Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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What is the integral of $e^{-\alpha|t|}e^{ivt}e^{-ipt}$ over $\mathbb{R}$? Given $\alpha, v > 0$, I'm supposed to get to the following result:
$$
g(p) = \int_{-\infty}^{\infty}e^{-\alpha|t|}e^{ivt}e^{-ipt}dt =\frac{2\alpha}{\alpha^{2}+(v-p)^{2}}
$$
I've tried calculating it as:
$$
\int_{-\infty}^{0} e^{((v-p)i+\alpha)t... | Note that: $$I_1 = \int_{-\infty}^{0} e^{((v-p)i+\alpha)t} dt = \lim_{b \to -\infty} \int_{b}^{0} e^{((v-p)i +\alpha)t} dt = \lim_{b \to -\infty} \frac{1}{((v-p)i +\alpha)}[1-e^{((v-p)i +\alpha) b}] = \frac{1}{((v-p)i +\alpha)}$$
Can you similarly see the next integral as well?
| {
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"timestamp": "2023-03-29T00:00:00",
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Finding the range of $\frac{x^2}{x^2-9}$ I am a student who is studying about functions (only basic ones) and was practicing till I found this -
Finding the range of
$$ \frac{x^2}{x^2-9} $$
Since I have only learnt the basics , I can only play around with the numbers and not use limits (which was what I found online)... | The domain is the set of $x$ values such that the function is defined. Here the expression exists for any $x\notin \{-3,3\}$.
The range is the set of $y$ values for which the equation $y=f(x)$ has solutions.
Indeed,
$$y=\frac{x^2}{x^2-9}$$ can be written
$$x^2=\frac{9y}{y-1}$$ and the RHS is only non-negative and def... | {
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"url": "https://math.stackexchange.com/questions/2553845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using the natural homomorphism $\mathbb Z$ to $\mathbb Z_5$
Prove that $x^4+10x^3+7$ is irreducible in $\mathbb Q[x]$ by using the natural homomorphism from $\mathbb Z$ to $\mathbb Z_5$.
So I would assume we should rewrite our polynomial, maybe as $(x + 10) x^3 + 7$? Then in terms of $\mathbb Z_5, (x+2\cdot 5)x^3+(5+... | The other answers do not address why there cannot be a quadratic factor. I don't know if there's a clever thing I'm missing, but here's one way to see. Assume the quadratic splits over $\mathbb{Q}$ as
$$
x^4 + 2 = (x^2+ax+b)(x^2+cx+d)
$$
Then, passing to $\mathbb{F}_5$ we get the equations
$$
\begin{eqnarray}
d+ac+... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can only the existence of the right and left derivatives imply continuity? If $f:X\rightarrow \mathbb{R} \,$ is a function with $x_0 \in \overline{X} \,\setminus \partial(\overline{X}) $ such that :
$$\exists \,\,\,\,f'_-(x_0)=\lim_{x\rightarrow x_0^-}\dfrac{f(x)-f(x_0)}{x-x_0},$$
$$\exists \,\,\,\,f'_+(x_0)=\lim_{x\... | Suppose that$$\lim_{x\to{x_0}^+}\frac{f(x)-f(x_0)}{x-x_0}$$exists (in $\mathbb R$). Then\begin{align}\lim_{x\to{x_0}^+}f(x)&=f(x_0)+\lim_{x\to{x_0}^+}f(x)-f(x_0)\\&=f(x_0)+\lim_{x\to{x_0}^+}\left((x-x_0)\frac{f(x)-f(x_0)}{x-x_0}\right)\\&=f(x_0)+\lim_{x\to{x_0}^+}(x-x_0)\lim_{x\to{x_0}^+}\frac{f(x)-f(x_0)}{x-x_0}\\&=f(... | {
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Is every commutative ring contained in a field? I imagine the answer to this question is very simple, but I haven't been able to locate it.
Can every commutative ring be imbedded in a field? This seems very plausible to me, for it seems we could just take the "closure" under inverses.
Thanks!
|
A commutative ring can be embedded in a field iff it is an integral domain.
Indeed, if a ring can be embedded in a field then it cannot have zero divisors because fields cannot have zero divisors.
Conversely, every integral domain can be embedded in a field, namely, its field of fractions.
| {
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Differentiable functions $f'(x)=f(-x)^4f(x)$ Find all differentiable functions $f\colon \mathbb{R}\to\mathbb{R}$ with $f(0)=1$ such that $f'(x)=f(-x)^4f(x)$, for all $x \in \mathbb{R}$.
| The defining relation $f'(x)=f^4(-x)f(x)$ implies that $f$ is continuously differentiable (in fact it's even $C^\infty$).
Setting $-x$ in the relation $f'(x)=f^4(-x)f(x)$, one gets $\forall x, f'(-x) = f^4(x)f(-x)$.
Multiplying $f'(x)=f^4(-x)f(x)$ by $f^3(x)$ yields $$\forall x, f'(x)f^3(x) = f^4(-x)f^4(x)=[f^4(x)f(-x)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Homomorphisms S4 to Z Find all homomorphisms f: S4 -> Z.
I know that Ker(f) must be a normal subgroup in S4. So, Ker(f) must be {e}, V4, A4 or S4.
However, I don't know how to use this information.
Thanks.
| There is only one such homomorphism, the zero homomorphism (mapping everything to 0).
The reason is there every non-zero element of $\mathbb{Z}$ has infinite order, and the order of the image (under a homomorphism) of any element of finite order must be finite.
| {
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Interchanging two columns of a matrix will lead to change in the sign of determinant Interchanging two columns of a square matrix changes the sign of the determinant.
I know that this is true, and I do understand how it works. But is there any proof for this statement?
| Any proof of this result depends on a definition of determinant. Let us define it using permutations: $\det(A) = \sum_{\tau \in S_n}\operatorname{sgn}(\tau)\,a_{1,\tau(1)}a_{2,\tau(2)} \ldots a_{n,\tau(n)},\;$ where the sum is over all $n!$ permutations of the columns by elements in the symmetric group $S_n.\;$ See the... | {
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Writing in CNF that only one statement can be true
Consider 5 Boolean variables $x_1, x_2, x_3, x_4, x_5$.
*
*Write a propositional formula that expresses the fact that one and at most one among the Boolean variables $x_1, x_2, x_3, x_4, x_5$ is true.
*Compute a conjunctive normal form of this formula.
Your answer... | The proposition for "none are true" is: $(\neg x_1\wedge\neg x_2\wedge\neg x_3\wedge\neg x_4\wedge\neg x_5)$
The proposition for "only $x_1$ is true" is: $(x_1\wedge\neg x_2\wedge\neg x_3\wedge\neg x_4\wedge\neg x_5)$
And so forth. Use that to build a DNF for "at most one from the five is true," and simplify it (hint:... | {
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Helen Borrows Money
Helen borrows a sum of money from a bank at 12% convertible monthly and wishes to repay it by 24 monthly payments. In total, she will pay 584 of interest. Determine the size of the loan.
I have started by doing this:
The total amount paid back is given by $Pi(1+i)^n/(1+i)^n-1$ so the total inter... | The monthly interest rate is $i=\frac{i^{(12)}}{12}=\frac{12\%}{12}=1\%$.
The total interest is $I=nP-L$, where $P$ is the monthly payment, $n$ is the number of months and $L$ is the loan.
Then
$$\left\{
\begin{align}
I&=P(n-a_{\overline{n}|i})\\
L&=Pa_{\overline{n}|i}
\end{align}\right.\qquad \Longrightarrow\quad \b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2554925",
"timestamp": "2023-03-29T00:00:00",
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Suppose that $p$ is prime and $≡ 3\bmod4$ then $((p-1)/2)!≡-1\bmod p$ or $((p-1)/2)!≡1\bmod p$ Prove or disprove:
Suppose that $p$ is prime and $≡ 3\bmod4$ then $((p-1)/2)!≡-1\bmod p$ or $((p-1)/2)!≡1\bmod p$
After I checked it I see it is true statement
so by Wilson's theorem we have $(p-1)!≡-1\bmod p$
so $1\cdot2\cdo... | Your idea of using Wilson's theorem is correct, but when you get to
$$1\cdot2\cdot3\cdots\frac{p-1}2\cdot\frac{p+1}2\cdots(p-1) ≡ -1\bmod p$$
you need to take a different approach. Rewrite $p-1$ as $-1$, $p-2$ as $-2$ and so on until you get
$$1\cdot2\cdot3\cdots\frac{p-1}2\cdot\left(-\frac{p-1}2\right)\cdots(-1)≡-1\bm... | {
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Truth vs Lie possibility A, B and C tell the truth independently with probabilities $1/3, 1/4, 1/5$ respectively. C makes a statement and B says that C has lied , whereas A says that C has told the truth. Find the probability that C made a true statement.
I did
$$(1*1*3/3*4*5)/[(1*1*3/3*4*5)+(4*2*1/5*4*3)]$$
Is this co... | Your expression is a bit difficult to read, but yes, that is correct. What you've done I assume is work out the probability $p$ of A and C telling the truth and B lying, and the probability $q$ of A and C lying and B telling the truth. We know one of these two things has happened, and so the probability that it is the ... | {
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If $x,y\in E$ then $\frac{x+y}{2}\in E$. Prove that $E$ has an interior point Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Assume that if $x,y\in E$ then $\frac{x+y}{2}\in E$. Prove that $E$ has at least one interior point.
