Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Given a convex set in a normed vector space, take a neighbourhood of it. Is still convex? Consider a normed vector space and a set there, call it $\mathrm{E}.$ Define the neighbourhood $\mathrm{E}^\eta$ of $\mathrm{E}$ with radius $\eta > 0$ as the set of vectors $v$ whose separation from $\mathrm{E}$ differs less than... | Let $x,y\in E^\eta$. Then there exist $x_E,y_E\in E$ such that $\|x-x_E\|$ and $\|y-y_E\|$ are lesser than $\eta$. Now, set $t\in(0,1)$. Then $tx_E+(1-t)y_E\in E$ and
$$\|tx+(1-t)y-tx_E-(1-t)y_E\|\le t\|x-x_E\|+(1-t)\|y-y_E\|<t\eta+(1-t)\eta=\eta$$
So $tx+(1-t)y\in E^\eta$. This proves the convexity of $E^\eta$.
Perhap... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complex Roots with improper fraction I'm having trouble with the following:
$(-16i)^{5/4}$
My calculations for the Principal root is:
$32(\cos (3\pi/2) * 5/4) + i \sin (3\pi/2)* 5/4))$
$=32(Cis (15\pi/8))$
This answer does not agree with the online calculators. It gives a positive real value and the online calculators... | Let $z=-16i$ and $n=\dfrac54$. For solving you have to compute $r=|z|$ and argument $\theta$ where $\tan\theta=\dfrac{y}{x}$.
then
$r=|z|=|-16i|=16$ and argument $\theta=\dfrac{3\pi}{2}$. Then write
$$z_k=r^n(\cos n\theta+i\sin n\theta)$$
But argument adds with $2k\pi$ so we have
$$z_k=r^n\Big(\cos n(\theta+2k\pi)+i\si... | {
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"url": "https://math.stackexchange.com/questions/2093793",
"timestamp": "2023-03-29T00:00:00",
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Big-Omega Proof So I'm trying to figure this one out, suppose this problem A.
$f(n) = n^2 - 2n$ and $g(n) = n^2$.
I want to prove that $f(n) \in \Omega(g(n))$ by showing a set of inequalities between $f(n)$ to $g(n)$ to derieve the $c > 0$ and $n_0 > 0$.
For example, say
$f(n) = n^2 + 2n$ and $g(n) = n^2$ and $f(n) \in... | For large $n$,
$$f(n)=n^2-2n \ge \frac{1}{2} n^2 = \frac{1}{2} g(n).$$
To find $n_0$ explicitly, note
$$n^2-2n - \frac{1}{2}n^2= \frac{1}{2} n(n-4) > 0$$
for $n>4$.
| {
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"source": "stackexchange",
"question_score": "1",
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$A^H=A^{-1}$ implies $\|x\|_2 = \|Ax\|_2$ for any $x\in \mathbb{C}^n$ Show that the following conditions are equivalent.
1) $A\in \mathbb{C}^{n\times n}$ is unitary. ($A^H=A^{-1}$)
2) for all $x \in \mathbb{C}^n$, $\|x\|_2 = \|Ax\|_2$, where $\|x\|_2$ is the usual Euclidean norm of $x \in \mathbb{C}^n. $
I am totally ... | Hint: $||x||^2_2 = x^H x$. What is $(Ax)^H$?
| {
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Is this induction question wrong or am I going insane? Here's a question that I've come across: Prove by induction that for every integer $n$ greater than or equal to $1$,
$${\sum_{i=1}^{2^n}} \frac{1}{i} \ge 1 +\frac{n}{2}.$$
Now I know how to prove by induction, but wouldn't this fail $p(1)$ since
$$\frac{1}{1} \ge 1... | Neither: the question is not wrong and you are also not going insane ... You just made a little mistake. :)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2094051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that if the sum of components of one vector adds up to 1 then the sum of the squares of the same vector is at least 1/n (NOTE: Already posted here, but closed without an answer)
Hi, I've been trying to complete the following question:
Suppose we have two vectors of $n$ real numbers, $[x_1,x_2,⋯,x_n]$ and $[y_1,y_2... | You can prove it using Jensen's inequality for $f(x) = x^2$. Since $x^2$ opens upwards,
$$f(\text{avg}~ a) \le \text{avg}~f(a)$$
More specifically:
$$\left(\frac 1n \sum_{k=1}^n a_k\right)^2 \le \frac 1n \left(\sum_{k=1}^n a_k{}^2\right)$$
The rest is just algebra, use $\sum_{k=1}^n a_k = 1$ to establish $\sum_{k=1}^... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Natural deduction proof of $p \rightarrow q \vdash \lnot(p \land \lnot q)$ So yeah, the entire question is pretty much in the title.
$$p \rightarrow q \vdash \lnot(p \land \lnot q)$$
I've been able to derive the reverse, but I don't how to logically go from the premise to the conclusion using natural deduction only. I... | $1.$ $p \rightarrow q \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -- (Premise)
$2.$ $p \wedge \neg q \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -- (Assume the contrary to what has to be proved in the conclusion)
$3.$ $p \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ -- ($\wedge E$ on $2.$)
$4.$ $\neg q \ \ \ \ \ \ \ \ \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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finding $ \int^{4}_{0}(x^2+1)d(\lfloor x \rfloor),$ given $\lfloor x \rfloor $ is a floor function of $x$ finding $\displaystyle \int^{4}_{0}(x^2+1)d(\lfloor x \rfloor),$ given $\lfloor x \rfloor $ is a floor function of $x$
Assume $\displaystyle I = (x^2+1)\lfloor x \rfloor \bigg|^{4}_{0}-2\int^{4}_{0}x\lfloor x \rfl... | About your attempt to integrate by parts: instead of doubting about the legitimacy of the change of the "corresponding limits", I advise you take the following steps:
1). Try to understand the basic theory of Riemann Stieljes integration.
2). Find a proof of integration by parts for R-S integrals, be careful about the ... | {
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"url": "https://math.stackexchange.com/questions/2094459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Connection between rank and matrix product I have a problem understanding the following:
Let $A$ be an $m \times n$ matrix and let t $\in \mathbb{N}$. Prove that
$\operatorname{rank}(A)\leq t$ if and only if there exists an $m \times t$ matrix $B$ and a $t \times n$ matrix $C$ so that $A = BC$.
I know what a rank i... | Think of matrices as linear transformations: $A:F^n\to F^m$, $C:F^n\to F^t$, $B:F^t\to F^m$. If $\mathrm{rk}(A)\le t$, then $\mathrm{Im}(A)\le t$, so there exists a $t$-dimensional subspace $T$ of $F^m$, such that $\mathrm{Im}(A)\subset T$. Consequently, if we restrict $A$ to $A'=C:F^n\to T$, and let $B:T\to F^m$ be th... | {
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Two circles touches each other externally at the point O. .. I am stuck on the following elementary problem that says:
Two circles touches each other externally at the point O. If PQ and RS are two diameters of these two circles respectively and $PQ || RS$ (Where || indicates parallel),then prove that P,O,S are colli... | I do not think that your approach is valid, since you have assumed that $QOR$ and $POS$ are collinear and then calculated
$\angle POR=180^{\circ}-90^{\circ}=\angle QOS$
What you can do instead, is to draw the line between centers of circles. The center of the left circle is $C_1$ and the right one is $C_2$. Also, draw ... | {
"language": "en",
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Automorphisms of order 2 Let $G$ be a finite group with no element of order $p^2$ for each prime $p$. Does there always exist an automorphism $\phi$ of order 2 such that for at least one subgroup of $G$ say $H$, we have $\phi(H)\neq H$?
