Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Dense Subspace: ONB This might be a duplicate. If so, then please let me know. Thanks!
Given a Hilbert space $\mathcal{H}$.
Consider a dense subspace $\overline{Z}=\mathcal{H}$.
Then it provides an ONB: $\mathcal{S}\subseteq Z$
(I guess it can be shown by slightly adjusting the usual proof via Zorn's lemma...)
| I just realized that this cannot be true for the following reason:
Assume every dense subspace would provide an ONB for the Hilbert space:
$$\overline{\mathcal{S}}=\mathcal{H}\quad(\mathcal{S}\subseteq Z)$$
Then, it would serve as well as an ONB for the subspace:
$$\overline{\mathcal{S}}\supseteq Z$$
But there are preH... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/983109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the number 0.2343434343434.. rational? Consider the following number:
$$x=0.23434343434\dots$$
My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in decimal form.
Also, is $0.234$ rational or irrational?
| We have $100x=23.434343\cdots$ and $10000x=2343.434343\cdots$ so
$$9900x=2320$$
hence $x$ is rational and
$$x=\frac{2320}{9900}=\frac{116}{495}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/983198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Fallacy - where is the mistake? Could anyone help me to find the mistake in this fallacy? Because the actual result for $I$ is $\pi/2$
\begin{equation}
I = \int_{0}^{\pi} \cos^{2} x \; \textrm{d}x
\end{equation}
\begin{equation}
I = \int_{0}^{\pi} \cos x \cos x \; \textrm{d}x
\end{equation}
substitution:
\begin{equatio... | if $\sin(x)=u$ we get $x=\arcsin(u)$ and $dx=\frac{1}{\sqrt{1-u^2}}du$ thus we get
$\int \cos(x)^2dx=\int 1-\sin(x)^2dx=\int \sqrt{1-u^2}du$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/983295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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"Evaluated at" or "at" notation Normally a variable that is a function another variable would be represented as in the following fashion: $ V(t) $ (voltage as a function of time). However, my engineering professor (who also wrote the textbook) likes to use the vertical bar notation instead, so the flux at $ r = R_0 $ i... | I've usually seen that notation mean "restricted to", which isn't too far from "evaluated at" in meaning, but it is more general. For instance,
$$
\left. f\right|_{[0,1]}
$$
could be interpreted to be a function with values that agrees with $f$'s values, but is only defined on the interval $[0,1]$.
One advantage is tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/983398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 0
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Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
Suppose that $f: \mathbb R \to\mathbb R$ satisfies $f(x + y) = f(x) + f(y)$ for each real $x,y$.
Prove $f$ is continuous at $0$ if and only if $f$ is continuous on $\mathbb R$.
Proof: suppose $f$ ... | First observe that $f(0) = f(0) + f(0)$, so that $f(0) = 0$.
Now suppose $f$ is continuous at $0$. Let $x \in \mathbb{R}, \epsilon > 0$. Let $\delta > 0$ be such that $|f(t)| < \epsilon$ whenever $|t| < \delta$.
If $|y - x| < \delta$, then setting $t = y -x$ we have $|f(y) - f(x)| = |f(t)| < \epsilon$, thus completing... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show $P^1\times P^1$ (as projective variety by Segre embedding) is not isomorphic to $P^2$? I am a beginner.
This is an exercise from Hartshorne Chapter 1, 4.5. By his hint, it seems this can be argued that there are two curves in image of Segre embedding that do not intersect with each other while in $P^2$ any ... | The canonical bundles of these varieties both only have the zero section, so these sections cannot be used to prove the non-isomorphism of the varieties.
The anticanonical bundles $K^\ast=\Lambda^2T$ of these varieties however have spaces of global sections of different dimensions $h^0$ over the base field and prove t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/983568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 9,
"answer_id": 4
} |
Solving for n in the exponent. Well, it's another question I feel like I should know. I'm trying to model the number of successes before the first failure. The probability of successes is given as $p$, which makes the probability of failure $(1-p)$.
The probability mass function, as I've calculated it, turns out to be ... | If $X$ is the number of successes (not trials) until the first failure, then $\Pr(X=n)=p^n(1-p)$ for $n=0,1,2,\dots$.
This cannot ever be equal to $1$ if $p\ne 0$.
If you want to solve $p^n(1-p)=a$, given $0\lt p\lt 1$, rewrite the equation as
$p^n=\frac{a}{1-p}$.
and take the logarithm of both sides.
Remark: As wa... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Convergence of the series $\sum\limits_{n=1}^{\infty}(-1)^{n}\frac{\ln(n)}{\sqrt{n}}$ I would like to see whether or not $$\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$$ is a convergent series.
Root test and ratio test are both inconclusive. I tried the alternating series test after altering the form of t... | $\dfrac{\ln(n)}{\sqrt{n}}$ is mono-tonic decreasing after $n=\lceil e^2 \rceil$ and remains bounded between $n=1$ to $\lfloor e^2 \rfloor$, so from alternating series test $\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$, must converge.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Group theory: subset of a finite group Given
*
*$G$ be a finite group
*$X$ is a subset of group $G$
*$|X| > \frac{|G|}{2}$
I noticed that any element in $G$ can be expressed as the product of 2 elements in $X$. Is there a valid way to prove this？
If the third condition was $|X| = \frac{|G|}{2}$ instead, does the... | Answer for the additional question: the statement need not be true if $|X|=|G|/2$.
For example, let $G$ be the cyclic group of order $2$ and $X$ consist of the non-identity element.
A (slightly) more general example: let $G$ be the cyclic group of order $2n$, let $g$ be a generator, and let
$$X=\{g^{2k}\mid k=0,1,\ldot... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Change of variable in double integrals I need help to solve the following question(s).
a) Evaulate the integral
$$\iint_D (x-y) \, dx \, dy,$$
where $D$ is the triangle with vertices $(0,0)$, $(-1,1)$ och $(4,2)$.
b) Evaulate the integral
$$ \iint_D (y-x) dx \, dy,$$
where $D$ is the triangle with vertices $(0,0)$, $(4... | So I may have found an method. Please comment if you have any suggestions of how it can be improved. So my remaining question is why the other change of variables (i.e. the change I mentioned in my first post) differ slightly from the following change of variables.
a) Evaulate the integral
$ \int \int_D (x-y) \, dx \, ... | {
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"question_score": "1",
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Calculating the hitting probability using the strong markov property ** This problem is from Markov Chains by Norris, exercise 1.5.4.**
A random sequence of non-negative integers $(F)n)_{n\ge0}$ is obtained by setting $F_0=0$ and $F_1=1$ and, once $F_0,\ldots,F_n$ are known, taking $F_{n+1}$ to be either the sum or the... | I also stumbled upon this problem. I did not solve (a) and (c) so far, but for you specific questions I have the answer.
If drawing the flow diagram of the Markov chain up to 4 steps from $(1,2)$ (i.e., about $2^4$ states can be reached), one notices, that one step into the wrong direction (e.g., from $(1,2)$ to $(2,3... | {
"language": "en",
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"question_score": "4",
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Solving Pythagoras Problem An aircraft hangar is semi-cylindrical, with diameter 40m and length 50 m. A helicopter places an inelastic rope across the top of the hangar and one end is pinned to a corner, a A. The rope is then pulled tight and pinned at the opposite corner, B. Determine the lenghth of the rope.
So, firs... | This question is a simple Pythagoras problem.
First, we find the length of the arc whose diameter is 40m.
Then we assume it to be a side of the quadrilateral whose side is 50 m.
Next, we use Pythagoras theorem to find the diagonal.
The answer will be near to 80.34..
| {
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"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Can you cancel out a term if equal to zero? quick question here:
In my proofs class we had a problem that after a little work we end up with:
$x(x-y)=(x+y)(x-y)$ where $ x = y $. Now, I know this is pretty basic, but my teacher said that for the next step, one cannot cancel out $(x-y)$ from both sides as $(x-y) = 0 $. ... | There is no such thing as "cancelling out".