Here is what I have done:
(1). By regularity, for any $\epsilon>0$ we can fi... | The following result is quite well-known:
If $E$ and $F$ are measurable with $m(E),m(F)>0$, then $$E+F = \{x+y\mid x\in E,y\in F\}$$
contains an interval.
Then the condition on your $E$ says $$\frac{E+E}{2} \subset E$$
since $E+E$ contains an interval, so is $E$.
| {
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Prove using lagrange's mean value theorem
*
*If f' is continuous on [a,a+h] and derivable on (a,a+h) prove that there exists a real number c between a and a+h such that $f(a+h)=f(a)+hf'(a)+{\frac{h^{2}}{2}}f''(c)$.
I used lagrange's mean value theorem $f(x)$ will be also be continuous in $[a,a+h]$ and differ... | It can be shown by MVT on the integral of the error.
https://brilliant.org/wiki/taylors-theorem-with-lagrange-remainder/
| {
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Finding an orthogonal vector to two vectors in $\mathbb{R}^4$ "Let $u_1$, $u_2$ be to vectors in $\mathbb{R}^4$ $$u_1=(1,0,1,1) \text{ and } u_2=(1,1,0,3)$$
Provide a real vector which is orthogonal to both $u_1$ and $u_2$
So, I kind of guessed a vector $u_3=(1,-1,-1,0)$ which must be orthogonal to both since $$u_1 \cd... | Since you have only two vectors, you can work in $\mathbb R^3$. Let $u_3 = (a, b, c, 0)$, so that $u_1 \cdot u_3$ and $u_2 \cdot u_3$ only depend on the first three components of $u_1$ and $u_2$. So, call $v_i$ the vector of the first three components of $u_i$, and compute $v_3 = v_1 \times v_2 = (a, b, c)$ (this would... | {
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"question_score": "3",
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Multiplicities in the Regular Representation of a Semisimple Algebra Let $R$ be a finite dimensional associative algebra over a field $k$ and suppose that $R$ is semisimple, i.e., that we can express $R$ as a direct sum of left $R$-modules $$R\cong \oplus_i S_i^{\oplus n_i},$$ where the $S_i$ are non-isomorphic simple ... | The general description of $R$ is given by the Artin-Wedderburn Theorem: $R$ is a product of matrix algebras over finite-dimensional division algebras over $k$.
If $k$ is algebraically closed, then the only finite-dimensional division algebra over $k$ is itself, and your theorem follows from the representation theory o... | {
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Find the fixed points of the difference equation $ \ a_n=\frac{2}{7} a_{n-1}-1 \ $ Find the fixed points of the difference equation $ \ a_n=\frac{2}{7} a_{n-1}-1 \ $ . Classify the fixed points whether stable, unstable or Neutral.
Answer:
Let $ \ x \ $ be the fixed point.
Then,
$ x=\frac{2}{7} x-1 \\ \Rightarrow 7x... | Hint:
Let $a_n=b_n-\frac75$. The recurrence is
$$b_n-\frac75=\frac27\left(b_{n-1}-\frac75\right)-1=\frac27b_{n-1}-\frac75,$$ or
$$b_n=\frac27b_{n-1}.$$
You should be able to conclude.
| {
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Find all the solutions to the equation: $e^z = e^{-1 + i\pi}$ Im slightly stuck on hiw to attampt this...
I thought to write both sides in polar form,
ie.
$$e^x (\cos(y) + i\sin(y)) = e^{-1}(\cos(\pi)+i\sin(\pi))$$
I could equate it from here i think, but it specifically says in the question find ALL the solutions to ... | The exponential map is 1-1 on the strip $\{x+iy : -\pi \le y < \pi \}$ and $e^z = e^{z+2n\pi i}$ for all $n \in \mathbb{Z}$. Since $z = -1 +i\pi$ is one solution of it. So, $z = -1 + i(\pi +2n\pi)$ where $n \in \mathbb{Z}$ are all the solutions of your equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2556023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Problem with Structure Theorem of PID modules proof I am using Lang's Algebra to prove the Structure Theorem for finitely generated modules over PIDs, and I am having difficulties understanding the proof of the existence of the decomposition for $E(p)$. $E$ is a torsion module over a PID $R$, $p \in R$ prime element, a... | *
*It's actually $\overline E_p$ and not $\overline{E_p}$ : you take $F= \overline{E}$ and then $F_p$, not $F=E_p$ and then $\overline F$.
*The induction is over the number of generators : you assume the result for all modules that have less than $r$ generators, and prove it for those that have $r$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Gamma functions and limits How can someone calculate the limit $\lim_{n\rightarrow \infty}\frac{\Gamma(n+p)}{n^p\Gamma(n)}$ ? Is there an article about it? Is $\frac{\Gamma(n+p)}{n^p\Gamma(n)}$ greater than unity?
| Stirling's approximation
$$\Gamma(z) \sim \sqrt{\frac{2 \pi}{z}} \left( \frac{z}{\mathrm{e}} \right)^z $$
does the trick:
$$\frac{\Gamma(n+p)}{n^p \Gamma(n)} \sim \frac{ \sqrt{\frac{2 \pi}{n+p}} \left( \frac{n+p}{\mathrm{e}} \right)^{n+p}}{n^p \sqrt{\frac{2 \pi}{n}} \left( \frac{n}{\mathrm{e}} \right)^n}= \frac{1}{e^p}... | {
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Equivalence class of a number on a relation?
Lets say there is an equivalence relation $x\sim y$ if and only if $x-y$ is an integer. Find the equivalence class of the number $\frac13$.
I came up with $\left[\frac13\right]=\left\{\frac13\right\}$ but I'm not sure if its right. Any tips?
| Hint: Let $S=\{(x,y)\mid x-y\in \mathbb{Z}\}$. Suppose that $(1/3,y)\in S$, then $1/3-y=n$ for some $n\in\mathbb{Z}$ and $y=1/3-n$. Moreover, if $y=1/3-n$ for some $n\in\mathbb{Z}$ then $(1/3,y)\in \mathbb{Z}$. Thus $(1/3,y)\in S$ if and only if $y=1/3-n$ for some $n\in\mathbb{Z}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Showing that $\{X_n, n\geq 1 \}$ are independent if for $n\geq 2$ we have $\sigma(X_1,...,X_{n-1}) \perp \sigma(X_n)$ Looking for hints to proceed or to corroborate my solution (I found it still weak).
So, as the title says:
We want to show that $\{X_n, n\geq 1 \}$ are independent random variables if for $n\geq 2$ we h... | I believe your perp stands fro independence, not orthogonality. By definition a sequence is independent if each finite subset is. So we have to show that $\{X_1,X_2,...,X_n\}$ is independent for each N. This means $P\{X_1^{-1}(A_1) \cap ...\cap X_N^{-1} A_N\}$ is the product of $P\{X_i^{-1} A_i\}$. Just note that $\{X... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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trying to find the general term for a serie in an easy way I have a general expression which is the following:
$$b_0=7$$
$$b_n = 2b_{n−1}+7\cdot4^n\quad∀n∈\Bbb N^+$$
and that I have to it resolve in the easiest way possible. I know that I could use the generative function technique, but there is probably quicker ways.
... | Here's another way to look at this. Consider the general form
$$f_n=A\cdot B^n+Cf_{n-1}$$
Then we can write
$$\frac{f_n-Cf_{n-1}}{f_{n-1}-Cf_{n-2}}=\frac{A\cdot B^n}{A\cdot B^{n-1}}=B\\
$$
or
$$
f_n=af_{n-1}+bf_{n-2}\\
$$
where
$$
a=B+C\\
b=-B
$$
This is now in a familiar form for which we know the characteristic roots... | {
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How to find the shortest distance from a line to circle while their equations are given Consider a line $L$ of equation $ 3x + 4y - 25 = 0 $ and a real circle $C$ of real center of equation $ x^2 + y^2 -6x +8y =0 $
I need to find the shortest distance from the line $L$ to the circle $C$.
How do I find that?
I am new t... | Hint:
Any point on the circle can be set as $$P(3+5\cos t,5\sin t-4)$$
The distance of this point from $L$ will be $$\dfrac{|3(3+5\cos t)+4(5\sin t-4)-25|}{\sqrt{3^2+4^2}}$$
$3(3+5\cos t)+4(5\sin t-4)-25=5(3\cos t+4\sin t)-32$
Now $-\sqrt{3^2+4^2}\le3\cos t+4\sin t\le\sqrt{3^2+4^2}$
$\iff-5\cdot5-32\le5(3\cos t+4\sin ... | {
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Prove that $ C_1'\cap Z=C_2'\cap Z $ if and only if $ C_1 $ and $ C_2 $ touch at $ \xi $. This is Exercise II.4.1 in Shafarevich's book Basic Algebraic Geometry, second edition.
Suppose that $\dim X = 2 $ and that $ \xi \in X $ is a nonsingular point. Let $ C_1, C_2 \in X $ be two curves passing through $ \xi $ and n... | I was also stuck on this problem for a while. I think I have a solution, but I find it a bit hand-wavey. Leaving it here for future visitors in hopes that somebody can improve it.