Update: What about if add the supposition that $G$ is not cyclic of prime order?
| No, take for example the cyclic group $C_2$. It certainly has no element of order $p^2$. Nevertheless, every automorphism is the identity, i.e., $\phi(H)=H$. So there exists no automorphism of order $2$ with $\phi(H)\neq H$ for some $H$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Equivalence classes in $\mathbb{R}^{3}$ such that $Mv_{1} = v_{2}$ for orthogonal $M$
Let $ A = \mathbb{R}^{3}$, let $R$ be an equivalence relation such that $\left(v_{1},v_{2}\right) \in R$ if and only if $\exists P \in M_{3 \times 3}\left(\mathbb{R}\right)$ such that $P$ is orthogonal and $Pv_{1} = v_{2}$
Describe a... | An orthogonal matrix is an isometry when viewed as an operator. So for any orthogonal matrix $P$ this holds: $|v|=|Pv|$. In particular if $(v_1, v_2)\in R$ then $|v_1|=|v_2|$.
The other implication also holds, i.e. if $|v_1|=|v_2|$ then you can find an orthogonal matrix $P$ such that $Pv_1=v_2$ (for explicit constructi... | {
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Prove that $\int_0^u\frac{\sin(mt)}{2\sin(t/2)}dt$ is bounded When I read the Fourier Series theory, a property show in the book but without the details.
That property is:
$\displaystyle\int_0^u\frac{\sin(mt)}{2\sin(t/2)}dt$ is bounded for
all $m$ and $0\leq u\leq\pi$
I want to ask how to prove it? Also, what is th... | Notice that, in general, for $n \ge 0$
$$\cos nx + \Bbb i \sin nx = \Bbb e ^{\Bbb i n x} = (\Bbb e ^{\Bbb i x})^n = (\cos x + \Bbb i \sin x)^n = \sum _{k=0} ^n \binom n k \Bbb i ^k \cos ^{n-k} x \sin ^k x$$
whence equating the imaginary parts in both sides gives us
$$\sin nx = \binom n 1 \cos ^{n-1} x \sin x - \binom n... | {
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consistency of numerical methods for ODE I was reading the Wikipedia article on numerical methods for ODEs
https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations#Consistency_and_order
and I saw that when it discusses "consistency and order", the consistency is defined as
$$\lim_{h\to 0} \fr... | Suppose you have $y(0)$ and your goal is to obtain $y(t)$ where $t$ is some small positive number. If $t$ is small enough and the method is consistent, you should be able to run your numerical method with $h=t/N$ for large $N$, and then as $N \to \infty$ you should get convergence to $y(t)$. This means that you will in... | {
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If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring. If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring.
My failed attempt:
Since $G$ is finite and $p$ divisde... | Hint:
Square $\sum_{g\in G}g$, and observe that it is central.
| {
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Factoring an expression with three variables the expression is $(a-b)c^3+(b-c)a^3-(a-c)b^3$
I've tried factoration by grouping, but it has been a trial and error situation. I don't know any strategies to factor expressions like that, so I'm basically just guessing.
There is a ton of questions like this one in the book... | One may first set the expression equal to $0$.
$$0=(a-b)c^3+(b-c)a^3+(c-a)b^3$$
One can then see that for this to hold, we have one solution $a=b$, $a=c$, and $b=c$. Turning this into "factors" that we can use, we get, as a polynomial of $a$,
$$(a-b)(a-c)(b-c)P(a)=(a-b)c^3+(b-c)a^3+(c-a)b^3$$
whereupon one will find t... | {
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Eigenvectors of complex matrix I'm working on a problem where I am trying to find the eigenvectors of a pretty complicated matrix, and I am in need of some assistance. The matrix in question is:
$$A =\begin{bmatrix}
\sin(x) & \cos(x)\cos(y) - i\cos(x)\sin(y)\\
\cos(x)\cos(y) + i\cos(x)\sin(y) & -\sin(x)... | Note that
$$
A=DQD^{-1},\ D=\pmatrix{1\\ &e^{iy}},\ Q=\pmatrix{\sin x&\cos x\\ \cos x&-\sin x}.
$$
It follows that if $v$ is an eigenvector of $Q$, then $Dv$ is an eigenvector of $A$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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A Ramanujan infinite series $$ 1-5\left(\frac{1}{2}\right)^3+9\left(\frac{1 \cdot 3}{2 \cdot 4}\right)^3-13\left(\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^3+\cdots $$
I went on evaluating the above series and encountered that solving $\displaystyle \sum_{n\ge 0}\left(\binom{2n}{n}\right)^3x^n$ would suffice.
B... | Modifying @Claude Leibovici's answer a little.
$\frac{1}{\left(1-z\right)^{a}}=\,_{1}F_{0}\left(a;;z\right)={\displaystyle \sum_{n=0}^{\infty}\frac{a_{n}}{n!}z^{n}}$
and
$\frac{1}{\sqrt{1-4\cdot x}}={\displaystyle \sum_{n=0}^{\infty}\left(\begin{array}{c}
2n\\
n
\end{array}\right)x^{n}}$
Let $z=4x,x=\frac{z}{4}$
$\frac... | {
"language": "en",
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Proving a limit equals zero I have to prove, without L'hopital rule, the following limit:
$$\lim_{x \to \infty}\sqrt{x} \sin \frac{1}{x} =0$$
I tried doing a variable change, setting $t=\frac{1}{x}$ and reaching the following: $$\lim_{t\to 0} \sqrt{\frac{1}{t}} \sin t $$
But I can't prove neither. Tried the second ver... | $$\lim_{t \to 0} \frac{1}{\sqrt{t}} \sin(t) = \lim_{t \to 0} \frac{1}{t} \sin(t^2) = \lim_{t \to 0} \frac{\sin t^2 - \sin 0^2}{t} = f'(0)$$
where $f(x) = \sin x^2$.
By the chain rule, $f'(x) = 2x \cos x^2$, which is $0$ at $x=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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suggestion for implicit equation problem Hye guys
Given $$a = b\ e^{mt}-c\ e^{kt},$$
where $a,b,c,m,k$ are constants.
I am trying to solve this implicit equation to find for $t$. Can anyone here suggest me any theorem/method which can help me to solve this?
Thanks
| By setting $x=e^{mt}$, you can put the equation in the form
$$x^\alpha=px+q,$$
which is the intersection of a power law and a straight line.
In a few particular cases $\alpha=2,3,4,\frac12,\frac13,-1,-2,\cdots$ you can use the formulas for the polynomials up to quartic. But in general, there is no closed form, and you ... | {
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How do I solve an ellipse with three chord lengths and angles? I have a plane on which is a circle, there are three arbitrary points on the circle ($A$, $B$ and $C$) of which the relative angles are known. The plane the circle is on is then rotated arbitrary line on the plane (through the centre of the circle) to crea... | If I understand the question properly,
You have 3 points on a circle given by the angles $(t, t_1, t_2) $ which the points make with (say) the x-axis.
Now the circle has undergone a stretch:
$$ (r\cos t,r\sin t) \mapsto \begin{bmatrix} a/r &&0 \\ 0 && b/r \end{bmatrix} (r \cos t,r \sin t)$$
Which makes it an ellipse. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why isn't the golden ratio defined as the points where $f(x)=x^3-x^2-x$ are zero? I was messing around on desmos, and when I plugged in $f(x) = x^2 - x - 1$, I get two points where $f(x)$ is zero, which are answers to the golden ratio. Why is this not used in the definition? It seems so much clearer to me.
Link: https:... | It is more clear to you, maybe. There are LOTS of equations whose solutions may be the golden ratio.
But it's definition comes from geometry, like many other mathematical constants like $\pi$ and $\sqrt{2}$.