In this case you want to divide both sides of the equation by (x-y). But always remember before dividing: you can't divide by zero. But if x=y then in fact (x-y) = 0 and so you would in fact divide by zero which you are not allowed to. But only professors are allowed to divid... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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"answer_id": 7
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Twisted logarithm power series I recently encountered a power series similar to the one of the $\log(1-x)$ of the form
$$
F(x)= \sum_{n=1}^\infty \frac{\psi(n)x^n}{n},
$$
where $\psi$ is some Dirichlet character. Has anyone here seen a function like this? Here are some observations I have made:
1) The radius of converg... | Such a function is actually a combination of logarithms. Suppose $\psi$ is a Dirichlet character mod $N$, and let $\zeta$ be a primitive $N$th root of unity. The functions $a \mapsto \zeta^{ka}$, for $k=0, ..., N-1$, form a basis of $L^2(\mathbb Z/N\mathbb Z, \mathbb C)$, hence there exist numbers $b_k \in \mathbb C$ ... | {
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closed but not exact I saw several times that $\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$ is closed but not exact. Closed, is obvious but I can't prove non exactness, can one please help me ?
My attempt, let $f\in \omega^{0}(U)$ and $df=\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy=\frac{\partial f}{\partial x}dx+\frac{\part... | You know that if you integrate over a loop into which there is no singularity, as the form is exact the integral is 0.
Then let us try with a loop containing 0 inside: let us take the circle $C(0,1)$.
$$\int_{C(0,1)} -\frac y{x^2 + y^2}dx + \frac x{x^2 + y^2}dy
= \int_0^{2\pi} -\frac {r\sin\theta}{r^2} (-r \sin\theta ... | {
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Beautiful little geometry problem about sines Given triangles ABC and $A_1B_1C_1$ such that $\sin A = \cos A_1, \sin B = \cos B_1, \sin C = \cos C_1$. What are the possible values for the biggest of these 6 angles?
I tried some stuff like sine theorem but can't derive from it? How do we do this one? Assume the closest ... | Short answer: $\frac{3\pi}{4}$.
Detailed answer: $A_1,B_1,C_1<\frac\pi2$. The equality
$$\sin A=\cos A_1$$
gives
$$A=\frac\pi2-A_1\text{ or }A=\frac\pi2+A_1,$$
and similarly for $B$ and $C$. Working out the cases, the only possibility is
$$A=\frac\pi2+A_1,B=\frac\pi2-B_1,C=\frac\pi2-C_1.$$
Since $A+B+C=A_1+B_1+C_1=\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/984587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$ I have a very short question: is there a faithful projective representaion
$\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$?
Thanks!
| Following Derek's hint, we first pick a faithful representation $\mathbb{Z}/3\mathbb{Z} \rightarrow GL(2, \mathbb{R})$.
Send $1$ to
$$\left(\begin{array}{cc}
1 & -1\\
1 & 0 \end{array}\right)$$
You can check that this is an element of order 3, so it gives us a representation. It's also obviously faithful.
Then, defin... | {
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Inversion of n x n matrix A matrix F is given:
$$
F = [e^{i\frac{2\pi kl}{n}}]_{k,l=0}^{n-1}
$$
Find $$ F^{-1} $$
I know Gaussian method for inverting matrices but I suppose it doesn't apply to matrices with not given exact n value. Could you tell me what are the methods for inverting matrices like this?
| HINT:
This is a famous matrix, $1/\sqrt{n}\cdot F$ is unitary. One can check this directly or look at
http://en.wikipedia.org/wiki/Discrete_Fourier_transform#Orthogonality
| {
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"timestamp": "2023-03-29T00:00:00",
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Integrating algebraic functions The function $y = f(x)$, restricted on the domain $ 0 < x < 1$ and satisfying
$$y^{5}+y^{4} + x = 0,$$
seems to be well-defined and smooth. So how does one integrate this thing? That is, what is $\int_{0}^{1} f(x) dx$?
Of course, one can use Newton's method to approximate $f(x)$ for any... | As MPW says, in general there is no reason to expect that the integral should be easy to compute. However, you can do the following. Differentiate the equation once to obtain $$0=5y'y^4+4y'y^3+1=y'y^3(y+4(y+1))+1. $$ Then use the equation to write $$ y+1 = -\frac{x}{y^4}$$ and plug this into the above to get $$-1= y'y^... | {
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Without actually calculating the value of cubes find the value of $(1)^3+(2)^3+2(4)^3+(-5)^3+(-6)^3$. Also write the identity used Without actually calculating the value of cubes find the value of $(1)^3+(2)^3+2(4)^3+(-5)^3+(-6)^3$.
Also write the identity used
| You can use the identity that if $a + b + c=0$, then $a^3 + b^3 + c^3 = 3abc$. So we can say that:
*
*$2^3 + 4^3 + (-6)^3 = 3\cdot 2\cdot 4\cdot -6 = -144$.
*$1^3 + 4^3 + (-5)^3 = 3\cdot 1\cdot 4 \cdot -5 = -60$.
Adding both, we get:
$1^3 + 2^3 + 2 (4)^3 + (-5)^3 + (-6)^3 = -204$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Real Analysis alpha Holder condition Holder condition
A function $f:(a,b)\rightarrow R$ satisfies a Holder condition of $\alpha$ order if $\alpha > 0$, and for some constant $H$ and for all $u,x \in (a,b)$,
$$|f(u)-f(x)| \leq H|u-x|^\alpha.$$
Prove that an $\alpha$ - Holder function defined on $(a,b)$ is uniformly con... | Take a Cauchy sequence that converges to $a$, say $\{a_n\} \subset (a,b)$. By uniform continuity (or Holder continuity if you prefer), $\{f(a_n)\}$ is also a Cauchy sequence, hence it has a limit, say $f_a$.
It is easy to prove that this limit does not depend on the Cauchy sequence $\{a_n\}$, hence we can say that $\li... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Fejer's theorem with Riemann integrable function
If $f$ is integrable and $f(x+), f(x-)$ exists for some $x$, then
$$
\lim_{N \rightarrow \infty} {\frac{1}{{2\pi }}\int_{ - \pi }^\pi {f\left( {x - t} \right){K_N}\left( t \right)dt} } = \frac{1}{2}[f(x+) + f(x-)]
$$ where $K_N(t)$ is Fejer kernel.
Here is my naive ... | The Fejer kernel $K_{N}(t)$ has several nice properties:
*
*$K_{N}(t) \ge 0$
*$K_{N}(t)=K_{N}(-t)$
*$K_{N}(t+2\pi)=K_{N}(t)$
*$\int_{0}^{\pi}K_{N}(t)\,dt = \pi$
*$K_{N}(t)$ tends uniformly to $0$ for $0 < \delta \le |t| \le \pi$ as $N\rightarrow\infty$.
Those are the only properties that you need. For example,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Triangulation of 4 points why Delaunay maximizes the minimum? I have been going through the chapter on Delaunay triangulations from the book by DeBerg (http://www.cs.uu.nl/geobook/interpolation.pdf). In lemma 9.4, he simply says that "from Thales theorem" we can show that of the two possible triangulations of four poin... | I guess that by “Thales' theorem” he's essentially referring to the inscribed angle theorem, probably in the form of Theorem 9.2 from that chapter.
Consider circles through $p_i$ and $p_k$. Concentrate on the arc above that line. Any point on such a circle will form the same angle with $p_i$ and $p_k$. The centers of a... | {
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"url": "https://math.stackexchange.com/questions/985415",
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"source": "stackexchange",
"question_score": "1",
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feedback on my solution (improper integral) i have done this improper integral but i am not sure if i have followed the correct procedure or my answer is correct. Please help!
| Your answer appears to be correct, though how you got there is unclear. Where you split the integral in two is technically correct, though it is unclear what purpose you felt that may have served. From then on, it's fine until you try to plug in, at which point you should have
$$5\times0^{1/5}-5\times(-1)^{1/5}+5\tim... | {
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Minimal $T_0$-topologies Let $X$ be an infinite set and let $\tau$ be a $T_0$-topology on $X$. Does $\tau$ contain a $T_0$-topology that is minimal with respect to $\subseteq$?
| Larson’s example in the paper cited by Tomek Kania and its verification are simple enough to be worth giving here (in very slightly modified form) for easy reference.
Let $\tau$ be the cofinite topology on an uncountable set $X$, let $\tau_0\subseteq\tau$ be a $T_0$ topology on $X$, and let $\tau_0^*=\tau_0\setminus\{... | {
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Solve $x^3 - x + 1 = 0$ Solve $x^3 - x + 1 = 0$, this cannot be done through elementary methods.