$X$ is 2-dimensional. We can choose $u_i$ so $X$ is locally given by $u_3=\ldots=u_N=0 ,$
$C_1$ given by $u_1=0$, $C_2$ by $F(u_1,u_2)=0$. O... | {
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Determine if $x=0$ is a point of relative extremum for $f(x)= \sin(x) + \frac{x^3}{6}$
Determine if $x=0$ is a point of relative extremum for $f(x)= \sin(x)
+ \frac{x^3}{6}$
I am trying to use this test
Here,
$f(x)= \sin(x)
+ \frac{x^3}{6}$
$f'(x)=\cos(x) + \frac{x^2}{2} \Rightarrow f'(0)=1 \neq 0$
So I am unable to... | Since the first derivative is positive, the function increases at that point.
| {
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} |
Calculate the limit using de L'Hopital's rule Calculate the following limit:
$\lim_{x \to +\infty}(\sqrt{x}-\log x)$
I started like this:
$\lim_{x \to +\infty}(\sqrt{x}-\log x)=[\infty-\infty]=\lim_{x \to +\infty}\frac{(x-(\log x)^2)}{(\sqrt{x}+\log x)}=$
but that's not a good way...
I would be gratefull for any tips.... | Hint
If you have to use l'Hôpital; this limit is easier to find (*):
$$\lim_{x \to +\infty} \frac{\sqrt{x}}{\log x}
= \lim_{x \to +\infty} \frac{\frac{1}{2\sqrt{x}}}{\frac{1}{x}} =\lim_{x \to +\infty}\frac{\sqrt{x}}{2} = +\infty$$
Can you see how this would help for your limit as well?
If not (hoover over), rewrite:
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof of $ \cos \alpha + \sin \beta + \cos \gamma = 4* \sin( \frac\alpha2 + 45°)* \sin \frac\beta2 * \sin (\frac\gamma2 + 45°) $ Proof of
$$ \cos \alpha + \sin \beta + \cos \gamma = 4* \sin( \frac\alpha2 + 45°)* \sin \frac\beta2 * \sin (\frac\gamma2 + 45°) $$
if $ \alpha + \beta + \gamma = \fracπ2 $
I tried to simpli... | use that $$\sin\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}\left(\left(\sin(\frac{x}{2}\right)+\cos\left(\frac{x}{2}\right)\right)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Divergence theroem I have a problem applying the divergence theorem in two problems.
The first asks me to calculate $\int \int F · N dS$ where $F(x,y,z)=(x^2 + \sin z, x y + \cos z, e^y)$ in in the cylinder $x^2 + y^2=4$ limited by the planes $XY$ and $x+z=6$.
I compute divergence and $\mathop{div}{F} = 3 x$.
To comp... | Your coordinates should be
\begin{align*}
x &= r \cos \alpha \\
y &= \color{red}{r \sin \alpha} \\
z &= t
\end{align*}
You have a different expression for $y$. I don't think it matters in the integration, however, as a coincidence.
The volume integral is
$$
\iiint_E 3x\,dV = \int_0^{2\pi}\int_0^2 \int... | {
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"source": "stackexchange",
"question_score": "1",
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$U$ is timelike if its orthogonal complement $U^\perp$ is spacelike Consider the bilinear form $\left<x,y\right>_{n,1} = \sum_{j=1}^n x_j y_j - x_{n+1} y_{n+1}$ on $\mathbb{R}^{n+1}$.
A vector $x \in \mathbb{R}^{n+1}$ is said to be timelike if $\left<x,x\right>_{n,1} > 0$ while $x$ is called spacelike if $\left<x,x\rig... | Suppose that $U^{\perp}$ is spacelike. Because $U^{\perp}$ nondegenerate, then the whole space $V = U \oplus U^{\perp}$. Suppose the contrary that $U$ is not timelike (i.e there is no timelike vector in $U$). Then for any $u \in U$, $g(u,u) \leq 0$. So for any $v \in V$ can be expressed as $v = u + w$ where $u \in U$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2557500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Isomorphic Groups Example I've been learning about isomorphisms (Two groups $(G, \cdot)$ and $(H, \circ)$ are isomorphic if there exists a one-to-one and onto map $\phi : G \rightarrow H$ such that the group operations is preserved; that is $\phi(a \cdot b) = \phi (a) \circ \phi (b)$ for all $a$ and $b$ in $G$.) but I ... | The group $\langle \{1\}, \times\rangle$ is isomorphic to $\langle \{0\}, +\rangle$ because they are both groups containing only the unit of the operation. The isomorphism $0\leftrightarrow 1$ preserves the unit, and every possible application of the group operation $0+0 = 0 \iff 1\times 1 = 1$.
Other examples include ... | {
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How to draw a triangle with knowing the length of 3 heights only???? I tried it,then I found that the product of the two parts of same height will be equal to the other 2 products getting in the same way,Then I tried to do it by drawing it as the bisecting chords of a circle,the product will be same,but not getting the... | OK let's define some variables first. Let $[ABC]$ be the area of the triangle, and $h_A,$ $h_B,$ and $h_C$ be the heights, respectively. Also, define $a,$ $b,$ and $c$ to be the lengths of segments $AB,$ $AC,$ and $BC,$ respectively.
Note that, trivially, $a=\frac{2[ABC]}{h_a},$ and so on. Therefore, the sides must be... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Reciprocal derivative of function: $(f^{-1})'(\frac{1}{2} \sqrt{2})$ for $f(x)=\cos x$
"The function $f(x)=\cos(x)$ has an inverse/reciprocal (Not sure which one is the correct translation) function arccos in the interval $[0;\pi]$
Determine the derivative of $$(f^{-1})'(\frac{1}{2} \sqrt{2})$$
So, I guess the step... | Use the definition $$(f^{-1})’(x)=\frac{1}{f’(f^{-1}(x))}$$ and see what you get. Note that $f(x)=\cos x$ and $f’(x)=-\sin x$. Also, that $f^{-1}(\frac{1}{\sqrt{2}})=\frac{\pi}{4}$.
EDIT:
We need $(f^{-1})’(\frac{1}{\sqrt{2}})$. Using the definition, we get, $$(f^{-1})’(\frac{1}{\sqrt{2}})=\frac{1}{f’(\frac{\pi}{4})}=... | {
"language": "en",
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Show that $(K,\circ)$ is a group. Let $K$ be the set of functions defined by : $f : \mathbb C \times \mathbb C \to \mathbb C \times \mathbb C$, such that $\exists a \in \mathbb C, \exists b \in \mathbb C$, $a,b$ not simultaneously equal to zero with : $f(u,v) = (au+bv,-\bar b u+\bar a v$)
Q:Show that $(K,\circ)$ is a ... | According to you, you have already shown the closure of such functions. Also, associativity is always true for composition of functions, no matter what. To prove that $K$ is a group under composition of functions, we have to show the existence of an identity element in the group, and the existence of the inverse of any... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why are there two terms for the values of a function but only one for its inputs? Every function has a domain, a codomain and an image, the last being a subset of the codomain.
My question, for which I ask for your opinions, is "Why is the domain not split into two parts in a similar fashion to the codomain?".
I believ... | Short answer: It's the definition of a function.
Long answer: Because a function maps everything in its domain to somewhere in the codomain - in order to define a function, you need to say where every point in its domain gets mapped to in its codomain, otherwise it's not a well-defined function. If a function only mapp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2558514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For $f$ in dual space, there exists $x$ with norm 1 and $f(x)=\|f\|$ if space is reflexive (and nontrivial) Let $X\ne\{0\}$ be a reflexive space and let $f\in X^*$, where $X^*$ is the dual of $X$. I want to know: in general, does there exist an $x\in X$ with $\|x\|=1$, and $f(x)=\|f\|$, where $\|f\|$ is defined as $\su... | If the space is reflexive than the immersion
$$ \iota:X\to X^{**}, x\mapsto x(L):=L(x) $$
is a linear bijection. Now let $f\in X^*$ and define the following map
$$ L: \mathbb{R}f:=\{g\in X^*: g=\alpha f, \alpha\in \mathbb{R}\}\to \mathbb{R}, g=\alpha f\mapsto \alpha\|f\|. $$
It is well defined and continuous, moreover ... | {
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"timestamp": "2023-03-29T00:00:00",
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Ratio of an inscribed circle's tangent to original square In the diagram, the circle is inscribed within square $PQRS,$ $\overline{UT}$ is tangent to the circle, and $RU$ is $\frac{1}{4}$ of $RS.$ What is $\frac{RT}{RQ}$?
| Suppose that the circle touches $RS$, $TU$ and $RQ$ at $X$, $Y$ and $Z$ respectively.
If $RQ=4$, $RU=1$. Let $RT=x$.
Then $TU=TY+YU=TZ+XU=2-x+1$.
So $1^2+x^2=(3-x)^2$ and thus $1+x^2=9-6x+x^2$.
$\displaystyle x=\frac{4}{3}$.