The golden ratio is defined in this way: it's the ratio of two numbers which is also equal to the ratio between ... | {
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Quotient Modules of a polynomial ring Let $R = K[x], K$ a field. Define for $a \space \epsilon \space K$ the ideal $ I_a := (x-a)$ in $K[x]$ and see $I_a$ as an $R$ module. Using that for $ a \space \epsilon \space K$ and $ b \space \epsilon \space K$, $I_a$ and $I_b$ are isomorphic, prove that the quotient modules $R... | Suppose you have a isomorphism $f:R/I_b \to R/I_a$ as $R$ modules. Observe that $x=b$ in $R/I_b$ and $x=a$ in $R/I_a$. Then you have
$$ bf(1)=f(b)=f(x)=xf(1)=af(1) $$
Now recall that the module structure in $R/I_a$ comes from the projection map $\pi$. Now by assumption $a\neq b$. And this means $(a-b)f(1)=0$ implies t... | {
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Solving equation over $\mathbb{Z}$ involving squares
Solve the equation $$a^2 - 10b^2 = 2$$ for $a,b \in \mathbb{Z}$.
I tried to consider the equation as a polynomial in the indeterminate $a$ and what I get is $$a_{1,2} = \pm \sqrt{10b^2 + 2}$$ which does not really help (I would have to find $b \in \mathbb{Z}$ such ... | If you consider the equation modulo $5$, you get $$a^2\equiv2\pmod5.$$ Checking the squares modulo $5$, we see that $2$ isn't a square, hence this equation has no solutions in the integers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2096378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to approximate 1/3 by only add/subtracting powers of 2 How approximate $\frac{1}{3}$ up to four significant digits by using only $\pm2^n$ where $n$ is a negative integer.
Preliminary attempt/example:
$$0.33\approx0.25+0.0625=0.3125$$
$$=2^{-2}+2^{-4}$$
| Hint: The series
$$\frac12 - \frac1{2^2} + \frac1{2^3} - \cdots + \frac{(-1)^{n-1}}{2^n} + \cdots$$
converges to $\dfrac13$.
| {
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What does the following statement in the definition of right inverse mean? ("For $b\in B$, $b\neq a\alpha$ for any $a$, define $b \beta=a_{1}\in A$") Question:
Let $A$ and $B$ be arbitrary sets, with $\alpha:A\rightarrow B$ an injection. Show how to define $\beta:B\rightarrow A$ such that $\alpha \beta$ is the identity... | I’ll rephrase the solution in what I hope is a more understandable way.
We need to define $b\beta$ for each $b\in B$. There are two kinds of elements of $B$: those that are in the range of $\alpha$, and those that are not. If $b=a\alpha$ for some $a\in A$, we define $b\beta=a$. Now fix a particular $a_1\in A$. If the... | {
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Is this a trick question? Balls into urns probability There are two urns. Urn 1 contains 3 white and 2 red ball, urn 2 one white and two
red. First, a ball from urn 1 is randomly chosen and placed into urn 2. Finally, a ball from
urn 2 is picked. This ball be red: What is the probability that the ball transferred from
... |
A Pretty Straightforward question. Please go through the diagram. Another way to think is if 1W transferred to U2= (1W + 2R) makes U2 = (2R + 2W). Now, # White Balls = # Red Balls . Then P(W)= P(R) = 1/2 for Urn2.
| {
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Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?
I'm trying to prove whether or not the Laurent polynomials $\mathbb{Z}[x, x^{-1}]$ with coefficients in $\mathbb{Z}$ form a principal ideal domain.
I know that $\mathbb{F}[x, x^{-1}]$ is a PID when $\mathbb{F}$ is a field, but clearly $\math... | Hint: Consider the ideal $I = (2, 1+x)$. Can you find a single generator for $I$?
Full solution:
Consider the ideal $(2, 1+x)$; I claim that it is not principal. Note that
$$(0) \subsetneq (2) \subsetneq (2, 1+x)$$ is a chain of prime ideals of length $2$, so $(2, 1+x)$ has height $\geq 2$. A principal ideal has he... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Derivative of Function with Cases: $f(x)=x^2\sin x^{-1}$ for $x\ne0$ So far I assumed the derivative of a function with cases such as this one:
$$f(x) = \begin{cases}x^2\sin x^{-1} & \text{ if } x \neq 0\\ 0 & \text{ else }\end{cases}$$ would be the cases of the derivatives.
So, for $f'$ I would get:
$$f'(x) = \begin{c... | How you concluded $f'(0)=0$ you made no attempt to explain. Remember that
$$
f'(0) = \lim_{h\to0} \frac{ f(0+h) - f(0) } h.
$$
You'll probably need to squeeze in order to find the limit. Next you have
$$
f''(0) = \lim_{h\to0} \frac{f'(0+h) - f'(0)} h.
$$
And again you'll probably have to squeeze. However, it's not hard... | {
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If $f'(a)$ exists, does $f'(a^+)$ and $f'(a^-)$ exist? Is it true that
If $f(x)$ is differentiable at $a$, then both $f'(a^+)$ and $f'(a^-)$ exist and $f'(a^+)=f'(a^-)=f'(a)$.
My answer is NO.
Consider the function
$$
f(x)=\begin{cases}
x^2\sin\dfrac{1}{x}&\text{for $x\ne0$}\\[1ex]
0&\text{for $x=0$}
\end{cases}
$$... | Yes, your answer is correct. The existence of the derivative of a function at a point does not always mean that the derivative will be continuous at that point. The condition $f′(a+)=f′(a−)=f′(a)$ implies continuity of the derivative at $x=a$ which is clearly not true for the function you mentioned at $x=0$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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$\left[\begin{array}{cc} P & A^T\\ A & 0\end{array}\right]$ is non-singular if and only if $\mathcal N(P) \cap \mathcal N(A)=\{0\}?$ Suppose $P\succeq 0$ and $A$ is of full row rank. I want to show that $\left[\begin{array}{cc} P & A^T\\ A & 0\end{array}\right]$ is nonsingular if and only if $\mathcal N(P) \cap \mathc... | Let $A$ be an $n \times n$ matrix and the rank of $A$ is $r$ where $r < n$.
There is an $n \times n$ matrix $U=[U_1|U_2]$ with orthogonal columns such that
$$
AU = [AU_1| AU_2] = [AU_1 | 0]
$$
where $AU_1$ and $U_1$ are $n \times r$ matrices and $AU_1$ is full rank.. The matrix $U$ may be computed using the singular v... | {
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"source": "stackexchange",
"question_score": "8",
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$\lim b_n=\frac{n^n}{(n+1)(n+2)\dots(n+n)}.$ $$b_n=\frac{n^n}{(n+1)(n+2)\dots(n+n)}.$$
Now, there is this theorem for sequences that if $\lim_{n\to ∞} a_{n+1} /a_n =l$, $|l|<1$ then $\lim_{n\to ∞} a_n=0$.
so, $\lim_{n\to ∞} b_{n+1} /b_n =e/4$ which is less than $1$, so $\lim_{n\to ∞} b_n$ should be equal to zero.
B... | One approach:
$$b_n=\frac{n^n}{(n+1)(n+2)\dots(n+n)}=\prod_{i=1}^n\frac{n}{n+i}$$
$$\implies \log b_n = n \cdot\left[ \frac{1}{n}\sum_{i=1}^n f\left(\frac{i}{n}\right)\right] \text{, where $f(x)=-\log(1+x)$}$$
So, the bracketed expression ought to tend to $\int_0^1f(x)\,dx=1-2\log2<0$
As such, $\log b_n = (1-2\log2)(n+... | {
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"source": "stackexchange",
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Which of the following value(s) of $t^3$ are not possible? Let $t$ be a real number such that $t^2 = at + b$ for some positive integers $a$ and $b$. Then for any choice of positive integers $a$ and $b$, $t^3$ is not equal to -
1) $4t+3$
2) $8t+5$
3) $10t+3$
4) $6t+5$
This question is from a prestigious Indian Scholarsh... | Assume that $t^3$ does equal $ut+v$ (where $u=4$, $v=3$ for the first part, etc.).