Although, this is way out of my capabilities, I would love to see a solution (closed form only).
Thanks!
| Let $x=u+v$ and note that $(u+v)^3=3uv(u+v)+(u^3+v^3)$
This has the required form if $3uv=1$ and $u^3+v^3=-1$
So $u^3v^3=\frac 1{27}$, and $u^3, v^3$ are roots of $y^2+y+\frac 1{27}=0$
$u,v= \sqrt[3] {\frac {-1\pm \sqrt{1-\frac 4{27}}}2 }$
$x=u+v=\sqrt[3] {\frac {-1+ \sqrt{\frac {23}{27}}}2 }+\sqrt[3] {\frac {-1- \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/985750",
"timestamp": "2023-03-29T00:00:00",
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Question about $e^x$ Let $ p(x)=1+x+x^2/2!+x^3/3!+....+x^n/n!$ where $n$ is a large positive integer.Can it be concluded that $\lim_{x\rightarrow \infty }e^x/p(x)=1$?
| No.
$$\frac{e^x}{p(x)}=1+\frac{\sum\limits_{k=n+1}^\infty \frac {x^k}{k!}}{p(x)}\stackrel{x\to\infty}\to\infty$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving a Extended Euclidean Algorithm related problem Alex has some (say, n) marbles (small glass balls) and he has going to buy some boxes to store them. The boxes are of two types:
Type 1: each box costs c1 Taka and can hold exactly n1 marbles
Type 2: each box costs c2 Taka and can hold exactly n2 marbles
He wants e... | The following does not answer absolutely your question, but I think that it may help.
Lemma: Let $a$ and $b$ be coprime integers greater than $1$.
*
*The equation $ax+by=n$ has nonnegative integer solution for $n\geq ab-a-b+1$
*The equation $ax+by=ab-a-b$ has no nonnegative integer solution.
... | {
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"url": "https://math.stackexchange.com/questions/985881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find series expansion of 1/cosx Find the series expansion of 1/cosx from basic series expansions.
I tried to find 1/cosx from the expansion of cosx but was unsure how to continue. When I found 1/cosx from the basic formula for finding series expansions I didn't get the same answer.
| (going to the fifth term for an example purpose)
Using the basic expansions of cos(x) gives us
$$
\frac{1}{\cos(x)} = \frac{1}{1-\frac{x^2}{2}+\frac{x^4}{24} + \cdots}
$$
of the form $ \frac{1}{1-X} $ which has a known and easy expansion :
$$
1+X+X^2+X^3+X^4+X^5+\cdots
$$
where $ X = \frac{x^2}{2} + \frac{x^4}{24} $ (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/985986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Find this maximum of this $\frac{\int_{0}^{\pi}f(x) \, dx}{\int_{0}^{\pi} f(x)\sin x\,dx}$ Question:
Assmue that $\int_0^\pi f(x)\,dx$ and $\int_0^\pi f(x)\sin x\,dx$ is convergence,and $f(x)>0,\forall x\in(0,\pi)$ Find this maximum as possible for all function $f$
$$I=\dfrac{\int_0^\pi f(x)\,dx}{\int_0^\pi f(x)\s... | Hint: let
$$f_n(x):= n\chi_{(0,1/n)} + n^{-1}\chi_{[1/n,\pi]}, $$
with $\chi$ denoting the indicator (i.e. for any set $E$, $\chi_E=1$ on $E$, $\chi_E=0$ outside $E$).
You get
$$\int_0^\pi f_n(x)dx = 1+ (\pi-1/n)/n ,$$
while (for large $n$)
$$\int_0^\pi |f_n(x)\sin x| dx \leq n\int_0^{1/n} x dx + (\pi-1/n)/n ... $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/986074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Are there mathematical contexts where "finite" implicitly means "nonzero?" I recently gave my students in a discrete math class the following problem, a restatement of the heap paradox:
Let's say that zero rocks is not a lot of rocks (surely, 0 is not a lot of rocks) and that if you have a lot of rocks, removing one r... | Maybe this is too applied for your question, but there is a context in which finite means "large" or possibly large. That is in elasticity theory, to distinguish the linear theory of small strain from the nonlinear theory of "finite" strain.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/986137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 5
} |
Writing the integral $ \int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$ in simpler form? I was wondering if
$\int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$ can be written more simply, where $x,r\in \mathbb{R}$ ?
wolfram al... | This can be simplified a lot. (I assume that $t \gt 0$ as well.) Take out the constants to get
$$\frac1{2 \sqrt{\pi}} \frac{r}{|x|} (|x|-r) \int_t^{\infty} ds \, s^{-3/2} e^{-(|x|-r)^2/(4 s)}$$
Sub $s=1/u^2$ to get that the integral is equal to
$$\frac1{\sqrt{\pi}} \frac{r}{|x|} (|x|-r) \int_0^{1/\sqrt{t}} du \, e^{-(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/986246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges? For real valued function $f$ define
$$S(f)=\{x:x>0,f(x)=x\}$$
For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges?
$\tan x,\tan^2x,\tan{\sqrt{x}},\sqrt{\tan x},\tan 2x$
I do not have any idea to solve this pr... | For $f(x)=\tan x$, there is a fixed point of $f$ in each interval $((n-\frac12)\pi,(n+\frac12)\pi)$, hence the series in question is essentially the harmonic series, diverging.
For $f(x)=\tan^2 x$ and $f(x)=\tan 2x$, the same argument applies.
$f(x)=\sqrt{\tan x}$ makes no difference compared to $\tan x$ (except that t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/986375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this formula satisfiable? I am confused whether or not my explanation for whether or not this formula is satisfiable is correct. Note that the question state it should be Brief and it should not
be necessary to write down a full truth table.
I asked someone and they said my explanation was not ok and one was only gi... | Possibly you are mixed up between satisfiable and tautology.
A statement is a tautology if it is always true. You have shown that the statement may be false, so it is not a tautology. But this is not what was asked.
A statement is satisfiable if it is sometimes true. So you need to find one example of truth values f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
} |
No simple groups of order 9555: proof While reading through crazyproject, I came across the following proof that there were no simple groups of order $9555$. However, I do not understand the step that says:
"Moreover, since 7 does not divide 12 and 13 does not divide 48, $P_7P_{13}$ is abelian."
I don't quite follow th... | The idea is to show that elements of $P_7$ and $P_{13}$ commute. The proof seems to be using that $|\mathrm{Aut}(P_7)| = 48$ but this is wrong. Because $|P_7| = 49$ we know that $P_7$ isomorphic to one of $\mathbb{Z}_{49}$ or $\mathbb{Z}_{7} \times \mathbb{Z}_{7}$ and so $|\mathrm{Aut}(P_7)|$ is either $42$ or $48 \cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/986763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$ I need help understanding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$ or $\displaystyle \sum_{i=0}^{180} \sin(i)$
This might be related to a formula to find the average voltage from a g... | Wikipedia says
$$\sum_{k=0}^n \sin{(\gamma + k\alpha)} = \frac{\sin\tfrac{(n+1)\alpha}{2} + \sin{(\gamma + \tfrac{n\alpha}{2})}}{\sin\tfrac\alpha2}$$So we may write $\gamma = 0$, $n=180$ and $\alpha=\frac{\pi}{180}$ to get $$\sum_{k=0}^{180} \sin{\tfrac{k\pi}{180}} = \frac{\sin\tfrac{181\pi}{360} \sin\tfrac{\pi}{2}}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/986835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
Area and integration question, is this area under the curve? Find the exact area between $x$ and the graph $f(x)=(x-1)(x-2)(x-3)$.
$$f(x) = x^3-6x^2+11x-6$$
I found that this is an odd shaped positive polynomial with a maxima between 1 and 2 and minima between 2 and 3.
I am confused to what the question wants. I natura... | $$A= \int_{1}^{2}x^3-6x^2+11x-6-\int_{2}^{3}x^3-6x^2+11x-6$$
$$A= \left[\frac{x^4}{4}-{2x^3}+\frac{11x^2}{2}-6x\right]_{1}^{2}-
\left[\frac{x^4}{4}-{2x^3}+\frac{11x^2}{2}-6x\right]_{2}^{3}=\frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/986917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that g is differentiable Question: Suppose f, g, and h are defined on (a,b) and $a < x_0 < b$. Assume f and h are differentiable at $x_0$, $f(x_0) = h(x_0)$, and $f(x) \le g(x) \le h(x)$ for all x in (a,b). Prove that g is differentiable at $x_0$ and $f'(x_0) = g'(x_0) = h'(x_0)$.