$\displaystyle \frac{RT}{RQ}=\frac{1}{3}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intersection of two x powers Many months ago in class I came up with the problem: $$x^{(x+1)} = (x+1)^x$$
Using the solve function on my calculator I have found that the answer is around 2.29... This is backed up by the graph. However I was determined to find the inverse function where: $$x = f(y) $$ or find the answer... | We can approximate the solution building around $x=2$ the Taylor expansion of function $$f(x)=(x+1)\log(x)-x \log(x+1)$$ the first derivatives of which being$$f'(x)=\frac{1}{x}+\frac{1}{x+1}+\log (x)-\log (x+1)$$
$$f''(x)=-\frac{x^2+x+1}{x^2 (x+1)^2}\qquad f'''(x)=\frac{(2 x+1) \left(x^2+x+2\right)}{x^3 (x+1)^3}$$ Thi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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The intersection of a sequence of bases of a Banach space Let $V$ be a Banach space
Let $\{B_n\}_{n \in \mathbb{N}}$ be a sequence of subsets of $V$ such that:
$$
B_{n+1}
\subsetneq
B_n
$$
and
$$
\bigcap_{n=1}^\infty B_n = \{b_0\}
$$
with $b_0 \in V$ and $\forall n \in \mathbb{N}: B_n$ is linearly independent.
I would ... | The answer is no, in general.
Let $V$ be any Banach space with $\dim V \ge 2$.
Let $x \in V$ be a nonzero vector and let $(x_n)_{n=1}^\infty$ be an injective sequence in $V$ which converges to $x$, such as $x_n = \frac{n}{n+1}x$ for $n \in \mathbb{N}$.
Also, let $y \in V$ be a vector such that $\{x, y\}$ is linearly in... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that there exists the limit $\lim\limits_{n \to \infty}\int_0^1\frac{e^x \cos x}{nx^2+\frac{1}{n}}dx$ Use the A-G inequality, we can easily have that $$\left|\frac{e^x \cos x}{nx^2+\frac{1}{n}}\right| \leq \frac{e^x}{2x}$$
however, $\int_0^1 g(x)dx= \int_0^1\frac{e^x}{2x}dx$ diverges ,thus we cannot apply the Leb... | By enforcing the substitution $x=\frac{z}{n}$, $dx=\frac{dz}{n}$ we have
$$ \int_{0}^{1}\frac{e^x\cos x}{nx^2+\frac{1}{n}}\,dx = \int_{0}^{n}\frac{e^{z/n}\cos(z/n)}{z^2+1}\,dz=\int_{0}^{+\infty}\frac{f_n(z)}{z^2+1}\,dz -O\left(\frac{1}{n}\right)$$
where $f_n(z)$ is defined as $e^{z/n}\cos(z/n)$ over $[0,n]$ and as $1$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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how to evaluate $\lim_{x\to0} (x^2)/(e^x-1) $ without L'Hospital So I'm trying to evaluate limit written in the title without L'Hospital nor series (cause its not introduced in our course),I tried to use this recognized limit $\lim_{x\to0} \frac{\left(e^x−1\right)}x =1$
so our limit equal $\lim_{x\to0} x\left(\frac{x}{... | $$
\begin{aligned}
\lim _{x \rightarrow 0} \frac{x^{2}}{e^{x}-1} &=\lim _{x \rightarrow 0} \frac{x}{\frac{e^{x}-1}{x}} \\
&=\frac{\displaystyle \lim _{x \rightarrow 0} x}{\displaystyle \lim _{x \rightarrow 0} \frac{e^{x}-1}{x}} \\
&=\frac{0}{1} \\
&=0
\end{aligned}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2559177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 6
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Why $\int_a^bf(x)\,dx=F(b)-F(a)$ I recently looked at the proof given for the fundamental theorem of calculus in this link:
Why is the area under a curve the integral?
It made perfect sense, except for one thing. The proof relies on creating a function $F(b)$ that gives the area under the curve $f(x)$ from $0$ to some ... | From my point.of view a simple way to see this fact is to consider the integral function:
$$F(x) = \int_{0}^{x}f(t)dt $$
that rapresent the area “under” the graph from 0 to x.
Now if we think to calculate its derivative is pretty clear that for a small change $\Delta x$ the area varies of the quantity:
$$\Delta F(x)=f(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find a function $f(x)$ such that$\int_0^txf(x)f(\sqrt{t^2-x^2})dx=\sin(t^2)$ I am looking for a function that satisfies the integral equation $\int_0^txf(x)f(\sqrt{t^2-x^2})dx=\sin(t^2)$. It's some sort of convolution.
If we denote the above equation by $ f(t)*f(t)=\sin(t^2)$ I was also able to show that $f(t)*(2\cos(... | $$\int_0^t f(x)f(\sqrt{t^2-x^2})xdx=\sin(t^2)$$
HINT :
Let $\quad f(x)=g(x^2)$
$$\int_0^t g(x^2)g(t^2-x^2)\frac{d(x^2)}{2}=\sin(t^2)$$
Let $\quad\begin{cases} x^2=X \\t^2=T\end{cases}$
$$\int_0^{\sqrt{T}} g(X)g(T-X)dX=2\sin(T)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Name for a polygon which is laying on a cylinder? Is there a name for a closed-loop polygon (2D, such as a circle, square, triangle, etc.) which is "draped" over a cylinder (3D)?
Picture Dali's famous painting, The Persistence of Memory. If those clocks were of zero thickness, just circles melted over some 3D object, ... | Just call it a polygon. Since the cylinder can be rolled up from a plane without changing any locally measured angles, your polygon will unroll into a polygon in the plane.
The mathematics behind this is that the cylinder has Gaussian curvature $0$ and is a developable surface.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solving a derivative through implicit differentiation
Find the derivative of $x^2+4xy+y^2=13$
So here is what I did:
$$2x+4\left(x\frac{dy}{dx}+y\right)+2y\frac{dy}{dx}=0$$
$$4x\frac{dy}{dx}+2y\frac{dy}{dx}=-2x-4y$$
$$\frac{dy}{dx}(4x+2y)=-2x-4y$$
$$\frac{dy}{dx}=\frac{-2x-4y}{4x+2y}$$
But this isn't the correct answ... | Given
$$
x^2+4xy+y^2=13
$$
and apply the derivative operator
$$
d(x^2+4xy+y^2)=d(13)\implies 2x+4x\frac{dy}{dx}+4y+2y\frac{dy}{dx}=0
$$
and so
$$
\frac{dy}{dx}=-\frac{2x+4y}{4x+2y}=-\frac{x+2y}{2x+y}
$$
and it appears you just didn't cancel the common factor 2 of the denominator and numerator.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2559672",
"timestamp": "2023-03-29T00:00:00",
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derivative caculation I met a problem that described as follow. I am not sure the title is suitable for my problem. If you have any advices about the title or the description, please comment below.
Aready known:
$$F_1(x)=\frac{F_0^\prime(x)}{F_0^\prime(1)}$$
$$G_0(x)=\frac{1}{F_0(u)}F_0(u+(x-1)F_1(u))$$
$$G_1(x)=\frac{... | We have $$G_0'(x)=\frac{F_1(u)}{F_0(u)}F_0'(u+(x-1)F_1(u))$$ and substituing $x=1$ gives $$G_0'(1)=\frac{F_1(u)}{F_0(u)}F_0'(u+(1-1)F_1(u))=\frac{F_1(u)}{F_0(u)}F_0'(u)$$ Hence $$G_1(x)=\dfrac{\dfrac{F_1(u)}{F_0(u)}F_0'(u+(x-1)F_1(u))}{\dfrac{F_1(u)}{F_0(u)}F_0'(u)}=\dfrac{\dfrac{F_0'(u+(x-1)F_1(u))}{F_0'(1)}}{\dfrac{F... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Markov Chains. How to find F with different shape of I and Q First for all I want to say that I deal with Markov Chains only second day so I have some problems with terminology. Sorry for that.
I need to solve following matrix:
Now in I need to compute FR. In order to compute FR I need to find F first which is (I-Q)^-... | $"I"$ that you see in $(I-Q)^{-1}$is an identity matrix and is of the same dimension as that of Q. The $"I"$ that you see in the image is also an identity matrix by construction and lets you know the number of absorbing states that you have in the system . This one and Q need not have the same dimension.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Differentiate the squared dot product I am new to Mathematics stack exchange community and has no experience in asking question so please bear with me. I am watching deep learning course from Coursera and encounter a question during the video.
$$\left|\frac d{d\vec x}(\vec x\cdot\vec x)^2\right|=?$$
$$\vec x=\begin{bma... | By definition, the derivative of a scalar $f(x,y)$ with respect to a vector $(x,y)$ is the following vector $$\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right).$$
We have $$f(x_1,x_2)=(\vec x \cdot \vec x)^2 =\left(x_1^2+x_2^2\right)^2.$$
So the derivative in question is the vector $$(4(x_1^2+x_2... | {
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Cardinality of $\Bbb N^{k}$ How can i determine the cardinality of $\Bbb N^{k}$ for $k \in \Bbb N$ ?
I know that $\Bbb{N} \times \Bbb{N}$ is of cardinality $\aleph_o$, is there any valid induction for $k\in \Bbb{N}$?
| Yes. Let $f:\Bbb N^2\to \Bbb N$ be a bijection. For $k\geq 2$ suppose there is a bijection $g: \Bbb N^k\to \Bbb N.$ For $x=(x_1,...,x_{n+1})\in \Bbb N^{n+1}$ let $h(x)=f(g(x_1,...,x_n),x_{n+1}).$ Then $h:\Bbb N^{n+1}\to \Bbb N$ is a bijection.
