\
\end{bmatrix}
$$
and image(A) contains the ve... | How about $\begin{pmatrix}1&-1&0&0\\1&-1&0&0\\1&0&-1&0\end{pmatrix}$?
Not sure about how else to go about it, than providing an example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3105669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that the following is an inner product Currently, I've started to take Analysis 2 in school and we are doing Euclidean Spaces. There is an example that I'm trying to prove but I can't wrap my mind around.
$x, y \in \mathbb R^d$
$\langle x,y \rangle:= \sum^d_{j=1}x_j-y_j$
So, there are 4 rules for this to be an ... | There is a very easy way to see that this is not an inner product. Note that it often happens that inner products are negative for given choices of $x$ and $y$. However, what happens when you compute $\langle x,x\rangle$ for any $x$? Is this a positive number for all $x\neq 0$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3106098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Will a path between $(x, y)$ and $(-x, -y)$ always intersect a 90 degree rotated copy? Suppose we have a path between two points $(x, y)$ and $(-x, -y)$. If we rotate it by 90 degrees around the origin, will the copy intersect the original? (You can add any number of assumptions to avoid pathological cases.)
It seems o... | This proof assumes there is no "backtracking" or radial movement (i.e. every line through the origin intersects the curve in exactly one point, with the obvious exception of the line through $(x, y)$ and $(-x, -y)$, which intersects the curve twice). Also, the curve is continuous and goes counterclockwise around the or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3106208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Relation between surfaces of an infinitesimal tetrahedron Let $d\sigma_1,d\sigma_2, d\sigma_3$ denote the areas of the faces perpendicular to the axes $x_1,x_2,x_3$ and let $d\sigma_n$ denote the area of the inclined face with unit exterior normal n. My book says that this relation holds:
$d\sigma_i = d\sigma_n \cos(\m... | $\let\a=\alpha \let\s=\sigma$
You rightly tagged "geometry" your question. And no limit or
infinitesimals are needed as far as the surface is plane. Consider $\s$ (finite, not infinitesimal) and $\s_1$. They are triangles sharing a base. Can you see what's the ratio of their heights?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3106351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Linear Dependence lemma - an unclear moment from the proof I am reading linear dependence lemma, namely:
If $(v_1,v_2,\dots,v_m)$ is linearly dependent and $v_1\neq 0$, there exists an index $j\in \{2,\dots,m\}$ such that:
$v_j\in \text{span} (v_1,\dots,v_{j-1}).$
Proof: Since $(v_1,\dots,v_m)$ is linearly dependent th... | The contradiction is that for $a_2=\dots=a_m=0$ and $a_1\ne0,\ v_1\ne 0$, we get
$$0=a_1v_1+\dots +a_mv_m=a_1v_1\ne 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3106459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find basis of fundamental subspaces with given eigenvalues and eigenvectors Let $\Lambda_1=0,\Lambda_2=1,\Lambda_3=2$ and $x_1=\begin{bmatrix}
1 \\
0 \\
0 \\
\end{bmatrix},x_2=\begin{bmatrix}
0 \\
1 \\
2 \\
\end{bmatrix},x_3=\begin{bmatrix}
0 \\
1 \\
1 \\
\end{bmatrix}$ a... | Your reasoning is correct; by what we are given the null space of the matrix must be generated by the vector $x_1$, and it follows that the column space must be generated by the vectors $x_1$ and $x_2$. The conclusion that $\operatorname{Ker}(A) = \operatorname{Ker}(A^T)$ is no contradiction; for example, this also hap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3106576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Particle on a line question with integrals calc2 I have a question for calculus two regarding a particle on a line.
Question:
A particle moves along a line with acceleration $a(t)=−1(t+3)~2\text{ft}/\text{sec}^2$. Find the distance traveled by the particle during the time interval $[0,1]$, given that the initial vel... | The velocity over $[0,1]$ is
$$
v(t) = v(0) + \int_{0}^{t} a(x) \ dx
$$
The distance (not displacement) over this interval is
$$
\int_{0}^{1} |v(t)| \ dt
$$
So you can find the distance with two integrals. It looks like your $a(t)$ is negative over $[0,1]$, but the initial velocity is much higher than $\int a(t) \ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3106679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Study the continuity of a function
I have the function $f:\mathbb{R}^2\rightarrow\mathbb{R}\ f(x,y) = \left\{\begin{matrix}
\sin\frac{x^3y}{x^4+y^4}, & (x,y) \in \mathbb{R}^2 \setminus\{(0,0)\}\\
0, & (x,y) = (0,0).
\end{matrix}\right.$ I need to study the continuity of the function $f$.
I tried to calculate the l... | HINT:
Note that along $x=0$, $\sin\left(\frac{x^3y}{x^4+y^4}\right)=0$. Along $x=y$, $\sin\left(\frac{x^3y}{x^4+y^4}\right)=\sin(1/2)$. What can you can conclude?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3106753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
What is the area of $\triangle ABC$ where $\triangle ADC$ is cyclic, point $P$ is on the circumference and $AD = AP$?
$\triangle ABC$ is a right angled triangle. The perpendicular drawn form $A$ on $BC$ intersects $BC$ at point $D$. A point $P$ is chosen on the circle drawn through the vertices of $\triangle ADC$ such... | Since $AC$ is the diagonal of square $APCD$, then $\angle{C}=45^{\circ}$. Then, we know $\triangle{ABC}$ is a 45-45-90 right triangle. Notice that $ABC$ is isosceles, so $BD=DC=CB=x$. Then, you can use Pythagorean Theorem to find $x$. Then, it is easily seen that $[ABC]=\frac{(\sqrt{2\cdot 70})^2}{2}=70$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3107015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Sum of Sine and Cosine to Higher Powers Is Constant (Recursion, Proofs) I have observed this empirically, but I have no idea how to prove it or if it has been proven before. If this has been proven before, in any form, either more or less generic, please point me to such a proof. If not, can you prove it?
Given:
$$ ... | The trick here is that they're all versions of the Pythagorean identity. Your iteration $f_n(x)=f_{n-1}^2(x)-c_n$ and $g_n(x)=g_{n-1}^2(x-\frac{\pi}{2^n})-c_n$ is a double-angle formula. We have $f_2(x)=\sin^2 x-\frac12 = -\frac12\cos(2x)$, $g_2(x)=\cos^2(x-\frac{\pi}{4})-\frac12=\frac12\cos(2(x-\frac{\pi}{4}))=\frac12... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3107141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find quantity of elements in group with given order
Let $G = ( \mathbb { Z } / 133 \mathbb { Z } ) ^ { \times }$ be the
group of units of the ring $\mathbb { Z } / 133 \mathbb { Z }$ . Find
the number of elements of $G$ of order $9 .$
133 cannot be divided by 9. So what is the solution to the problem? Or my consi... | The fact that $9\nmid133$ is irrelevant here, since the group $\mathbb{Z}_{133}^\times$ has $108$ elements. Since $9\mid108$, Lagrange's theorem is not an obstacle to the existence of elements of order $9$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3107305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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