Then $t$ is a root of the cubic polynomial
$$f(X)=X^3-uX-v \in\Bbb Z[X]$$
as well as of the quadratic polynomial
$$g(X)=X^2-aX-b \in\Bbb Z[X].$$
But then $t$ is also a root of
$$f(X)-Xg(X) =aX^2+(b-u)X-v$$
and also of
$$h(X)=(f(X)-Xg(X... | {
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Find $xyz$ given that $x + z + y = 5$, $x^2 + z^2 + y^2 = 21$, $x^3 + z^3 + y^3 = 80$ I was looking back in my junk, then I found this:
$$x + z + y = 5$$
$$x^2 + z^2 + y^2 = 21$$
$$x^3 + z^3 + y^3 = 80$$
What is the value of $xyz$?
A) $5$
B) $4$
C) $1$
D) $-4$
E) $-5$
It's pretty easy, any chances of solving this que... | Consider the polynomial
$$p(t) = (1-x t)(1-y t)(1-z t)$$
Let's consider the series expansion of $\log\left[p(t)\right]$:
$$\log\left[p(t)\right] =-\sum_{k=1}^{\infty}\frac{S_k}{k} t^k$$
where
$$S_k = x^k + y^k + z^k$$
Since we're given the $S_k$ for $k$ up to $3$ we can write down the series expansion of $\log\left[p(t... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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Does the series $\sum_{n\ge1}\int_0^1\left(1-(1-t^n)^{1/n}\right)\,dt$ converge? Here is a question that I've been working on, a few years ago. I do know how to solve it but I am convinced that it deserves at least another point of view ...
I will post my own solution soon (within a week, at most) and I hope that - mea... | Here is a rather elementary solution.
Notice that $u_n$ is the area of the region $D$ in the unit square $[0,1]^2$ defined by $x^n + y^n \geq 1$. Now let $a_n = 2^{-1/n}$ and we split $D$ into three parts,
*
*$ D_1 = \{(x, y) \in [0,1]^2 : x^n + y^n \geq 1 \text{ and } x \leq a_n \} $
*$ D_2 = \{(x, y) \in [0,1]^2 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Exercise on a fixed end Lagrange's MVT Given a function f with derivative on all $[a;b]$ with $f'(a) = f'(b)$, show that there exists $c \in (a;b)$ such that $f'(c) = \frac{f(c)-f(a)}{c-a}$.
This is some kind of MVT with constraint. I have a proof but it uses Darboux's theorem. Can you prove it without using it?
Sketc... | This paper https://arxiv.org/pdf/1309.5715.pdf contains two nice proofs of Flett's MVT, which is a statement of your question. Of course, Riedel and Sahoo book I pointed in the comment is also good.
| {
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Is $a>b\iff \neg(b\geq a)$ true in constructive maths?
*
*Is $a>b\iff \neg(b\geq a)\;$ true in constructive maths? Why (not)?
*Also: is $\neg(a > b) \iff b\geq a\;$ true in constructive maths? Why (not)?
| (Disclamer: This is based on what I remember about counterexamples involving creative subjects in intuitionism and my memory is a bit rusty. I may have made a mistake.)
I'm pretty sure that (1) is not true in intuitionism. Here is why:
Let $P$ be a mathematical proposition which has not yet been proved or disproved. Le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $(1-\rho(x,F)/\varepsilon)^+$ is uniformly continuous. Let $(M,\rho)$ be a metric space, $F\subset M$ a closed subset, $\varepsilon>0$.
On page 8 of the book "Convergence of Probability measures" by Patrick Billingsley one says that the function
$$f:M\to [0,1]\\ x\mapsto(1-\rho(x,F)/\varepsilon)^+$$
is unifo... | If $x$ is in the metric space and $z \in \mathrm{F}$ then $d(x, \mathrm{F}) \leq d(x, z) \leq d(x, y) + d(y ,z)$ so taking the infimum on $z$ on the left-most and right-most gives $d(x, \mathrm{F}) - d(y, \mathrm{F}) \leq d(x,y)$ and by symmetry, you can put absolute value. So $f(x) - f(y) = \dfrac{\rho(x, \mathrm{F}) ... | {
"language": "en",
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Question about reading proof notation When reading the following problem, do you assume that each premise is true? So since number 2 states ¬ B am I to assume that ¬ B is true? Which would mean B is false?
*
*A ∨ C → D Premise
*¬ B Premise
*A ∨ B Premise
*A 2, 3, Disjunctive Syllogis... | dictionary.com defines:
'Premise': a statement which is assumed to be true for the purpose of an argument from which a conclusion is drawn
| {
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Can you explain me this problem of combinatorics? Yesterday was my final test of Discrete Mathematics and the second question was:
Wath is the cardinality of these sets:
a) Integers with 4 different digits and its digits increase, decrease and then increase again.
Eg. 1308 is valid
1300 is not valid
1320 is n... | For (b), which is the easiest of the three problems:
Let the digits from left to right be $D(1),D(2),D(3),D(4).$
If each of the $3$ cases $D(4)=0$ or $D(4)=2$ or $D(4)=8,$ there are $3$ choices ($4,5$,or $6$) for $D(1)$. This gives $(3)(3)=9$ choices for the pair $(D(1),D(4)).$ For each pair there are $8$ choices of $... | {
"language": "en",
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Proving that $\cos(\arcsin(x))=\sqrt{1-x^2}$ I am asked to prove that $\cos(\arcsin(x)) = \sqrt{1-x^2}$
I have used the trig identity to show that $\cos^2(x) = 1 - x^2$
Therefore why isn't the answer denoted with the plus-or-minus sign?
as in $\pm \sqrt{1-x^2}$.
Thank you!
| Let $\arcsin x = \theta$. Then, by definition of the arcsine function, $\sin\theta = x$, where $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$, and
$\cos(\arcsin x) = \cos\theta$. Using the Pythagorean Identity $\sin^2\theta + \cos^2\theta = 1$, we obtain
\begin{align*}
\sin^2\theta + \cos^2\theta & = 1\\
\cos^2\th... | {
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"source": "stackexchange",
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Find all complex numbers $z$ satisfying the equation $z^{4} = -1+\sqrt{3}i$ Since any complex number can be of polar form. We set that $z = r\cos \theta +ir\sin \theta$. Now by de Moivre's Theorem, we easily see that $$z^{4} = r^{4}\cos4 \theta + ir^{4}\sin4 \theta$$
Since $$z^{4} = -1+\sqrt{3}i$$
We equip accordingly ... | Observe that $-1+\sqrt{3}i=2e^{\frac{2\pi i}{3}}$. Then the roots of the equation are
$$
z=\sqrt[4]{2}e^{\frac{\pi i}{6}+\frac{k\pi}{2}},
$$
where $k=0,1,2,3$. Thus $r=\sqrt[4]{2}$. Substitute $k$, then we get $z=\sqrt[4]{2}e^{\frac{\pi i}{6}}$, $z=\sqrt[4]{2}e^{\frac{2\pi i}{3}}$, $z=\sqrt[4]{2}e^{\frac{7\pi i}{6}}$, ... | {
"language": "en",
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Conditions for when a line in $\mathbb{C}$ is tangent to a point on a circle I am working on the following problem from Chapter 1, Section 5 of Conway's "Functions of One Complex Variable":
Let $C$ be the circle {z:|z-c|=r}, r>0; let $a=c+r\text{ cis }\alpha$ and put
$$
L_\beta=\left\lbrace z:\text{Im}\left(\frac{z... | Here is a "computational" proof.