This is a common technique in inductive proofs.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the method to show exactly one positive root of a Cubic equation? I have $a x^3 + b x^2 + c x + d =0$ and $a>0 , d<0$
$a, b, c, d$ is the function of all parameters. I'm looking for an analytic solution of this cubic equation. How to prove that is cubic equation has exactly one positive root?
| The standard method to define
that the cubic polynomial
\begin{align}
f(x)&=ax^3+bx^2+cx+d
\end{align}
has only one real root
is to check that
its discriminant
$\Delta<0$:
\begin{align}
\Delta &= 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2
,
\end{align}
Obviously, just the two conditions $a>0$, $d<0$
is not enough... | {
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"timestamp": "2023-03-29T00:00:00",
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$f$ continuous $\Leftrightarrow f\circ p$ continuous implies $p$ quotient map Let $X,Y,Z$ be topological spaces.
Let $p:X\rightarrow Y$ be a continuous surjection. Let $f:Y\rightarrow Z$ be continuous if and only if $f\circ p:X\rightarrow Z$ is continuous.
I want to prove that this makes $p$ a quotient map.
My thought... | I assume you want the property to hold for all spaces $Z$. In this case, pick $Z = Y$ as sets, endowed with the quotient topology for $p$. Let $f:Y \to Z$ be the identity map. We will show that $f$ is a homeomorphism, and hence that $Y$ also has the quotient topology.
First, $\tilde p =f \circ p$ is continuous because ... | {
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Proof of the equation of the fundamental matrix in absorbing Markov chains The standard or canonical form of the transition matrix of an absorbing Markov chain is
$$P = \begin{bmatrix}I & 0 \\ R & Q \end{bmatrix}$$
and the fundamental matrix is calculated as
$$F=(I - Q)^{-1}$$
such that $FR$ spells out the probability... | The submatrix $R$ contains the transition probabilities from transient to absorbing states, while $Q$ contains the transition probabilities from transient to transient states.
Powers of the transition matrix $P$ approach a limiting matrix with a pattern:
$$\begin{align}
P^2 &=\begin{bmatrix}I & 0 \\ R & Q \end{bmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2560653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $p$ is a prime such that $p\equiv 4 \mod 7,$ then $p \equiv 4\mod 14.$ Prove or disprove :
If $p$ is a prime such that $p\equiv 4 \mod 7,$ then $p \equiv 4\mod 14.$
I think it is a false statement because if $p=11$ then $p\equiv 4 \mod 7$ but $p \not \equiv 4\mod 14.$
Is my counterexample correct?
thanks
| Note that $n\equiv 4 \pmod {14}$ is an even number. Thus it can’t be a prime.
| {
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In the sequence $1,4,11,26$… each term is $2⋅(n-1)^{th}$ term $+ n$. What is the $n^{th}$ term? I readily see that it is $2^{n+1} - (n+2)$ but how can I deduce the $n^{th}$ term from the given pattern i.e. $2⋅(n−1)^{th}$ term $+n$ .
| Hint: $a_n = 2 a_{n-1} + n \iff a_n + n + 2 = 2 \left(a_{n-1} + (n-1) + 2\right)\,$, so $a_n+n+2$ is a geometric progression with common ratio $2$.
[ EDIT ] To followup on comments about doing it "by inspection", the heuristic would go like:
*
*try adding some multiple of $n$ on both sides of the given recurrence... | {
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How to write Euler product and Zeta function for $K = \mathbb{Q}(\sqrt{3})$ For general zeta number fields, I have found a carefully written statement of the Dedekind zeta function and it's Euler product:
$$ \zeta_L(s) = \sum_{\mathfrak{a}\subseteq \mathcal{O}_L} \frac{1}{N_{L/\mathbb{Q}}(\mathfrak{a})^s} = \prod_{\mat... | Dedekind zeta functions of non-imaginary quadratic number fields are more complicated. There is how you'll show $$\zeta_{\mathbb{Q}(\sqrt{3})}(s)= \sum_{I \subset \mathbb{Z}[\sqrt{3}]} N(I)^{-s}= \!\!\!\!\!\!\!\!\!\!\!\! \sum_{n+m\sqrt{3} \in \mathbb{Z}[\sqrt{3}]^*/\mathbb{Z}[\sqrt{3}]^\times}\!\!\!\!\!\! \!\!\! N(n+m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Kernel of injective group homomorphism If $\phi$ is a homomorphism from the group $G$ with identity $e$ to the group $G'$ with identity $e'$, then $\phi$ is injective if and only if $\ker\phi=$?
I think the $Ker\phi = \{x\in G:\phi(x)=1\}$
because you are looking at an identity then it is all reals $0$ does not work a ... | A homomorphism $ \phi $ is injective iff its kernel is {e}.
To prove this , suppose $\phi$ is injective . Then $\phi (x) = e' =\phi (e) $ means that $x=e$. Thus $ker \phi = {e} $.
Now suppose $ker \phi ={e}$. Then if $ \phi (a) =\phi (b)$ it means that $\phi (a b^-1) = e' $ ie $a b^-1 = e$ and thus a=b. Thus $\phi $is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find $\frac{d^n}{dx^n} e^x\cos x$ How can I get a formula for the n-th derivative of this function? I know that it cycles every 4 derivatives with a factor of $-4$. $e^x(\cos x-\sin x) \to e^x(-2\sin x) \to -2e^x(\sin x+\cos x) \to -4e^x\cos x$
| You can use the addition formula:
$$\cos x\cos y-\sin x\sin y=\cos(x+y)$$
Hence:
$$y=e^x\cos x\\
y'=e^x\cos x-e^x\sin x=\sqrt2\cdot e^x\left(\frac1{\sqrt2}\cos x-\frac1{\sqrt2}\sin x\right)=\\
\sqrt2\cdot e^x\left(\cos \frac{\pi}4\cos x-\sin \frac{\pi}4\sin x\right)=2^{\frac12}e^x\cos \left(x+\frac{\pi}{4}\right)\\
y''... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Set of functions from empty set to $\{0,1\}$ How does the set of all functions $\{f \,|\, f: \emptyset \to \{0,1\}\}$ look like? Is it empty or does it contain infinitely many functions? Does the definition $f: \emptyset \to \{0,1\}$ make sense at all?
I was wondering because we know that the two sets $\{0,1\}^X$ and $... | Yes, it makes sense. There is one and only one function from $\emptyset$ into $\{0,1\}$, which is the empty function. Think about the definition of function as a set of ordered pairs to see why.
| {
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"source": "stackexchange",
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Greens theorem on the circle $x^2 + y^2 = 16$.
Use Greens theorem to calculate the area enclosed by the circle $x^2 + y^2 = 16$.
I'm confused on which part is $P$ and which part is $Q$ to use in the following equation
$$\iint\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right){\rm d}A$$
| Hint: You want
$$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=1$$
so the integral is
$$\iint_{x^{2}+y^{2}\leq 16}{\rm d}A$$
Can you find $P$ and $Q$ that satisfy this? Notice that there is more than one choice.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find out all solutions for the system Given the system
$$ \left[
\begin{array}{ccc|c}
x_1&x_2&x_3&k\\
x_1&x_2&kx_3&1\\
x_1&kx_2&x_3&1\\
kx_1&x_2&x_3&1\\
\end{array}
\right] $$
I tried to solve this...It looks simple but I found a problem at the end...
$$ \left[
\begin{array}{ccc|c}
1&1&1&k\\
1&1&k&1\\
1&k... | The system has solutions only if
$\det(A|B)=0$, otherwise $\text{rank }(A|B)=4>\text{rank }(A)$
that is $k^4-6 k^2+8 k-3=0\to (k-1)^3 (k+3)=0$
for $k=-3$ and $k=1$
for $k=-3$ we get the system
$\begin {cases}
x+y+z=-3\\
x+y-3 z=1\\
x-3 y+z=1\\
-3x+y+z=1\\
\end{cases}
$
which has solution $(-1,-1,-1)$
for $k=1$ we get... | {
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"timestamp": "2023-03-29T00:00:00",
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Find all the equilibrium points of this second-order equation $x''+2x'=3x-x^3$ Find all the equilibrium points of this second-order equation $x''+2x'=3x-x^3$
I know that you must the roots of the equation, but when the equation isn't equal to $0$ I am confused on where to begin, thank you for any help!
edit: sorry, i d... | The equilibrium points are such that $x'=x''=0$. Hence $x=0,\pm\sqrt3$.
| {
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Integers and Sequences Is there a sequence of integers such that for ∀ k it contains an arithmetic subsequence of length k but it does have no infinitely long arithmetic subsequence?
| There's two approaches to constructing a sequence like this.
First, we might try to space out blocks of the sequence sufficiently far. For example,
$$
1, \qquad 3,4, \qquad 9,10,11, \qquad 23,24,25,26, \qquad 53,54,55,56,57, \dots
$$
where, whenever a block ends at $x$, the next block begins at $2x+1$.