I know the case where $g(x) = f(... | We have: $f(x_0) \leq g(x_o) \leq h(x_0)$, and $f(x_0) = h(x_0)$, this implies $f(x_0) = g(x_0) = h(x_0)$, you can continue. From this we have: $f'(x_0^{+}) \leq g'(x_0^{+}) \leq h'(x_0^{+})$, and also $f'(x_0^{-}) \geq g'(x_0^{-}) \geq h'(x_0^{-})$. But $f'(x_0^{+}) = f'(x_0^{-}) = f'(x_0) = h'(x_0) = h'(x_0^{+}) = h'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/987016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Prove that $(\mathbb{N},\le)$ and $(\mathbb{Z}, \le)$ are not order isomorphic I want to prove that $(\mathbb{N},\le)$ and $(\mathbb{Z}, \le)$ are not order isomorphic. So what I want to show is that the following is not true:
$$x \le_\mathbb{N} y \iff f(x) \le_\mathbb{Z} f(y)\qquad\forall x,y \in \mathbb{N}$$
But I th... | $\mathbb{N}$ has a minimal element; $\mathbb{Z}$ doesn't. An order isomorphism preserves minimality of an individual element. Can you show this?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/987123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate the limit $\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$ I have this assignment
Evaluate the limit:
5.$\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$
I don't think we are allowed to use L'Hopital, but I can't imagine how else.
| If you set $y=\ln x$, you can rewrite it as $\lim_{x\to-\infty}ye^y=-\lim_{y\to\infty}\frac{y}{e^y}$, and you probably know what that is. Of course that approach requires that you be allowed to use known limits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/987257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Positive definiteness of block matrices I have a $2 \times 2$ block matrix of the form $$M = \left[ {\begin{array}{*{20}{c}}
{\delta I}&A\\ {{A^T}}&kA \end{array}} \right]$$ where the matrix $A$ is positive definite and not symmetric, $I$ is the identity matrix, and $k > 0$ and $\delta > 0$.
*
*Can we choose $\delta ... | There is always a $\delta$ large enought that turns $M$ positive definite.
First, since $A$ is positive definit, there is $\alpha>0$ such that
$$
x^TAx \ge \alpha\|x\|^2 \quad \forall x\in \mathbb R^n,
$$
where I used the vector norm $\|x\|^2 = x^Tx$.
Let $x = \pmatrix{x_1\\x_2}\in \mathbb R^{2n}$. Then
$$
x^TMx = \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/987345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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uniqueness of joint probability mass function given the marginals and the covariance Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e.
$$
\mathbb{P}(X= k, Y= h)
$$
for some $k,h\geq 0$, given the marginals and the covariance?
Moreover, i... | In general $E[X]=\sum_x xp(x)$; So, obviously, for $Z=XY$ we would have
\begin{align*}
E[Z]=&\sum_z zP(Z)\\
=&\sum_x \sum_y xy P(X,Y)\\
=&~a
\end{align*}
where $a$ is a constant. But your question is how many coefficient sets $\{c_{x,y}\}$ we can find so that
\begin{align*}
\sum_x \sum_y x~y~c_{x,y}=a
\end{align*}
For ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Compute the degree of the splitting field I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial is irreducible in this field. So I thought we could consider some element $\al... | If $K$ is a finite field and $f \in K[X]$ is irreducible, then $K[X]/(f)$ is a splitting field of $f$. This follows from the fact that finite extensions of finite fields are always normal. If $\alpha$ is a root, the other roots are $\alpha^{p^n}$, $n \in \mathbb{N}$, where $p=\mathrm{char}(K)$. In particular, the degre... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational? $P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
| Proof: Assume, to the contrary, that $\sqrt{pq}$ is rational. Then $\sqrt{pq}=\frac{x}{y}$ for two integers $x$ and $y$ and we further assume that $gcd(x,y)=1$. Observe that $pqy^2=(qy^2)p=x^2$. Since $qy^2$ is an integer, $p\mid x^2$ and by Euclid's Lemma, $p\mid x$. Thus $x=pd$ for some integer $d$ and so $pqy^2=x^2=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $A,B,C,D$ be the vertices of a four sided polygon taken in anti clockwise. Given $|AB|=|BC|=3,|AD|=|CD|=4,|BD|=5$ , Find $|AC|$ Let $A,B,C,D$ be the vertices of a four sided polygon taken in anti clockwise.
Given $$|AB|=|BC|=3,|AD|=|CD|=4,|BD|=5$$
Find $|AC|$
My try:I have noticed trangles $ABD$ and $BCD$ are right... | You know the angle $C$ and $A$ are right and $BD$ must be perpendicular to $AC$(since the length of BC and AB are the same). Suppose the intersection point is $M$. Hence you can $AM$ by using BCM is similar to BDC.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which theorem could be used? I want to write the $p-$adic expansion of $6!$ in $\mathbb{Q}_3$.
I have to solve the congruence $x \equiv 6! \pmod {3^n}$, right?
I found the following:
$$x_0 \equiv 6! \pmod 3 \Rightarrow x_0 \equiv 0 \pmod 3$$
$$x_1 \equiv 6! \pmod {3^2} \Rightarrow x_1 \equiv 0 \pmod {3^2}$$
$$x_2 \equi... | From the comments, it seems you understand now that $720$, just like any other integer, has a finite $3$-adic representation, i.e. one of the form $$720=a_0 3^0+a_1 3^1+\cdots +a_n 3^n,$$ and you want to find the digits $a_0,\cdots, a_n$, where $0\leq a_i\leq 2$.
We will do it step by step. How do we find $a_0$? If w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I find the radius of a circle from a chord and a section of the radius? Draw a circle with center O.
Line AD is a chord that is 8cm long. The arc above is smaller than the one below.
B is the center of AD.
Line CB is a line that is 2cm long. It meets AD at 90°.
Diagram:
Given the facts above, is it possible to... | Hint: Suppose $OB=x$, then $OA=OC=x+2$. Now apply Pythagoras theorem in triangle $AOB$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/988011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integration with square root in denominator I am honestly embarrassed to ask this because i feel like i should know how to do this but:
$
\int \frac{x}{\sqrt{2x-1}}dx
$
Try to use u-substitution please
| Substitute $u=2x-1, du = 2dx$ so that you get
$$\int \dfrac{x}{\sqrt{2x-1}} \, dx = \frac12\int\frac{u+1}{2\sqrt{u}}\, du = \frac12\int\frac{\sqrt{u}}{2} + \frac{1}{2\sqrt{u}}\, du.$$
Do you see how to proceede?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Decomposition of $_1F_2(1+n;1,2+n;x)$ I am looking for a way to decompose $_1F_2(1+n;1,2+n;z)$ for $n\in\mathbb{N}$ into either Bessel J functions or regularized confluent hypergeometric functions $_0\tilde F_1(b(n),z)$. Mathematica seems to be able to do the decomposition for $n=1,2,3,4$ but I am trying to get a close... | I have found a somewhat simple relation based on the triangle of falling factorials, reading by rows (A068424). We have $$_1F_2(1+n;1,2+n;z)=\sum_{k=1}^{n+1}\begin{pmatrix}
n+1 \\
k
\end{pmatrix}k!(-1)^{k+1}\,_0\tilde F_1(;1+k;z),\,n\geq 0 \\
=\sum_{k=1}^{n+1}\begin{pmatrix}
n+1 \\
k
\end{pmatrix}k!(-1)^{k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/988232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that if p divides xy then p divides x or p divides y I am given that the following proposition is true. (Proved in class)
"Suppose that $x$, $y\in \Bbb Z$, not both zero. Then there exists $m$, $n\in\Bbb Z$ such that $$mx + ny = d$$ where $d$ is the greatest common factor of $x$ and $y$. Moreover, if $ax + by = ... | Hint $gcd(p,x)$ is a divisor of $p$. Therefore it is either $p$ or $1$.
If the gcd is $p$ it means $p|x$ and you already treated that case.