This issue is translation-invariant and enlargment (i.e., homothety)-invariant; you may thus assume $c=0$ and $r=1$.
Therefore,
*
*you can parametrize the circle as the set of $z$ such that $z=e^{i \theta}.$
*the "equation" of the straight line becomes
$$\Im(\dfrac{z-e^{i \alph... | {
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Four numbers are chosen from 1 to 20. If $1\leq k \leq 17$, in how many ways is the difference between the smallest and the largest number equal to k?
Four numbers are chosen from 1 to 20. If $1\leq k \leq 17$, in how many ways is the difference between the smallest and the largest number equal to k?
My Working: ... | I will assume that numbers may not be repeated and that order of selection of the numbers does not matter., i.e. we are counting how many subsets, $A$, of $\{1,2,\dots,20\}$ have the property that $max(A)-min(A)=k$
First, recognize that $max(A)-1\geq max(A)-min(A)=k$ implies $max(A)\geq k+1$, for example if the distanc... | {
"language": "en",
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Showing $\sum_{d\mid n} \mu(d)\tau(n/d)=1$ and $\sum_{d\mid n} \mu(d)\tau(d)=(-1)^r$ Need some help on this question from Victor Shoup
Let $\tau(n)$ be the number of positive divisors of $n$. Show that:
*
*$\sum_{d\mid n} \mu(d)\tau(n/d)=1$;
*$\sum_{d\mid n} \mu(d)\tau(d)=(-1)^r$, where $n=p_1^{e_1}\cdots ... | Using the fact that both $\mu$ and $\tau$ are multiplicative for the first one we get from first principles the value
$$\tau(n) \prod_{q=1}^r \left(1+(-1)\times\frac{e_q}{e_q+1}\right)$$
which simplifies to
$$\tau(n) \prod_{q=1}^r \frac{1}{e_q+1} = 1.$$
For the second one we may write
$$\prod_{q=1}^r (1+(-1)\times 2) ... | {
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"source": "stackexchange",
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Solve matrix exponential power series The matrix exponential is defined as
$$ e^A = \sum_{n=0}^\infty \frac{1}{n!} A^n $$
However I would like to solve something similar:
$$ B = \sum_{n=1}^\infty \frac{1}{n!} A^{n-1} $$
(NOTE: starting index and the power)
I can transform that into
$$ B = A^{-1}\sum_{n=0}^\infty \fr... | Basically, your question asks how to apply the function $(e^x-1)/x$ to a matrix. One way to do it is using the integral representation
$$\frac{e^x-1}{x}=\int_0^1 e^{tx}\ dt.$$
Thus we can define $B$ to be
$$
B:=\int_0^1 e^{tA}\ dt.
$$
Since you already know what the matrix exponential is, this expression makes sense: t... | {
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Which of the integers cannot be formed with $x^2+y^5$ So, I was asked by my teacher in school to solve this problem it really had me stumped.The problem is as follows:Given that $x$ and $y$ are integers, which of the following cannot be expressed in the form $x^2+y^5$?
$1.)\ 59170$
$2.)\ 59012$
$3.)\ 59121$
$4.)\ 59149... | I have found
$$59170=9^5+11^2$$
$$59012=8^5+162^2$$
$$59149=9^5+10^2$$
$$59130=9^5+9^2$$
and $59121$ can't be expressed in this form.
| {
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"timestamp": "2023-03-29T00:00:00",
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Finding the limit of the area of a Koch Snowflake this is my first question for this site and I made this account specifically for help with the following topic.
I am doing a research presentation on the Koch Snowflake, specifically, the area.
So far, I have been attempting to generalize a formula for finding the area... | The following forms a GP with $a=\frac{1}{9}$ and $r=\frac{4}{9}$.
$$\begin{align}
&\lim_{n\to\infty} \sum_{r=2}^{n} \frac{3 \cdot 4^{r-2}}{9^{r-1}} \cdot \frac{s^2 \sqrt{3}}{4}\\
=\ &\frac{ 3\sqrt{3}\cdot s^2}{4}\lim_{n\to\infty} \sum_{r=2}^{n} \frac{ 4^{r-2}}{9^{r-1}} \\
=\ &\frac{ 3\sqrt{3}\cdot s^2}{4}\cdot\frac{\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Relation between Diagonals in a pentagon i do need another pair of eyes to look at this stupid question:
Take a regular pentagon and draw diagonals from each vertex/point. By doing that, you create another pentagon in the middle of the old one. Regarding the n-Pentagon with sides $s_n$ and diagonals $d_n$, show the fol... | Given a regular pentagon $ABCDE$, draw diagonals $AC$ and $BE$ which intersect at $F$. $BCDF$ is a parallelogram because each diagonal is parallel to the side it does not meet at the vertices, thus $AF=BF=$(diagonal minus side).
Next draw all five diagonals of the pentagon. Let $F$ be the intersection of $AC$ and $BE... | {
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"timestamp": "2023-03-29T00:00:00",
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how to solve $min\{x \in \mathbb N_0 \quad |x \cdot 714\quad mod \quad 1972 \quad = \quad 1292 \quad mod \quad 1972 \} $ (modulo equation) Question: How can I solve: $min\{x \in \mathbb N_0 \quad |x \cdot 714 \equiv 1292 \mod 1972 \} $ ?
I only know about:
$x \cdot a \equiv _m b \Rightarrow m|x \cdot a - b$
different... | As $714=34\cdot21$, $1292=34\cdot 38$, $1972=34\cdot 58$, this is equivalent to solving $21 x\equiv 38\mod 58$.
We have to find the inverse of $21$ modulo $58$. The tool for this is the Extended Euclidean algorithm:
$$\begin{array}{rrrl}
\hline
r_i&u_i&v_i&q_i\\
\hline
58&0&1\\
21&1&0&2\\
\hline
16&-2&1&1\\
5&3&-1&3\\1... | {
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"url": "https://math.stackexchange.com/questions/2099165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof regarding logic and negation of quantifiers I've read about the following task, but don't know how to prove it:
Proof that $\neg(\forall x ( V(x)\rightarrow F(x))\iff \exists x( V(x) \land \lnot F(x)) $.
Maybe we start by proving "$\Leftrightarrow$" by proving the contraposition:
$\neg\exists x (V(x) \land \lnot... | Here is an approach to proving the biconditional using a Fitch-style proof checker. Using a proof checker makes sure that I am following the rules.
I used the following rules: change of quantifiers (CQ), universal elimination (∀E), universal introduction (∀I), De Morgan's laws (DeM), negation elimination (¬E), negatio... | {
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"url": "https://math.stackexchange.com/questions/2099256",
"timestamp": "2023-03-29T00:00:00",
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How are these two different general equations both represent a hyperboloid of 2 sheet? A. $\frac{x^2}{a^2}+\frac{y^2}{b^2}-z^2=k<0$
B.$\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
I don’t know how I can turn one into the other.
EDIT:Thanks to Bernard, I have somewhat of a clue as to where to begin. But Dividing t... | Rewrite the first equation as
$$\frac{z^2}{(\sqrt{-k})^2}-\frac{x^2}{(\sqrt{-k}\, a)^2}-\frac{y^2}{(\sqrt{-k}\, b)^2}=1.$$
| {
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How to calculate the number of states of a 7-segment LCD? A 7-segment LCD display can display a number of 128 states.