In particula... | {
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Find a sequence $f_n$ so that $\int_0 ^1 |f_n(x)| = 2$ and $\lim_{n \to \infty} f_n(x) = 1$. The entire problem reads:
(a) Find a sequence $f_n: [0,1] \rightarrow \mathbb{R}$ such that $\int_0 ^1 |f_n(x)| = 2$ for all $n \in \mathbb{N}$ and $\lim_{n \to \infty} f_n(x) = 1$ for all $x \in [0,1]$. (b) If the $f_n$ are as... | Choose $$f(x):=4n^2x+1 \text{ for }x\in \left[0,\frac{1}{2n}\right]\\
f(x):=-4n^2x+1+4n \text{ for }x\in \left[\frac{1}{2n},\frac{1}{n}\right]\text{and }f(x)=1\text{ otherwise}.$$
| {
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Show that $f(x)=\begin{cases}1/b& x=\frac{a}{b}\in [0,1],a,b\in\mathbb Z\\ 0&x\in \mathbb R\backslash \mathbb Q \cap [0,1]\end{cases}$ is integrable. Show that $$f(x)=\begin{cases}\frac{1}{b}& x=\frac{a}{b}\in [0,1],a,b\in\mathbb Z\\ 0&x\in \mathbb R\backslash \mathbb Q \cap [0,1]\end{cases}$$
is integrable on $[0,1]$.... | $f$ is integrable iff $\exists P$ for every $\epsilon >0$ such that $S^{\sigma}-S_{\sigma}<\epsilon$.
Choose $P=\{x_0, ....,x_n\}$ such that $x_i - x_{i-1} = \frac{b-a}{n}$.
$S_{\sigma} = 0$ on $P$ because for every interval $[x_i, x_{i+1}]$, $m_i=0$.
Also, since $f(x)<1$ for $x<1$, we also have $M_i < 1$ and so $$\sum... | {
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What is the domain and range of $f(x,y)=x^4/(x^4 + y^2)$? I believe the domain is:
x,y cannot equal to 0 OR x can be any real number but y cannot equal to 0
and the range, well I just have no approach in solving it. Are just supposed to know because it's a rational function? or is there a proper approach in finding th... | For the domain note that
$$
x^4 + y^2=0\iff(x,y)=(0,0).
$$
Hence the domain is $\mathbb{R}^2\setminus \{(0,0)\}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to find a closed form expression for the following summation? How to find a closed form expression for the following summation?
$$ \sum_{m\geq 0} \frac{m r^m \Gamma(m+c)}{\Gamma(m+1)} $$
| Since $\Gamma(m+c)=\int_{0}^{+\infty}t^{m+c-1}e^{-t}\,dt$, for any $r<1$ the given series can be written as
$$ \int_{0}^{+\infty}\sum_{m\geq 0}\frac{m r^m t^{m+c-1}}{m!} e^{-t}\,dt =\int_{0}^{+\infty} r t^c e^{(r-1)t}\,dt=\color{red}{\frac{r\,\Gamma(c+1)}{(1-r)^{c+1}}}.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to calculate the limit $\lim_{x\to +\infty}x\sqrt{x^2+1}-x(1+x^3)^{1/3}$ which involves rational functions?
Find $$\lim_{x\to +\infty}x\sqrt{x^2+1}-x(1+x^3)^{1/3}.$$
I have tried rationalizing but there is no pattern that I can observe.
Edit:
So we forget about the $x$ that is multiplied to both the functions a... | Hint: Your expression equals
$$x^2[(1+1/x^2)^{1/2} - (1+1/x^3)^{1/3}].$$
Now use the fact that $(1+h)^p = 1 + ph +o(h)$ as $h\to 0.$ (This fact is equivalent to the statement that the derivative of $(1+x)^p$ at $x=1$ is $p.$)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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On the volume of a non-right tetrahedron and the integral of a linear function over it I've solved quite a bit of these questions with tetrahedrons with a 90 degree angle, but I am not sure how to approach a non-right angle tetrahedron. I am unsure how to find the triple integral's bounds. Calculation itself is no prob... | Actually the integration of a linear function on a tetrahedron with a vertex at the origin does not require to find the volume of such tetrahedron, but we may solve both problems at once. $\mathbb{R}^3$ is spanned by $v_1=(0,2,0)^T$, $v_2=(1,3,1)^T$ and $v_3=(1,1,0)^T$ and the wanted volume is just
$$ \frac{1}{6}\left... | {
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Diffusion Equation with a noncontinuous Auxiliary Condition I am currently trying to show that if we are given the Diffusion Equation $u_t=cu_{xx}$ with Auxiliary Condition $u(x,0)=g(x)$ where $g$ is non continuous at a point $x_0$, we get the statement:
$$\lim_{t\rightarrow 0^+}u(x_0,t)=\frac{1}{2}[g(x_0+)+g(x_0-)]$$
... | First show that the result holds for the function
$$
H(x)=\begin{cases}0 & \text{if }x<0,\\1 & \text{if }x\ge0.\end{cases}
$$
Then consider the function $f(x)=g(x)+c\,H(x-x_0)$, where the constant $c$ is chosen so that $f$ is continuous at $x_0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can the interval category be expressed as a colimit? A category with two objects and no non-identity morphisms can be expressed as the colimit of a diagram consisting of two copies of the trivial category. Naively, I would think that the interval category might be the colimit (or 2-colimit) of a diagram consisting of t... | $\DeclareMathOperator*{\colim}{colim}$
You can obtain the category $[1]=\{0\to 1\}$ joining two copies of the terminal category $[0]=\{0\}$. A join is a colimit: given simplicial sets $X,Y$ you have to compute
$$
\colim_{[n]\to [p+ q+1]} X_p\times Y_q
$$ or, in a more elegant way, the convolution
$$
\int^{p,q} X_p\time... | {
"language": "en",
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Bayesian network I have this following question and i would like to know if anyone can answer it.
For a given Bayesian network where $P(a) =.6, P(b|a) =.8, P(b|-a)=.4, P(c|a)=.4$ and $P(c|-a) = .3$, compute $P(c|b)$. Note that $a, -a, b$, etc. are propositions: e.g.) $a \leftrightarrow A = true , -a \leftrightarrow A... | Begin with the definition of conditional probability.
$\hspace{25ex}\mathsf P(c\mid b) = \dfrac{\mathsf P(b\cap c)}{\mathsf P(b)}$
Now apply the law of total probability,
$\hspace{25ex}\mathsf P(c\mid b) = \dfrac{\mathsf P(a\cap b\cap c)+\mathsf P(\neg a\cap b\cap c)}{\mathsf P(a\cap b)+\mathsf P(\neg a\cap b)}$
Then... | {
"language": "en",
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Show that the sequence $S_n := \sum_{k=0}^{n} \lambda_k A^k$ has a limit in $ L(X)$. I have a simple question, but I do not know what to do with it:
The question is as follows:
Let $X$ be a Banach space. Consider an integral operator $A \in L(X)$ and a function $\varphi(t) = \sum_{k=0}^{+\infty} \lambda_k t^k$ for $\la... | The power series $\varphi(t)=\sum_{k=0}^{+\infty} \lambda_k t^k$ converges absolutely in each $t \in \mathbb R$.
Put $S_n:=\sum_{k=0}^{n} \lambda_k A^k$. For $n,m \in \mathbb N$ with $m>n$ we have
$||S_m-S_n||= ||\sum_{k=n+1}^{m} \lambda_k A^k|| \le \sum_{k=n+1}^{m} |\lambda_k| \cdot ||A||^k=|a_m-a_n|$, where $a_n:=\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2563223",
"timestamp": "2023-03-29T00:00:00",
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Euler method with infinite gradient at initial value The title itself is self explanatory - I am trying to numerically solve an ODE with an initial value that has an infinite gradient. It seemed problematic to me and I am not certain as to how I should approach this.
e.g. $\frac {dy}{dx} = \frac y{\sqrt x} , y(0)=1$
(O... | Others can probably give a more "conventional" approach but one thing you can do, because $y'(x)$ is independent of $y(x)$, is take the following approach for the first step (at least).
For the initial slope think rather angle. You want to proceed with a certain angle but you can't use $\theta_1=\frac{\pi}{2}$... which... | {
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How to prove if $\frac{20b}{19b-20}=a$ is possible or not, where $a$ and $b$ are positive integers? This is a follow up question to the one posted here. (The method posted here works for all cases where there isn't a coefficient in front of $b$ in the denominator)
Here is the problem:
$$\frac{20b}{19b-20}=a$$
$a$ and $... | If $b=1$, then we get $a=-20$ which is not a positive integer.
In the following, $b\ge 2$.
We have
$$a=\frac{20b}{19b-20}=\frac{19b-20+b+20}{19b-20}=1+\frac{b+20}{19b-20}$$
So, $\frac{b+20}{19b-20}$ has to be a positive integer.
Then, we have to have
$$\frac{b+20}{19b-20}\ge 1\implies b+20\ge 19b-20\implies 18b\le 40\i... | {
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$\operatorname{Ext}^1(M,R/m)=0$ iff $M$ is projective In this answer it is suggested that over a commutative local ring, a module $M$ is projective iff $\operatorname{Ext}^1_R(M,R/m)=0$. A similar result holds for flatness and Tor.