Otherwise, by the lemma
$$mp+nx=1$$
Multiply this by $y$, and use that $p|xy$.
P.S. The key to the proof is that $p$ is prime, otherwise the gcd could be something else but 1 or p.... | {
"language": "en",
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Repeated substitution gone wrong It was an exam question.
$$ f(n)=
\begin{cases}
0 & \mbox{if } n \leq 1 \\
3 f(\lfloor n/5 \rfloor) + 1 & \mbox{if } n > 1 \\
\end{cases}$$
So by calculating some I have $f(5) = 1$, $f(10) = 4$, $f(50) = 13$.
I had to solve this recurrence. So to get rid of the floor operator I said let... | I didn't check your whole argument, but assuming it's correct, you can use the fact that if $n$ is $5^k$, then $k$ is $\log_5 n$ to express the result in terms of $n$.
(Side Question: are the $x$'s in the function definition supposed to be $n$'s?)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/988504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the points at which a surface has horizontal tangent planes Find the points at which the surface
$$
x^2 +2y^2+z^2 -2x -2z -2 = 0
$$ has horizontal tangent planes. Find the equation of these tangent planes.
I found that $$ \nabla f = (2x-2,4y) $$ I'm thinking that the gradient vector must
be equal to $(0,0)$ so... | Your answer is correct, but the logic is wrong. You should consider the above surface as a level surface of the function:
$$ f(x, y, z) = x^2 + 2y^2 + z^2 - 2x - 2z - 2$$
Then the gradient should be a vector with 3 components.
$$ \nabla f(x, y, z) = (2x - 2, 4y, 2z - 2) $$
If the tangent plane is horizontal, the gradie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/988558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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} |
Limit of $(1-2^{-x})^x$ I am observing that $(1-2^{-x})^x \to 1$ as $x \to \infty$, but am having trouble proving this. Why does the $-x$ "beat" the $x$?
I thought of maybe considering that $$1-(1-2^{-n})^n = 2^{-n}(1+2^{-n}+2^{-2n}+\cdots+2^{-n(n-1)})\le2^{-n} \frac{1}{1-2^{-n}} \to 0,$$
as $n \to \infty$ and then not... | Look at the logarithm:
$$
\log(1-2^{-x})^x=x\log(1-2^{-x})\sim x\cdot(-2^{-x})\rightarrow 0.
$$
So
$$
(1-2^{-x})^x \rightarrow 1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/988672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why isn't the orthogonal vector to a direction vector of a plane not necessarily perpendicular to such plane? I had made a question, and the problem with my exercise was that I was trying to calculate a vector perpendicular to some plane in $\mathbb{R}^3$: given one line $L$ inside the plane, I grabbed the direction ve... | The vector perpendicular to your line will be just that, perpendicular to your line; nothing more is guaranteed. To be perpendicular to the plane, you must be perpendicular to all vectors (or lines) in the plane at once; there is only one direction (and its opposite) which does that. The vector you found perpendicular ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Optimization problem: rowing across a lake
A woman at a point A on the shore of a circular lake with radius $r=3$ wants to arrive at the point $C$ diametrically opposite $A$ on the other side of the lake in the shortest possible time. She can walk at the rate of 10 mph and row a boat at 5 mph. What is the shortest amo... | I would like to propose a calculus approach here. So the question is vague in the sense that, we don't really know if she has to walk first or row first. But it's actually symmetric in nature, if you consider walking (along the arc) first (say 'Q' miles) AND then row the remaining distance (say 'P' miles), it's literal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/988865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Find the sum of an infinite series of Fibonacci numbers divided by doubling numbers. How would I find the sum of an infinite number of fractions, where there are Fibonacci numbers as the numerators (increasing by one term each time) and numbers (starting at one) which double each time as the denominators?
I'm assuming ... | Here's a reasonably efficient low-tech approach (i.e., not using Binet's formula or generating functions).
Prove by induction that
$$ \sum_{k=0}^n \frac{F_k}{2^k} = 2 - \frac{F_{n+3}}{2^n} \tag{1} $$
and that
$$ nF_n \le 2^n \tag{2} $$
(For (2) you might find it convenient to assume $n\ge 6$.) From (2) it follows that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/988939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Rigorous proof that surjectivity implies injectivity for finite sets I'm trying to prove that, for a finite set $A$, if the map $f: A\rightarrow A$ is a surjection, then it's an injection. I've looked at this post: Surjectivity implies injectivity but the arguments there seem very round-about, proving that injectivity... | Let $f: A\to A$ be any function. For each $a\in A$, let $N(a)$ be the number of elements in $A$ that are mapped to $a$. Then $\sum_a N(a) = n = |A|$ because every element of $A$ is mapped to some element of $A$. (This is the statement that the fibers of $f$ partition $A$.)
If $f$ is surjective, then $N(a)\ge 1$ for all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/989043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
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A question on a nonnegative quadratic form Denote $x,y,z$ as variables, and $a,b,c$ as coefficients. Suppose $a\leq b\leq 0\leq c$ and $a+b+c=0$.
Could anyone help me prove whether the following quadratic form positive semi-definite?
\begin{equation*}
\begin{split}
I(x,y,z)=&(a^2+4b^2+4c^2)a^2x^2+(4a^2+b^2+4c^2)b^2y^2\... | Well, I showed you how to try to attack these problems a couple of days ago, and you can use exactly the same strategy here.
A problem on positive semi-definite quadratic forms/matrices
Once again using MATLAB Toolbox YALMIP to compute a sum-of-squares certificate. Ordering of the variables is not required, but the eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/989127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$? How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we don't integrate from minus infinity to plus infinity. Thus,... | Let $\Phi(t) = \int_{-\infty}^t e^{-x^2/2} \, dx$. Then
$$
\int_k^{\infty} x^2 e^{-x^2/2} \,dx = -\frac{d}{d\alpha}\Big|_{1/2}\int_k^{\infty}e^{-\alpha x^2}\,dx = - \frac{d}{d\alpha}\Big|_{1/2} \int_k^{\infty} e^{-(\sqrt{2 \alpha}x)^2/2}\,dx =\\ -\frac{d}{d\alpha}\Big|_{1/2} \Big( \frac{1}{\sqrt{2\alpha}} \int_{\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/989229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$ Please, help me to understand this problem:
Let $\alpha=\sqrt[3]{2}$ be a root of the polynomial $x^3-2$.
a) Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$;
b) If... | In a) how do you make the assumption that $f(\alpha)=\alpha$? In b) you are asked to compute the Galois group of $L$ over $\mathbb{Q}$.
Hint for both parts of the exercise: First show that any automorphism of the fields in question fixes $\mathbb{Q}$. Then try to find out where it could possibly map $\alpha$ and $u$; s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/989321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Show that ${n \choose k}{k \choose m} = {n \choose m}{n-m \choose k-m}$ Show that $\begin{align}{n \choose k}{k \choose m} = {n \choose m}{n-m \choose k-m}.\end{align}$
Not sure how to approach this exactly. I've tried to use the property $\begin{align}{n \choose k} = {n \choose n-k} \end{align}$, which seems like it c... | Hint: $$\binom{a}{b} = \frac{a!}{b!(a-b)!}.$$
| {
"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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Trace of $L^p$ function For $U$ a bounded domain in $\mathbb{R}^n$, why is it that, in general, an $L^p$, $1\leq p<+\infty$, function does not have a trace on the boundary of $U$?
Thanks in advance.
| Here is a counterexample for $n=1$, which can me easily modified to hold in higher dimensions:
Consider the sequence $f_n(x):=(1-nx)_+$ for $x\in (0,1)$. $(\cdot)_+$ denotes the positive part. By dominated convergence $f_n\to 0$ in $L^p(0,1)$. Assume there exists a contiunous and linear trace operator $T$ on $L^p(0,1)$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question regarding notation in algebraic topology My class has not been following a book and my professor's last bit of notation is a bit confusing to me.