The following image shows the 16x8-grid with all the possible states:
How can you calculate the number of states?
| The combinations of segments can be calculated with the binomial coefficient:
$$
_nC_k=\binom nk=\frac{n!}{k!(n-k)!}
$$
I came across this problem evaluating the different number of segments lightened:
No segment $= 1$
One segment $= \binom 71 = \frac{7!}{1!(6)!} = 7$
Two segments $= \binom 72 = \frac{7!}{2!(5)!} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2099422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to calculate the limit $\displaystyle \lim_{x\rightarrow 0} (\sin(2\phi x)-2\phi x)\cot(\phi x)\csc^2(\phi x)$ How to calculate the limit?
$$\lim_{x\rightarrow 0} (\sin(2\phi x)-2\phi x)\cot(\phi x)\csc^2(\phi x)=-\frac{4}{3}$$
where $\displaystyle \phi$ is a real number.
| First see that
$$\sin(x)=x-\frac16x^3+\mathcal O(x^5)$$
So,
$$\sin(2\phi x)-2\phi x=\color{#4488dd}{-\frac43}\phi^3x^3+\mathcal O(x^5)$$
Similarly,
$$\cot(\phi x)\csc^2(\phi x)=\frac{\cos(\phi x)}{\sin^3(\phi x)}=\frac{\cos(\phi x)}{\phi^3x^3-2\phi^5x^5+\mathcal O(x^7)}$$
And combining all of this,
$$\begin{align}(\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2099542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find a subsequence whose limit exists
Let $\{a_n\}$ be a sequence of non-zero real numbers.
Show that it has a subsequence $\{a_{n_k}\}$ such that $\lim \dfrac{a_{n_{k+1}}}{ {a_{n_k}}}$ exists and belongs to $\{0,1,\infty\}$.
I am finding the above problem false.
If I take $(a_n)_n=(e^{-n})_n$ then any sub-sequence o... | Notice that $n_k+1 = n_{k+1}$ is false most of the times so your counterexample does not work. Take $n_k = 2^k$, then
$$ \frac{a_{n_{k+1}}}{a_{n_k}} = \frac{e^{-2^{k+1}}}{e^{-2^k}} = \exp(-2^k) \to 0
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2099666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Definition of significant figures Normally when we are taught how to add numbers with regards to significant figures, we are told to round the result to the rightmost place of the least precise digit. $3.55 + 4 = 7.55$, for example, would be rounded off to $8$. But for my argument I will be considering the addition of ... | There is some inconsistency in usage. Some people count significant digits from the decimal point, which works well enough when the possible range of values is reasonably small. For the rule you cite (and others related to error analysis and propagation) to be applied consistency, one must instead start counting signif... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2099735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Does $\int_{0}^{\infty}{\cos(10x^2\pi)\sin(6x^2\pi)\over \sinh^2(2x\pi)}\mathrm dx={1\over 16}?$ How do we prove these two results?
$$\int_{0}^{\infty}{\cos(10x^2\pi)\sin(6x^2\pi)\over \sinh^2(2x\pi)}\mathrm dx={1\over 16}\tag1$$
$$\int_{0}^{\infty}{\sin(10x^2\pi)\sin(6x^2\pi)\over \sinh^2(2x\pi)}\mathrm dx={1\over 8\p... | Maybe the most effective way to calculate these integrals is not to start from scratch and use residue theory, but to apply the general formula proved in Ramanujan's Lost Notebook, part IV, formula 14.4.14. In another words there is no need to reinvent the wheel. I will demonstrate the method and its effectiveness by c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2099800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 2,
"answer_id": 0
} |
solve system of two trigonometric equations I have two equations:
$$f(x) = x^2/k^2$$
and
$$ g(x) = a \cdot sinh(bx)$$
The derivatives are therefore
$$ f'(x) = 2x/k^2$$ and $$g'(x) = ab \cdot cosh(bx)$$
Now for given values of $k$ and $x$ (for example: $k=0.42$ and $x=7$) I want to find values of $a$ and $b$ such that ... | Since you have $a$ in both equation, you can eliminate it and the equation becomes
$$
\frac{x}{\sinh(bx)}=\frac{2}{b\cosh(bx)}
$$
that is,
$$
bx=2\tanh(bx)
$$
Consider the function
$$
f(t)=2\tanh t-t
$$
that we can study for $t\ge0$. The derivative is
$$
f'(t)=\frac{2}{\cosh^2t}-1=\frac{2-\cosh^2t}{\cosh^2t}
$$
which v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2100016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solutions of $x^ y = y ^ x$, with rational x and irrational y It seems to me that the equation $x^y = y^x$ has no solution in which $x$ is rational and $y$ irrational, or vice versa.
I could not get any counterexample.
| The equation is equivelant to $y\log x = x\log y$ or ${y\over\log y}={x\over\log x}$. As $x/\log x$ is convex, for any rational number $x$, there exists a real number $y$ such that $x^y = y^x$.
Let $y$ be such that $3^y=y^3$. Suppose $y$ is rational, $y=\frac mn$. Then,
$$n^{3n}3^m=m^{3n}$$
Which implies $n=1$, as $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2100147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Fundamental group of $S^2\cup\{xyz = 0\}$ We want to compute the fundamental group of $S^2 \cup \{xyz=0\}$. It is easy to see that it retracts to a sphere joined with three disks. How can I show that its fundamental group is trivial?
| Note that
\begin{align*}
&\ \{(x, y, z) \in \mathbb{R}^3 \mid xyz = 0\}\\
=&\, \underbrace{\{(x, y, z) \in \mathbb{R}^3 \mid x = 0\}}_{yz-\text{plane}}\cup\underbrace{\{(x, y, z) \in \mathbb{R}^3 \mid y = 0\}}_{xz-\text{plane}}\cup\underbrace{\{(x, y, z) \in \mathbb{R}^3 \mid z = 0\}}_{xy-\text{plane}}.
\end{align*}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2100383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What's the name of this fractal?
I created it with some simple chaos game restrictions:
*
*You can always move towards the point you moved to on the last step
*When on the top left corner, you can go to the right corners
*When on the top right corner, you can go to the left corners
*When on the bottom corners, y... | Your figure can be constructed by a graph directed iterated function system. An iterated function system typically has no restrictions on which transforms may follow each other. A graph-directed IFS has restrictions like the one you have imposed: there is a directed graph in which the transformations correspond to ed... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2100526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find $\lim \limits_{x\to 0}{\sin{42x} \over \sin{6x}-\sin{7x}}$ I want to find
$$\lim \limits_{x\to 0}{\sin{42x} \over \sin{6x}-\sin{7x}}$$
without resorting to L'Hôpital's rule. Numerically, this computes as $-42$. My idea is to examine two cases: $x>0$ and $x<0$ and use ${\sin{42x} \over \sin{6x}}\to 7$ and ${\sin{42... | Hint:
$$\frac{\sin42x}{\sin6x-\sin7x}=\cfrac{\frac{\sin42x}{42x}}{\frac17\frac{\sin6x}{6x}-\frac16\frac{\sin7x}{7x}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2100637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Given a word, how to tell whether it is a Roman numeral? If I have a word that consists of letters I, V, X, L, C, D, M, how can I tell whether it is a valid roman numeral? For example, how do I tell that IXXL is not valid?
| To ease the description you can consider the pairs "IV", "IX", "XL", "XC", "CD" and "CM" as single symbols.
That is, the roman numerals are sequences made of these symbols: I, IV, V, IX, X, XL, L, XC, C, CD, D, CM, M.
The sequences must hold these rules:
*
*The symbols I, X, C, M can be repeated up to three consecu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2100834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Uniform Boundedness principle for bounded linear maps from Frechet Space into a Banach Space I am looking for a proof of the uniform boundedness principle where the domain is a Frechet space, instead of the usual setting of a Banach Space.
This is used in proving the space of tempered distributions is complete but I c... | Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X \rightarrow Y$ be a continuous linear map for each $a\in A$, and let $q: Y \rightarrow [0,\infty)$ be a continuous semi-norm.