In the case of Tor, I can find a proof in Robert Ash's Commutative Algebra, which invol... | This is answered in the question you link to. It is clear that if $M$ is projective then $\text{Ext}_R^1(M,k)$ vanishes where $k= R/m$ is the residue field of $R$. To show that if $\text{Ext}_R^1(M,k)$ vanishes then $M$ is in fact free, you use the fact
modules over local rings admit minimal free resolutions, i.e. ther... | {
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When finding the maximum and minimum points of a function on an interval what do you do if the derivative is undefined at zero Say I have a function and I want to find the maximum and minimum on an interval, I would first differentiate the function and equate it to zero. Then use this value and the two ends of the inte... | The candidates for maximum and minimum of a continuous function on an interval are the zeros of the derivative, the endpoints, and the points where the derivative does not exist.
Whether the term "critical point" includes a point where the function is not differentiable is a question of convention: most authors do no... | {
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Finding the number of possible sequences(of any length) Given $x$ & $y$. Count the number of distinct sequences $a_1, a_2, \dots, a_n$ $(\forall a_i \ge 1)$ consisting of positive integers such that $\gcd(a_1, a_2, \dots, a_n) = x$ and $\sum_{i=1}^n a_i =y$.
| First, it is obvious that if $x\not | \ \ y$, then such sequence does not exist. Therefore we suppose $x | y$.
We first enumerate the number of positive sequences $a_1, \cdots, a_n$ such that $x|a_i$ for all $i$ and $\sum_{i=1}^n a_i = y$. By taking $a_i' = \frac{a_i}{x}$ for each $i$ this means a positive sequence $a_... | {
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How to show that $\frac{dy}{dx}=\frac{dy}{d(x-c)}$? It seems intuitive to me that $\frac{dy}{dx}=\frac{dy}{d(x-c)}$ (the derivative of $y$ with respect to $(x-c)$, where $c$ is a constant), since subtracting a constant from $x$ doesn't change the slope of $y$, but how can I show it? Thanks in advance.
| Note that if $z=x-c$ then the chain rules says that
$$\frac{dy}{dz} = \frac{dy}{dx}\cdot \frac{dx}{dz} = \frac{\frac{dy}{dx}}{dz/dx}=\frac{dy}{dx}.$$
| {
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Left-multiplication of a finite field element - Matrix representation Can someone explain me why and how a left multiplication of an element of a finite field GF(2^k) can be seen as a linear transformation on GF(2^k) over GF(2)?
I read this https://www.maa.org/sites/default/files/Wardlaw47052.pdf but it is not clear t... | The field $\operatorname{GF}(2^k)$ is a finite dimensional vector space over $\operatorname{GF}(2)$ for $k$ any integer greater zero, so we can talk about linear transformations on this vector space. Let $\alpha \in \operatorname{GF}(2^k)$ and define the following map
$$
\begin{align}
T_\alpha: \operatorname{GF}(2^k) &... | {
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"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Find a geometrical description of the set of all vectors of the form av+bw where a+b=1 My first intuition for solving this problem was to solve for $a$, therefore $a = 1-b$,
and letting $v$ and $w$ be arbitrary vertices $v =(1,3)$ and $w = (5,2)$. Since the original question is $av+bw$ I just substitute $(1-b)*(1,3)+... | $1-b$ is a number, there is no dot product for us to take.
$$(1-b)\cdot(1,3) + b\cdot (5,2)= (1,3) +b((5,2)-(1,3))$$
Try to simplify the expression above and think of which object is it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving $e^x\leq e^a\frac{b-x}{b-a}+e^b\frac{x-a}{b-a}$ I'm trying to prove
$$e^x\leq e^a\frac{b-x}{b-a}+e^b\frac{x-a}{b-a}$$
for any $x\in[a,b]$. Since this looks reminiscent of the mean value theorem or linear approximations I jotted down some equations relating to those, but didn't see any way of making progress w... | We can consider the function $$g(x) =f(b) - f(x) - \frac{f(b) - (a)} {b-a} (b-x) $$ where $f(x) = e^{x} $. We have to prove that $g(x) \geq 0$ for all $x\in[a, b] $. We have via mean value theorem $$f'(c) =\frac{f(b) - f(a)} {b-a} $$ for some $c\in(a, b) $ and since $f'(c) =f(c) $ we get $$g(x) =f(b) - f(x) - (b-x) f(c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Terms for 3 aspects of a function If, for some function $f$ and some value $p\ne 0$, $$\forall x, f(x)=f(x-p),$$ then $f$ is periodic and $p$, if $>0$ and minimal, is $f$'s period. If, instead, for some values $u\ne 0$ and $v$,
$$\forall x, f(x)=f(x-u)+v,$$
then what is the correct adjective to describe $f$, and what a... | Not an authoritative answer by any stretch, but I'd call it maybe "staircase-like function", or perhaps "linearly augmented periodic function". The latter coming from the observation that $f(ux) - v x$ is in fact periodic: $$\require{cancel}\;f\big(u(x+1)\big) - v(x+1) = f(ux+u)-vx-v = f(ux)+\bcancel{v}-vx-\bcancel{v}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Mathematical induction proof that $f(n)=\frac{1}{2}+\frac{3}{2}(-1)^n$
The function $f(n)$ for $n=0,1...$ has the recursive definition $$f(n)= \begin{cases} 2 & \text {for n=0} \\ -f(n-1)+1 & \text{for n=1,2...} \end{cases}$$
Prove by induction that the following equation holds: $$f(n)=\frac{1}{2}+\frac{3}{2}(-1)^n$... | Well, you have come close in proving the correct form of $f(n+1)$. We then just need to do one more step. Notice that: $$f(n+1)= \frac12+ \frac32(-1)^{n+1}= (1)-(\frac12 +\frac32(-1)^n) =1-f(n)$$ proving that $f(n+1)$ is also true. The proof is thus finished!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Solving $z^2 - 8(1-i)z + 63 - 16i = 0$ with reduced discriminant $z^2 - 8(1-i)z + 63 - 16i = 0$
I have come across a problem in the book Complex Numbers from A to Z and I do not understand the thought process of the author when solving:
The start to the solution is as follows:
$\Delta$ = discriminant
$\Delta ' = (4 - 4... | Note that solving quadratic equations involving complex numbers follows the same procedure as solving those involving real numbers.
Consider a quadratic equation with real coefficients: $$F(x)=ax^2+bx+c=0$$ How would you solve it? Well, the first step would involve solving out for the discriminant, right?
Let us now co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proof by mathematical induction in Z Is it possible to proof the following by mathematical induction? If yes, how?
$a\in \mathbb{Z} \Rightarrow 3$ | $(a^3-a)$
I should say no, because in my schoolcarrier they always said that mathematical induction is only possible in $\mathbb{N}$. But I never asked some questions why... | Since $\mathbb{Z}$ is countable as $\mathbb{N}$ we can extend induction over $\mathbb{Z}$.
BASE CASE:
$$a=1 \implies 3|0$$
INDUCTIVE STEP 1 "UPWARD"
assume: $3|a^3-a$
$$(a+1)^3-(a+1)=a^3+3a^2+3a+1-a-1=a^3-a+3a^2+3a\equiv0\pmod 3$$
thus
$$3|(a+1)^3-(a+1)$$
INDUCTIVE STEP 2 "DOWNWARD"
assume: $3|a^3-a$
$$(a-1)^3-(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 7
} |
Expectation of the minimum of two first passage times of a standard Brownian motion Let $W_t$ be a standard Brownian motion. For any real constant $a$, $T_a=\min(t≥0:W_t=a)$.
I know how to derive the distribution of $T_a$.
Now given $a>0$ and $b<0$, I want to compute $E[\min(T_a,T_b)]$.
Any help? Are $T_a$ and $T_b$ i... | Hints:
*
*Show that $\min\{T_a,T_b\} = T := \inf\{t \geq 0; W_t \notin (b,a)\}$.
*Show that $W_T \in \{a,b\}$ and $|W_{t \wedge T}| \leq \max\{|a|,|b|\}$ almost surely.
*Use the optional stopping theorem for the martingale $(W_t)_{t \geq 0}$ to prove that $$\mathbb{E}(W_{T \wedge t})=0$$ for any $t \geq 0$. Conclu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
What's wrong with this equation involving cumulative distribution functions? Suppose $X$ is a continuous random variable with finite mean $\mu = 1$ and standard deviation $\sigma$. Suppose also that it's pdf is symmetric about $x=\mu$. Show that $P(|X-\mu|\le 2\sigma) = 2P(X\le\mu + 2\sigma) - 1$.
My solution is as fol... | Alright I realized the error, I thought I'd include it for anyone else curious:
$$
F_Z(2\sigma) - F_Z(-2\sigma)\neq 2F_Z(2\sigma)
$$
This is plainly false. What is true is
$$
F_Z(2\sigma) - F_Z(-2\sigma) = 2(F_Z(2\sigma) - F_Z(0))
$$
and $F_Z(0) = \frac12$ by symmetry, which introduces a $-1$ term, as expected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Domain of $y=\arcsin\left({2x\sqrt{1-x^{2}}}\right)$ How do you find the domain of the function $y=\arcsin\left({2x\sqrt{1-x^{2}}}\right)$
I know that the domain of $\arcsin$ function is $[-1,1]$
So, $-1\le{2x\sqrt{1-x^{2}}}\le1$ probably?
or maybe $0\le{2x\sqrt{1-x^{2}}}\le1$ , since $\sqrt{1-x^{2}}\ge0$ ?