This is the goal. We are given a path-connected space $Y$ and $H$ a subgroup of $\pi_1(Y,y)$. We want to find a cover $X$, of $Y$ such that $X$ is path-connected, $p... | That notation doesn't mean much to me, either. But to try to help:
I would guess that perhaps the prof has defined an action of $\pi_1(Y)$ on the fiber $p^{-1}(y)$, an action denoted by "*". So this says that the path $\gamma$, acting on $x'$, gives $x$. (The definition of the action would be "lift the path to one in X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/989626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum of coefficients in binomial theory. While trying to get introduced to binomial theory at university's website, I learned about the sum of binomial coefficients, and they showed me some of the features, and one of them was the pyramid of coefficients sum which is:
$$1 = 1$$
$$1+1 = 2$$
$$1+2+1 = 4$$
$$1+3+3+1 = 8$$
... | There are a variety of different ways to understand this. One of those is the combinatorial way of understanding it, which says that binomial coefficients count subsets of specified sizes (i.e. "combinations") and the sum of all the coefficients in one row counts all subsets of all sizes, of a set of a specified size.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Show that $\boldsymbol{\mathrm{F}}$ is independent of path.
Consider a vector field $\boldsymbol{\mathrm{F}}(x,y) = \langle 2xy,
x^2 \rangle$ and three curves that start at $(1, 2)$ and end at
$(3,2)$. Explain why $$\int\limits_{C}\boldsymbol{\mathrm{F}}\cdot
\text{ d}\boldsymbol{\mathrm{r}}$$ has the same value ... | As you showed, the vector field is conservative, so it doesn't matter which path you take, the only thing you need are the starting and end point.
First, as $\mathrm{F}$ is conservative, you have to calculate a function $f$ such that $\nabla f=\mathrm{F}$. An easy way to do this is using this formula:
$$\displaystyle f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/989767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate limit $\lim_{n\rightarrow\infty}\frac{(4n-100)^{4n-100}n^n}{(3n)^{3n}(2n)^{2n}}?$ The limit $$\lim_{n\rightarrow\infty}\dfrac{(4n)^{4n}n^n}{(3n)^{3n}(2n)^{2n}}$$ can be calculated as $\lim_{n\rightarrow\infty}\dfrac{4^{4n}}{3^{3n}2^{2n}}=\lim_{n\rightarrow\infty}\left(\dfrac{4^4}{3^32^2}\right)^n=\infty$.
Wha... | Hints:
Can you see that $(4n-100)^{4n-100} \sim (4n)^{4n}$ as $n\to \infty$?
Then $$\lim_{n\rightarrow\infty}\dfrac{(4n-100)^{4n-100}n^n}{(3n)^{3n}(2n)^{2n}}=\lim_{n\rightarrow\infty}\dfrac{(4n)^{4n}n^n}{(3n)^{3n}(2n)^{2n}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/989857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Percentages not adding up I have a series of percentages:
132/220 (60%) and 88/220 (40%).
Now when I break them down into subcategories and then recalculate the percentages them come out 5% different.
81/140 (58%) and 59/140 (42%) (percentages ROUNDED).
21/40 (52.5%) and 19/40 (47.5%).
14/20 (70%) and 6/20 (30%).
16/2... | If the samples/subcategories have different sizes, the average of the averages is not the same as the average of all numbers.
You have to make a weighted average: multiply each percentage by the size of the group, add them together and divide by the total size.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/989956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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An equation which has solution modulo every integer In the book Abstract Algebra by Dummit and Foote he remarks that there is an equation which has solutions modulo every integer but has no integer solutions. The equation he gives is $$3x^3+4y^3+5z^3=0$$My question is how do we prove that this has solutions modulo ever... | Here is another
$$ x^2 + y^2 + z^9 = 216, $$ where we allow $z$ negative or positive or zero as desired. Same conclusion for
$$ x^2 + y^2 + z^9 = 216 p^3, $$ with prime $p \equiv 1 \pmod 4.$
See PDF
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why doesn't $\arccos x = -\tfrac12\sqrt{3}$ have any solutions? I have this exercise with an unclear answer. The question is this:
$$\arccos x = -\frac{\sqrt3}{2}\,.$$
The answer is this:
$$\begin{gather*}
\varphi(x)= \arccos x\\
V_\varphi = [0,\pi]\\
-\frac{\sqrt3}{2}\notin V_\varphi\,.
\end{gather*}
$$
can someone th... | As a couple of others have already pointed out, $-\sqrt{3}/2$ is simply not in the range of the arccosine. Here's an explanation as to why that's true.
Here's the graph of the cosine function over the interval $[-\pi,2\pi]$:
The issue is that this function is not one-to-one. As a result, we must restrict it to an ap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/990230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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When to use Integral Substitution? $e^x$$(1+e^x)^{1\over{2}}$
why can't use integration by part,
What is meant by in the form of f(g(x))g'(x)?
Can you give a few example?
Thank you
| Integration by substitution says that
$$
\int f(g(x))g'(x) \; dx = \int f(u)\;du
$$
where $u = g(x)$. This is a tool that will let you compute the integral of all functions that look exactly like $f(g(x))g'(x)$. When you first see this, you might be thinking that this seems like a very specialised rule that only seem g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/990285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Square root and principal square root confusion A few months ago I asked a question about the $\pm$ symbol because I was confused about it... I still carry the same confusion (which really bugs me) but I think the real confusion has to do with the square root and principal square root. I hope I can finally grasp the co... | Q1: If $a,b$ are nonnegative, then $\sqrt{ab}=\sqrt a\sqrt b$. If $a<0$ or $b<0$ (and you are not into complex numbers), you better not talk about the square roots on the right.
Q2:
In general $a=b\implies f(a)=f(b)$.
Since $(x+8)^2$ and $1$ are both nonnegative, we can ltake $\sqrt{\ }$ for the function $f$ and obtai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/990394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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General solution of recurrence relation if two equal roots Consider the recurrence relation
$$
ax_{n+1}+bx_n+cx_{n-1}=0
$$
If the characteristic equation
$$
a\lambda^2+b\lambda+c=0
$$
has two equal roots, then the general solution is given by
$$
x_n=(A+nB)\alpha^n.
$$
Could you please explain that to me, I do not see t... | Let $D$ be defined as $(Dx)_n = x_{n+1}$. Assuming $a \neq 0$, we have
$x_{n+2}+{b \over a} x_{n+1} + {c \over a}x_n = 0$, or
$(D^2 +{b \over a} D + {c \over a}I)x = 0$. Suppose $\lambda$ is the repeated root of the quadratic $y^2+{b \over a} y + {c \over a} = 0$, then we can write the difference equation as
$(D-\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/990551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Expected value with two uniformly distributed random variables A surveyor wishes to lay out a square region with each side having length L. However, because of a measurement error, he instead lays out a rectangle in which the north–south sides both have length X and the east–west sides both have length Y. Suppose that ... | Hint:
If random variable $U$ is uniformly distributed over interval $[-1,1]$ then $U$ and $-U$ have the same distribution, so that $\mathbb{E}\left(U\right)=\mathbb{E}\left(-U\right)=-\mathbb{E}\left(U\right)$ hence: $$\mathbb{E}\left(U\right)=0$$
Note that $X=L+A.U$ is uniformly distributed over interval $[L-A,L+A]$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does induction procedure of Euler characteristic fail for non-convex polyhedra? What am I missing? Euler characteristic of convex polyhedra is always $V-E+F=2$.
Induction procedure reduces edges and vertices until we are down to one vertex whose $V-E+F=2$ and hence you are done.
The same induction procedure should... | The proof I know that $V-E+F=1$ for convex polyhedra starts by removing a face, then stereographically projecting the complement into the plane, to get a convex polygon in the plane subdivided into smaller polygons. Then one removes pairs of edges and vertices or edges and faces until you get down to a point.
The thin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/990731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\int_0^{2\pi}\frac{3a\sin^2\theta}{(1-a\cos \theta)^4}$ or $\int_0^{2\pi}\frac{\cos\theta}{(1-a\cos\theta)^3}=\frac{3a\pi}{(1-a^2)^{5/2}}$ While doing some mathematical modelling of planetary orbits I have come up with two definite integrals $D_1$ and $D_2$ which appear to produce the same result R when tested w... | With
$$
\int_{0}^{2\pi} \frac{1}{1-a \cos \theta} d \theta=\frac{2\pi}{\sqrt{1-a^2}}
$$
evaluate the following integrals successively
\begin{align}\int_{0}^{2\pi} \frac{1}{(1-a \cos \theta)^2} d\theta
=&\frac{d}{da} \int_{0}^{2\pi} \frac{a}{1-a \cos \theta} d \theta=\frac{2\pi}{(1-a^2)^{3/2}}\\
\int_{0}^{2\pi} \frac{\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/990813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Multivariable limit .... no L'Hopital rule? I am looking a bit at limits for multivariable functions by myself, and I can't figure it out; my book only mentions them shortly, but now I am looking at an "assignments for those interested" and it says
$$\lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy} \cos(x+y)$$
But that's a ... | Hint: $$\lim_{(x,y)\to(0,0)} \frac{\sin (xy)}{xy} = 1$$
And $\cos (x + y)$ is continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/990915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Rules of i ($\sqrt -1$) to a power $i^{2014}$ power =?