If $\sup \{q(T_a x):a\in A\} <\infty$ $\forall x\in X$ (pointwise bounded), then $x\mapsto \sup \{q(T_a x):a\in A\}$ is a continuous sem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2100943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate $\int^4_0 g(x) \text { d}x$ Evaluate $\int^4_0 g(x) \text { d}x$
Breaking this up into separate integrals, I got:
$$\int^4_0 g(x) \text { d}x = \int^1_0 x \text { d}x + \int^2_1 (x-1) \text { d}x + \int^3_2 (x-2) \text { d}x + \int^4_3 (x-3) \text { d}x$$
$$ = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1... | You are most right.
As a result of its definition, the riemann integral does not change its value if you change a function in a finite set of points, because you can just choose a partition to have those troublesome points as boundaries of the intervals of the partition. Likewise, integrating $f $ over the four interv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Identifying left- and right-Riemann sums of $\int_9^{14}e^{-x^4}\ dx$
My attempt:
Relooking at it, I think $L_{20}$ would be the highest, so like
$R_{1200} < L_{1200} < L_{20}$, but I have no way to justify it, any help is appreciated.
| No matter what $n$ and $m$ are, $R_n<L_m$ based on your knowledge that $R_n<A$ and $A<L_n$. So $R_{1200}$ should be the smallest of the three: $0.33575$
Now both $L_{20}$ and $L_{1200}$ overestimate the value of $A$. Informally, $L_{1200}$ is closer to $A$, because $A=\lim_{n\to\infty}L_{n}$. The function is not partic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A tough integral : $\int_{0}^{\infty }\frac{\sin x \text{ or} \cos x}{\sqrt{x^{2}+z^{2}}}\ln\left ( x^{2}+z^{2} \right )\mathrm{d}x$ Recently, I found these two interesting integrals in Handbook of special functions page 141.
$$\mathcal{I}=\int_{0}^{\infty }\frac{\sin(ax)}{\sqrt{x^{2}+z^{2}}}\ln\left ( x^{2}+z^{2} \rig... | For the second integral
Note that
$$ K_\nu(az)=\frac{\Gamma(\nu+1/2)(2z/a)^\nu}{\sqrt{\pi}}\int_0^\infty\frac{\cos at }{(t^2+z^2)^{\nu+1/2}} dt$$
By differentiation with respect to $\nu$
\begin{align}
\frac{\partial K_\nu(az)}{ \partial \nu} &=(\Gamma'(\nu+1/2)+\log(2z/a) )K_\nu(az)\\&-\frac{\Gamma(\nu+1/2)(2z/a)^\nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Endomorphisms whose matrices relative to every basis are triangular Let $E$ be a vector space and $u\in{\cal L}(E)$.
Given $\beta$ a basis of $E$, we denote by $M_\beta(u)$ the matrix of $u$ relative to $\beta$.
If we suppose that, for every basis $\beta$, the matrix $M_\beta(u)$ is upper triangular, then it is easy to... | For any given basis $(e_1,\ldots,e_n)$, we can show that $e_1$ or $e_n$ is an eigenvector. So we obtain one eigenvector $f_1$.
Now, choose a basis $(g_1,\ldots,g_n)$ such that $f_1\notin \text{span}\{g_1,g_n\}$. Thus, we obtain a second eigenvector $f_2$ (which is $g_1$ or $g_n$) linear independent with $f_1$. We can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Idea for deriving Green's functions - Helmholtz equation Let $x,y\in\mathbb{R}$. Then the Green's function for the Helmholtz equation is given by
$$\left(\Delta+\frac{\omega^{2}}{c^{2}}\right)G(x,y,\omega)=\delta(x-y).$$
Now what is the idea for deriving the Green's function here? Intuitively, I would take Fourier tran... | I don't see where you get the convolutions on the LHS. The FT would give you (up to some multiplicative constants from FT normalisation, who cares)
$$(\omega^2/c^2-|\xi|^2) G(\xi,y,\omega ) = \exp(i y \xi),$$
which should be easy to manipulate - I think a residue theorem would do the trick.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Weak derivative: recursive definition, or confusing notation? According to the Wiki article, if $u$ and $v$ are locally integrable functions on some open subset of $\mathbb{R}^n$, then $v$ is the weak derivative of $u$ if, for any infinitely differentiable function $\varphi$ on $U$ with compact support, we have
$$\int_... | Yes $D^{\alpha}u$ is a notation for the weak derivative that satisfies the "partial integration-equation" and $D^{\alpha}\varphi$ denotes the classical derivative which is of course a weak derivative too, because it satisfies
$\int_U\phi D^{\alpha}\psi=(-1)^{|\alpha|}\int_U D^{\alpha}\phi \psi$
for all smooth $\psi$ wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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What is the slope of a line half way between two lines of slope m1 and m2? I have two lines of slope m1 and m2 respectively. What is the slope of the line bisecting these two lines? Clearly there are 2 different lines which can be constructed, one perpendicular to the other. I am looking for the one which bisects th... | We have:
$$m_1=\frac{y_2-y_1}{x_2-x_1}=\tan{(\alpha)}$$
And:
$$m_2=\frac{y_3-y_1}{x_3-x_1}=\tan{(\beta)}$$
Therefore,
$$m_3=\tan\left(\beta+\frac{\alpha-\beta}{2}\right)=\tan\left(\frac{\alpha+\beta}{2}\right)$$
Where $m_3$ is the slope of the bisector.
Now, we will use this result to find an expression in terms of $m_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Is it possible to simplify the complexity $O(n\log{}n + \frac{n^2}{m}\log{}m)$? The question is in the title: is it possible to simplify the complexity $O(n\log{}n + \frac{n^2}{m}\log{}m)$ ? $n$ and $m$ are two variables, and you know that $n > m$ (by the way, what if we don't know that?).
What I first thought was that... | In general, if you have two parameters $m$ and $n$ that can be independently adjusted, you are stuck with two parameters. Sometimes $m$ and $n$ are naturally tied together (maybe they're roughly proportional in most cases, or maybe $m$ is generally $O(1)$ and its typical values don't depend much on $n$) and then you c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$ Prove that $$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert.$$
My try : $ f(x) =\cos x$ and use the mean value theorem.
| since $\left| \sin { x } \right| \le \left| x \right| $
$$|\cos x-\cos y|≤\left| 2\sin { \frac { y-x }{ 2 } \sin { \frac { x+y }{ 2 } } } \right| \le 2\left| \sin { \frac { x-y }{ 2 } } \right| \left| \sin { \frac { x+y }{ 2 } } \right| =2\left| \frac { x-y }{ 2 } \right| \cdot 1=\left| x-y \right| $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
asymptotics on series Define
$$
f(x)=\sum_{k=1}^\infty \frac{x^k}{k\cdot k!}.
$$
Is there a way to find the asymptotics of $f(x)$ as $x\rightarrow \infty$? What I suspect is
$$
f(x)\sim \frac{e^x}{x},
$$
because
\begin{align*}
f(x)&=\sum_{k=1}^\infty \frac{x^k}{k\cdot k!}\\
&\geq \sum_{k=1}^\infty \frac{x^k}{(k+1)\cdo... | Here it is a (very!) brute-force approach. Both $\frac{e^t-1}{t}$ and its primitive are entire functions, hence
$$ g(x)\stackrel{\text{def}}{=}\frac{1}{x\,e^x}\sum_{k\geq 1}\frac{x^k}{k\cdot k!} = e^{-x}\sum_{k\geq 0}\frac{x^k}{(k+1)(k+1)!} = \sum_{j,k\geq 0}\frac{(-1)^j x^{k+j}}{(k+1)^2 j!k!}$$
leads to:
$$ g(x) = \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Can someone give an example of an ideal $I \subset R= \Bbb{Z}[x_1,...x_n]$ with $R /I \cong \Bbb{Q}$? Question in the title. I have never seen a quotient of $R$ by a maximal ideal $I$ that is an infinite field, so I would also be interested in the case that $R/I$ is any infinite field. If no such examples exist, I woul... | You can't find such an example: a maximal ideal $\mathfrak m$ in $\mathbf Z[x_1,\dots,x_n]$ has a non-zero intersection with $\mathbf Z$, which is a prime ideal $p\mathbf Z$, hence the quotient $\mathbf Z[x_1,\dots,x_n]/\mathfrak m$ is a finitely generated algebra over the finite field $\mathbf F_p$, which has characte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Is there a additive non-linear non-complex map?