EDIT: So ... | A proof without calculus or trigonometry.
$$\begin{aligned}
(1-2x^2)^2&\geq0
\\
(1-4x^2+4x^4)&\geq0
\\
x^2-x^4&\leq\frac{1}{4}
\\
0\leq \sqrt{x^2({1-x^2})}&\leq\frac{1}{2}
\\
0\leq |x\sqrt{1-x^2}|&\leq\frac{1}{2}
\\
-1\leq 2x\sqrt{1-x^2}&\leq1
\end{aligned}$$
Then, since $\arcsin:\,[-1,1]\mapsto\left[\frac{-\pi}{2},\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Composition of Proximal Operators Consider a proximable function $f$ where the proximal operator is defined as follows,
$$\operatorname{prox}_{\lambda f}(x) = \arg \min_z \frac{1}{2\lambda}\left\| z - x \right\|_2^2 + f(z)$$
$\lambda \geq 0$. With an additional constraint the problem is
$$\arg \min_{z \ge 0} \frac{1}{... | You are indeed right and can read about it in the paper - On Decomposing the Proximal Map.
Also have a look on the answers in Proximal Mapping of Least Squares with $ {L}_{1} $ and $ {L}_{2} $ Norm Terms Regularization (Similar to Elastic Net).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Applicability of L'Hôpital rule on infinite sum. $$\lim_{n \to \infty} \left( \frac{n}{n^2+1} + \frac{n}{n^2+2} + \frac{n}{n^2+3} + \space ... \space + \frac{n}{n^2+n}\right) $$
As $n$ is not $\infty$ but tends to $\infty$ I can split the limit of sums into sum of limits. i.e.
$$\lim_{n \to \infty} \frac{n}{n^2+1} +\... | Clearly the problem is way before L'Hopital's, splitting the limit into a sum of limits is not allowed (precisely because the number of terms tends to infinity, even though it is not infinite). For instance, consider:
$$ 1 = \lim_{n \to \infty} \left( n \frac{1}{n} \right)= \lim_{n \to \infty} \underbrace{\left( \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find $\int_\gamma \frac{dz}{(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)}\, $ Let $f(z)=\frac{1}{[(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)]}$ and let $\gamma$ be the polygon $[0,2,2+2i,2i,0]$. Find $\int_{\gamma}^{} f$ .
I'm trying to use the partial fractions decom... | By the residue theorem: $\int_{\gamma}f(z)dz=2\pi i\sum_i\textrm{res}_{z_i}$. So the problem essentially is to evaluate the residue of each pole. The poles are at:
$1/2+i$, $1+3i/2$, $1+i/2$, $3/2+i$.
Simply check which are in your boundary and evaluate the residues.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integrating $\int_0^{\frac{\pi}{2}} x (\log\tan x)^{2n+1}\;dx$ Does anybody have any thoughts about how to integrate
$$I=\int_0^{\frac{\pi}{2}} x (\log\tan x)^{2n+1}\;dx$$
for integral $n$ where $n\ge 1$
In the case $n=0$
$$\int_0^{\frac{\pi}{2}} x \log\tan x \;dx=\lambda(3)=\frac{7}{8}\zeta(3)$$
I have managed to inte... | By setting $x=\arctan u$ we are left with
$$ \mathcal{I}(n) = \int_{0}^{+\infty}\frac{\arctan u}{1+u^2}\left(\log u\right)^{2n+1}\,du =\left.\frac{d^{2n+1}}{d\alpha^{2n+1}}\int_{0}^{+\infty}\frac{u^\alpha \arctan u}{1+u^2}\,du\right|_{\alpha=0}$$
but the integral in the RHS is related to the Beta function. By un-doing ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
MIT PRIMES Question, polynomial satisfying conditions There was an MIT PRIMES application problem that goes like this: (don't worry, the admission ended on Dec 1, so I'm not cheating or anything)
For all $d\geq 0$, determine if there is a polynomial $p(x)$ with degree $d$ such that for all $n$ in $1\ldots 99$ inclusive... | From the given condition $nP(n)=n^2+1$ for $n=1,2,\cdots,99$. So the polynomial $P^{\star}(x)=xP(x)-x^2-1$ of degree $d+1$, has 99 zeros. If $d<98$ then $P^{\star}(x)$, despite of being of degree $d+1$, would have $>d+1$ roots, and hence should be identically equal to zero for every $x$. Then $P(100)$ would be simply $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Counting Multi-Sets for Donuts The problem is:
Suzy is selecting 20 donuts to bring to her club meeting. The donut store sells 7 varieties of donuts. Donuts of the same variety are all the same. There is a large supply of each variety of donuts, except for jelly donuts. There are only 5 jelly donuts available. How many... | Christian Blatter has explained how to correct your approach and why your approach was incorrect. Here is another method.
Let $x_j$ denote the number of jelly donuts. Let $x_k$, $1 \leq k \leq 6$, be the number of donuts of type $k$ that Suzy purchases. Then
$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_j = 20 \tag{1}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Probability that $x$ divides $k \leq n$ and an equation question. The exact probability that a fixed positive integer $x \leq n$ divides a randomly selected positive integer $k \leq n$ is $\dfrac{\lfloor \dfrac{n}{x} \rfloor}{n}$ which is the same as $\dfrac{n - n_{(x)}}{xn}$ where $n_{(x)}$ is defined to be the smalle... | *
*Your expression for $P(x, y)$ does not hold generally. To be more precise, it should be
$$
P(x, y) = P(x) + P(y) - P(\mathsf{lcm}(x,y))
$$
where $P(x) = \left(n - n_{(x)}\right)/{nx}$, and $\mathsf{lcm}(x, y)$ is the least common multiple of $x$ and $y$.
*Any combination with $x = 1$, $1 \leq y \leq n$ is a soluti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
the characteristic function of Levy Distribution The standard Levy distribution has the PDF:
$$f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2x}}\frac{1}{x^{3/2}},$$
where $x\geq0$.
My question is how to compute its characteristic function:
$$ \phi(t)=\int_{-\infty}^{\infty} e^{jtx}f(x)\mathrm{d}x. $$
I tried to use the Resi... | I can't find the question this duplicated, so I'll write the answer here. Define $a:=(1-i)\sqrt{t}$ so $a^2=-2it$. Since $\phi(-t)=\phi^\ast(t)$, once we've proven that $t>0\implies \phi(t)=\exp -a$ we'll know that more generally $\phi(t)=\exp -(1-i\operatorname{sgn}t)\sqrt{|t|}$. For the $t>0$ case$$\phi(t)=\int_0^\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Does $f(x)=O(\log x)\implies f'(x)=O(1/x)$? Does $f(x)=O(\log x)\implies f'(x)=O(1/x)$ ?
My attempt:
Since $f(x)=O(\log x)$, then there exists $C>0$ such that $f(x)<C\log x$ for sufficiently large $x$.
What I'm proposing is that $f'(x)=O(1/x)$, but this would mean there exists $C>0$ such that
$$f'(x)<\frac{C}{x},$$
and... | Consider $f(x)=\sin(x^{2})$. Then $f(x)=O(1)$ so $f$ is certainly $O(\log(x))$. But
$$f'(x) = 2x\cos(x^{2}) = O(x)$$
which is certainly not $O(x^{-1})$. It is a general principle that bounded functions can oscillate very heavily (imagine any curve you like, perturbed by lots of small-amplitude but high-frequency wiggle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Given 8 cube vertices and a point inside, find two points that a line going from inside point with vector V cuts the surface of the cube. I have the coordinates of a vertices of a cube and a point inside the cube. Now I draw a line from that point with vector V. How can I find the coordinates in which that line cuts t... | Let $p$ be the point inside the cube, and $\textbf{v}$ the vector.
For each of the six faces of the cube, find the point where the ray from $p$
in direction $\textbf{v}$ intersects the plane. The ray can be represented as $p + t \,\textbf{v}$, where $t \ge 0$ is a parameter. Solve for the $t$ that places
$p + t \,\text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can we use simultaneous row and column operations in solving same determinant? Please help.. Can both(row and column) operations be used simultaneously in finding the value of same determinant means in solving same question at a single time?
| A row operation corresponds to multiplying a matrix $A$ on the left by one of several elementary matrices whose determinants are easy to compute to get a matrix $B = EA$. For instance, swapping the rows of a 2x2 matrix is done with
$$
\pmatrix{0 & 1 \\ 1 & 0 } \pmatrix{a & b \\ c & d}
$$
The determinant of the resulti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Degree of chromatic polynomial We have graph with $n$ vertices. Chromatic number of this graph is $\chi(G)=3$.
What degree has a chromatic polynomial of this graph? I ended up with degree of size $n$, but I can not find proof for that.
| The degree of the chromatic polynomial is equal to the number of vertices. Here is a simple proof from first principles.
Let $G=(V,E)$ be a (simple finite) graph of order $n.$ For $x\in\mathbb N$ let $P(x)$ be the number of proper colorings $\varphi:V\to\{1,2,3,\dots,x\}.$ For $e=uv\in E,$ let $A_e$ be the number of ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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