A. $i^{13}$
B. $ i ^{203}$
C. $i^{726}$
D. $i^{1993}$
E. $i^{2100}$
I don't understand the concept that powers of i repeat in fours and that "two powers of i are equal if their remainders are equal upon division by four". I especially don't understand the second pa... | You know that $i^4=1$, because
$$
i^4=(i^2)^2=(-1)^2=1
$$
Now $i^5=i^4\cdot i=i$, $i^6=i^4\cdot i^2=-1$, $i^7=i^4\cdot i^3=-i$ and finally $i^8=i^4\cdot i^4=1$. You can go on forever, the powers of $i$ will repeat the same pattern
$$\dots\quad i\quad {-1} \quad {-i}\quad 1 \quad\dots$$
If $n=4q+r$ with $0\le r<4$, that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/990994",
"timestamp": "2023-03-29T00:00:00",
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Coprime numbers and equations Suppose $~m~$ and $~n~$ are coprime and both of them are greater than one. Is it right that equation $~mx + ny = (m-1)(n-1)~$ has solutions over non-negative integers?
For example $~ (x,y) = (6,0) ~$ satisfies equation $~3x + 10y = 18~$.
Thanks in advance.
| Yes, this equation has a (unique) solution in non-negative integers.
Suppose $m > n$. Looking at it modulo $n$, if that equation holds, we have
$$mx \equiv (m-1)(n-1) \equiv -(m-1) \pmod{n}.$$
There is a unique integer $x_0 \in [0,n-1]$ with $mx_0 \equiv 1-m \pmod{n}$. Then $(m-1)(n-1) - mx_0$ is a multiple of $n$, and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Distance between two skew lines I have 2 skew lines $L_A$ and $L_B$ and 2 parallel planes $H_A$ and $H_B$.
The line $L_A$ lies in $H_A$ and $L_B$ in $H_B$. If the equations of $H_A$ and $H_B$ are given like this:
$x+y+z = 0$ (for $H_A$)
$x+y+z = 5$ (for $H_B$)
Can I just simply say that the distance between two lines ... | Yes,the shortest distance between the skew lines will equal the distance between the planes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/991154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $S$ is a subring of $\mathbb{Z}_{28}$ Question: $S=\{0,4,8,12,16,20,24\}.$ Prove that $S$ is a subring of $\mathbb{Z}_{28}$
Confusion 1: This might be a dumb question, but when we refer to $[4]$ in $S$, for example, is that the congruent class of $4$ modulo $28$ in our case? Because I assume $[4]=\{x\in\math... | Since you are in $\mathbb{Z}_{28}$, everything is modulo $28$ i.e. $a=[a]=\{a+28k:k\in\mathbb{Z}\}$ for all $a\in\{0,2,\dots,27\}$.
Now you have noticed that all elements in $S$ are multiples of $4$, then it follows that if you add any two elements in $S$, the answer must still be a multiple of $4$, which must be an el... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/991241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Recursion and Time Complexity Concept The question prompt is as follows:
Consider the function $f(n)$ defined as:
$$f(n) = \begin{cases}n(n-1)f(n-2) & n > 1\\1 & n=0,\; n=1\end{cases}$$
How may be $g(n)$ be defined to make $f(n) \in \mathcal O(g(n))$? (I'm supposed to choose one response from $1$ and a response fro... | Your problem is that you're seeking the wrong thing limit. In order for $f(n)$ to be $\mathcal O(g(n))$, you want:
$$\lim_{n\to\infty} \frac{f(n)}{g(n)} = c, \quad 0\leq c < \infty$$
Clearly, if we choose $g(n) = n!\cdot g(0)$ and $g(0) = 100$, the resulting value is $c=\frac{1}{100}$.
You probably were confused with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/991351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Riemann Zeta Function at $s = 1 + 2 \pi i n / \ln 2$ I am aware that the function defined by
$$ \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots $$ for $Re(s)>1$ can be extended to a function defined for $Re(s)>0$ by writing
$$ \zeta(s) = \frac{1}{1-2^{1-s}} \cdot \left( \frac{1}{1^s} - \frac{1}{2^s... | This issue has been adressed by J. Sondow (2003) here: Zeros of the alternating zeta function on the line R(s)=1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/991498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to prove the amounts of dominos with x+y=n+k = x+y=n-k? I've been trying to answer this question for hours with no luck at all.. The question is the following:
Question
*
*Imagine we have domino blocks of the following shape: [ x | y ] with x, y ∈ [0..n].
*Imagine 0 ≤ k ≤ n.
Show that the amount of blocks with
... | For each $x$ from $0$ to $n-k$ let $y=n-k-x$. This makes $n-k+1$ blocks with $x+y=n-k$. But if $n-k+1$ is even, we must count only a half of them, because the block $[x|y]$ is the same as $[y|x]$. And if $x-k+1$ is odd, there is a block that has no pair, namely $[(x-k)/2, (x-k)/2]$.
For each block $[x|y]$ with $x+y=n-k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/991610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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What is the generalized form of this identity and how to interpret it? I have learnt that for any inner product space of $\mathbb{C}$, we have
$$\langle f,g\rangle=\frac{1}{4}\Big[||f+g||^2-||f-g||^2+i\big(||f+ig||^2-||f-ig||^2\big) \Big]$$
I know how to prove it, but I have a hard time to remember the formula. I think... | This was an answer to the original question, which specifically asked for help memorizing the identity.
You could try to remember it "in parts"
*
*$[f,g] = \|f+g\|^2-\|f-g\|^2$
*$\langle f,g \rangle = \frac{1}{4}([f,g]+i[f,ig])$
If someone can offer a geometric interpretation of $[f,g],$ that would certainly hel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/991714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
solve this simple equation:$ax^2+byx+c=0$ I need help solving the diophantine equation:$$ax^2+bxy+c=0$$
The quadratic formula does not seem to help much. Please help.
| Usually we solve Diophantine equations for positive integer solutions. Therefore I think you need positive integer solutions.
Consider the equation $ax^2+bxy+c=0$ as a quadratic equation of $x.$ For integer values of $x$ discriminant of this should be a perfect square. $$△_x=(by)^2-4ac=z^2$$ for some integer $z.$ Since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/991773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $A$ is connected, is at least one of the sets $\mathrm{Int}A$ and $\mathrm{Bd}A$ connected? If $A$ is a connected subset of $X$, does it follow at least one of the sets $\mathrm{Int}A$ and $\mathrm{Bd}A$ are connected?
I have found counterexamples showing that they not both need to be connected, and was wondering wh... | Thanks to Daniel Fischer for his correction!
Consider the union $X$ of two full triangles $T,T'$ in the plane that meet at a vertex, and remove a small disk from the interior of $T$. Then $X$ is connected, and neither the interior nor the boundary of $X$ are connected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/991879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why can't prime numbers satisfy the Pythagoras Theorem? That is, why can't a set of 3 prime numbers be a Pythagorean triplet? Suppose $a$, $b$ and $c$ are three prime numbers.
How to prove that $a^2 + b^2 \neq c^2$?
| The sum of two odd numbers are even, so one of the numbers must be $2$.
If $a$ or $b$ are $2$ we have $a^2+4=c^2$ or $4=(c+a)(c-a)$ Since $c-a$ and $c+a$ have the same parity, this is impossible.
If $c=2$ we have $a^2+b^2=4$ but since $a$ and $b$ are positive, both must be $1$, but $1^2+1^2=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/991947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 7,
"answer_id": 0
} |
Show complex solutions exist Let A be a complex number and B a real number. Show that the equation
$\,\lvert z^2\rvert+ \mathrm{Re}\, (Az) + B = 0\,$ has a solution iff $\,\lvert A^2\rvert \geq 4B$. If this is so, show that the solution set is a circle or a single point.
Well i am trying to do the first part first. So... | We have
$$
0=\lvert z^2\rvert+ \mathrm{Re}\, (Az) + B =
z\overline{z}+\frac{1}{2}(Az+\overline{Az})+\frac{1}{4}\lvert A\rvert^2-\frac{1}{4}\lvert A\rvert^2+B=\left\lvert z+\frac{1}{2}A\right\rvert^2-\frac{1}{4}\lvert A\rvert^2+B
$$
If $4B>\lvert A\rvert^2>0$, then $\lvert z^2\rvert+ \mathrm{Re}\, (Az) + B=\left\lvert z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
If I roll three dice at the same time, how many ways the sides can sum up to $13$? If I rolled $3$ dice how many combinations are there that result in sum of dots appeared on those dice be $13$?
| At the lower division math level we can do the following easily given the low number of combinations:
1)list the number of potential combinations
116
265
355
364
454
2) Now we find out the numbers of way that we can arrange the listed numbers in which it is:
3
6
3
6
3 respectively
thus when we add the numbers we get 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 5
} |
How to find the value of a function where a relation is given If $g(x)$ is a polynomial satisfying
$$g(x)g(y)=g(x)+g(y)+g(xy)-2$$
For all real value of $x$ and $y$. And
$$ g(2)=5$$
then find
$$g(3)=?$$
| We have that $\big(g(x)-1\big)\big(g(y)-1\big)=g(xy)-1$.
Set $f(x)=g(x)-1$, then
$$
f(xy)=f(x)f(y)\tag{1}
$$
Also, $f(x)\ge 0$, for all $x>0$, as $f(x)=(f(\sqrt{x}))^2$
Thus $f(x^n)=(f(x))^n$, as as $f(x)\ge 0$, then $(f(x^{1/n}))^n=f(x)\to f(x^{1/n})=(f(x))^{1/n}$. Altogether $f(x^{m/n})=(f(x))^{m/n}$, and as $f$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the power set $P(S)$ for $S=\{\emptyset, \{\emptyset\}, \{\emptyset \{\emptyset\}\}\}$ Find the power set $P(S)$ for $S=\{\emptyset, \{\emptyset\}, \{\emptyset \{\emptyset\}\}\}$
OK this problem confuses me for many reasons, but here is what I know. The cardinality of a set is $2^n$ where $n$ is the number of ele... | Every set has the empty set as an element. However #(S) should be zero where #(S) is the cardinality of set S.
The power set of the empty set or sets of empty sets should be simply the empty set.
Remember however that the power set of a set is not a cardinality. It is a set constructor that builds a set from the eleme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Integrate without substitution I'm wondering how I could integrate the following without substitution:$$\int \frac{4}{1 + e^{-x}}dx$$
I know we can factor out the constant so that $4 * \int \frac{1}{1 + e^{-x}}$ but I'm stumped as of what to do next. Could anyone help me out?
| $$\int \frac{1}{1+\frac{1}{e^x}} dx = \int \frac{e^x}{1+ e^x} dx = \ln (1 + e^x) + C $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/992366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$f(x)=x^6(x-7)^3$ over the range $[-14, 10]$ Where does the function achieve its global minimum?
I've identified the global min as $14/3$ and was told this was incorrect. I tried its corresponding y value as well and received the same answer. Other critical points are $7$ and $0$ and they are also incorrect.
| In $x=\frac{14}{3}$, $f$ achieve local minimun, note that
$f(-4)=(-4)^{6}(-4-7)^{3}= -5. 451\,8\times 10^{6}$
$f(14/3)=(14/3)^{6}(14/3-7)^{3}=-\frac{2582\,630\,848}{19\,683}=-1.312\,1\times 10^{5}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/992477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Usage of law of sines The vertex angle of an isosceles triangle is 35 degrees. The length of the base is 10 centimeters. How many centimeters are in the perimeter?
I understand the problem as there are two sides with length 10 and one side of unknown length.
I used laws of sines to find the side corresponding to the 35... | The other answer does not use the law of sines, as your title states.
You're given that the side of $10$ lies opposite the vertex angle of $35^\circ$.
First find the other two angles (the "base angles"). As the triangle is isoceles, they are equal. Denote one base angle as $\theta$.
By the angle sum of the triangle, $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\sum_{i=0}^n4^i$ = $1/3(4^{n+1} - 1)$ $\sum_{i=0}^n4^i$ = $1/3(4^{n+1} - 1)$
Attempt:
Let $n =0$ $4^0 = 1 \text{ and } (4^{0+1} -1)/3 = 1$
Assume true at $n = k \text{ so we have} \sum_{i=0}^k4^i = 1/3(4^{k+1} -1)$
The part I'm stuck at is the 3rd step. Can someone point me in the right direction after this... | Hint: $\displaystyle \sum_{i=0}^{k+1} 4^i = \displaystyle \sum_{i=0}^k 4^i + 4^{k+1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/992746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inverse of an infinitely large matrix? This is probably a trivial problem for some people, but I've spent quite some time on it:
What is the inverse of the infinite matrix $$
\left[\begin{matrix}
0^0 & 0^1 & 0^2 & 0^3 & \ldots\\
1^0 & 1^1 & 1^2 & 1^3 & \ldots\\
2^0 & 2^1 & 2^2 & 2^3 & \ldots\\
3^0 & 3^1 & 3^2 & 3^3 & \... | Numerically, we get some very interesting results for the matrix $M_{ij}=(i-1)^{j-1}$ if it is expressed in the Fourier basis with alternating row signs,
$$A := \begin{bmatrix}1 \\ & -1 \\ && 1 \\&&& -1 \\ &&&& \ddots \end{bmatrix}\mathcal{F}^{-1} \left[\begin{matrix}
0^0 & 0^1 & 0^2 & 0^3 & \ldots\\
1^0 & 1^1 & 1^2 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
How can i prove this interesting limit Was just playing around any thought it was interesting.
knowing now that $a>b>0$
$\lim\limits_{x\rightarrow \infty} (a^x-b^x)^{\frac{a+b}{x}} = a^{a+b}$
| It can be done as follows:
$$
\lim_{x\to\infty} \left(a^x-b^x\right)^{\frac{a+b}{x}}=\lim_{x\to\infty} a^{a+b}\cdot\left(1-\left(\frac{b}{a}\right)^x\right)^{\frac{a+b}{x}}=
$$
Now we have $\lim_{x\to\infty} \left(\frac{b}{a}\right)^x=0$ and $\lim_{x\to\infty} \frac{a+b}{x}=0$ and therefore:
$$
\lim_{x\to\infty} a^{a+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to show that there is no odd $n$ bigger than 100 such that the group $U(n)$ will contain $2^n$ number of elements? Consider $U(n):=\{1\leq r\leq n: (r, n)=1\}$. Under multiplication modulo $n$ it forms an abelian group. Its order will be $\varphi(n)$ i.e. $\varphi(n)=|U(n)|$.
Assume that $n\geq 100$. Then my quest... | You want positive integers $n$ such that $\varphi(n)$ is a power of $2$. But if $n=p_1^{a_1}\cdots p_m^{a_m}$, then $\varphi(n)=\displaystyle\prod_{i=1}^m (p_i^{a_i}-p_1^{a_1-1})$.
Therefore, the only primes you can have are $2$ and those such that $p_i-1$ is a power of $2$. The only such primes which are known are $3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/992966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$ If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that:
$$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$
I started with Cauchy–Schwa... | Observe that the inequality follows first from Cauchy-Schwarz inequality and then by AM-GM inequality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/993084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How prove $\pi^2>2^\pi$ show that
$$\pi^2>2^\pi$$
I use computer found
$$\pi^2-2^\pi\approx 1.044\cdots,$$
can see this
I know
$$\Longleftrightarrow \dfrac{\ln{\pi}}{\pi}>\dfrac{\ln{2}}{2}$$
so let
$$f(x)=\dfrac{\ln{x}}{x}$$
so
$$f'(x)=\dfrac{1-\ln{x}}{x^2}=0,x=e$$
so $f(x)$ is Strictly increasing
on $(2,e)$, and i... | Hint:
$$\frac{\ln 2}{2}=\frac{\ln 4}{4}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/993170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
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