Let $V$ and $W$ be vector spaces over a field $\mathbb{F}\neq \mathbb{C}$. Give an example of a non-linear map $T:V\to W$ such that \begin{equation}T(x+y) = T(x)+T(y), \forall x,y\in V.\end{equation}
I asked myself this question when I was resolving an excercise list os... | A discontinuous additive function $a\colon\Bbb R\to\Bbb R$ is non-linear map of a linear space $\Bbb R$ over a field $\Bbb R$ to itself. Observe that any additive map is linear, if the field of scalars is $\Bbb Q$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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A question on Boundary Value Problem asked in CSIR(India)-Dec16 The boundary value problem $x^2y''-2xy'+2y=0$ subjected to boundary conditions$y(1)+\alpha y'(1)=1 ; y(2)+\beta y'(2)=2$ has unique solution if
$$1.\alpha=-1,\beta=2\\2.\alpha=-2,\beta=2\\3.\alpha=-1,\beta=-2\\4.\alpha=-3,\beta=2/3.\\$$ Kindly suggest fro... | Given the boundary value problem
$$\tag 1 x^2y''-2xy'+2y=0$$
We see that this is an Euler-Cauchy type equation, so we try
$$y = x^m \implies y' = mx^{m-1}, y''=m(m-1)x^{m-2}$$
It is worth noting that as an alternate approach, we can make the substitution $x = e^t$ and then simplify to arrive at the same result.
We sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to determine what components make up a complex mixture based on a summary of that mixture? Any wine can be made up of many components. Each component is basically a bunch of grapes from a specific place and time, so each component has three attributes: vintage, varietal, and appellation.
While grapes come in to a... | One solution is take all $3\times 6 \times 5 = 90$ possible combinations and simply multiply the relevant percentages together to give an overall percentage for each particular combination. For your example of a 2016 Cabernet Sauvignon from Russian River Valley, that would give $0.50\%\times 78.95\% \times 37.02\% = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Using an Increasing Differentiable Function to Show A Sequence is Increasing I am taking my first real analysis course, and we are talking about sequences of real numbers. I have a question where as part of a proof I want to show that a sequence $a_n=\frac{n}{n+1}$ for $n\in\Bbb Z_{>0}$, is strictly increasing. Note $\... | Suppose we manage to construct, $f$ such that $f(n)=a_n$ and $ \forall x >0, f'(x) >0$, then we know that $f$ is an increasing function.
That is $x>y$ then $f(x)> f(y)$.
In particular since $n+1 > n$, we have shown that $a_{n+1} > a_n$ and hence the sequence is increaseing.
(remark: if we have further information such ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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The general solution of $y''=-\sin y$ When I asked Mathematica to solve the ODE
$$y''=-\sin y \tag{1} $$
I got the solutions
$$y=\pm 2 \text{am}\left(\frac{1}{2} \sqrt{\left(c_1+2\right) \left(t+c_2\right){}^2}|\frac{4}{c_1+2}\right), \tag{2} $$
where $\text{am}(u|m)$ is the Jacobi Amplitude function. I wonder why ther... | $$y=\pm2 \text{am}\left(c_1t +c_2\bigg|\frac{1}{c_1^2}\right).$$
The Jacobi amplitude function is symmetrical : $\quad\text{am}(-x|k)=-\text{am}(x|k)$
$$2\text{am}\left(c_1t +c_2\bigg|\frac{1}{c_1^2}\right)=-2\text{am}\left(-(c_1t +c_2)\bigg|\frac{1}{c_1^2}\right) = -2\text{am}\left(C_1t +C_2\bigg|\frac{1}{C_1^2}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Existence of $\alpha\in \Bbb C$ such that $f(\alpha)=f(\alpha+1)=0$.
Let $f(x)\in \Bbb Q[x]$ be an irreducible polynomial over $\Bbb Q$.
Show that there exists no complex number $\alpha$ such that $f(\alpha)=f(\alpha+1)=0$.
Following @quasi;
Let $f(x)=a_0+a_1x+a_2x^2+\dots +a_nx^n$.
Define $g(x)=f(x+1)-f(x)$
Then $g(... | This is a conceptual proof: if $f(a)=f(a+1)=0$, then $a$ is a root of both $f(x)$ and $f(x+1)$. Both polynomials are irreducible, hence minimal polynomials of $a$. By uniqueness, we obtain $f(x)=f(x+1)$ as polynomials, hence by plugging in $x=a+1$ we get $0=f(a+1)=f(a+2)$. Continue with $a+2,a+3, \dotsc$ and you will o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Cofinite topology on $X \times X$ with $X$ an infinite set Let $X$ be an infinite set. And consider $(X \times X)_{cof}$ and $X_{cof} \times X_{cof}$. I can see that $(X \times X)_{cof}$ is not finer than $X_{cof} \times X_{cof}$.
MY QUESTION: But is $X_{cof} \times X_{cof}$ finer (hence strictly finer) than $(X \time... | Maybe is slightly simpler to think about closed sets. For any pair of points $x,y\in X$, both $\{x\}$ and $\{y\}$ are closed in $X_{cof}$. By definition of product topology, both $\{x\}\times X$ and $X\times \{y\}$ are closed in $X_{cof}\times X_{cof}$. Therefore their intersection
$$(\{x\}\times X)\cap (X\times \{y\})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Similarities between triangles that share the same incircle I know this question may be based on programming but its core is geometry hence I am posting it here.
Okay what I am trying to do is to write a small program to find out all the possible triangles that have the same incircle radius.
One important condition is ... | The triangles you are looking for are either Pythagorean triangles or a sum of two (possibly scaled) Pythagorean triangles.
To see that, one can show first of all that any Pythagorean triangle has an integer inradius. Any primitive Pythagorean triple $(a,b,c)$ can be obtained from two coprime integers $m$ and $n$ (not ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Doubt in Moorey's ineqality given in Sobolev space . I am going through the statement of Moorey's inequality given in chapter $5$ of evans pde book on page $266$.
Statement is: Assume $p > n$. Then there exist a constant $C$, depending on $p, n$ such that $$\|u\|_{C^{0, \gamma}(\mathbb{R}^n)} \leq C \|u\|_{W^{1,p}(\... | The theorem states that if $u\in C^1(\mathbb{R}^n)$ is no element of $C^{0,\gamma}(\mathbb{R}^n)$(which might be the case) it is not in $W^{1,p}(\mathbb{R}^n)$ (under the given circumstances)So
$||u||_{C^{0,\gamma}(\mathbb{R}^n)}=\infty$ implies $||u||_{W^{1,p}(\mathbb{R}^n)}=\infty$(this for (1)). As for (2) You are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2102982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equivalence of two smooth curves in the plane having the same image The book I read is complex analysis by stein, he first define what is the equivalence of two parametrized curve, and then define the complex intagral on a smooth curve which is indepentent of our choice of parametrization.
And, my question is about a b... | Hint: Consider the following cases; in each case the images of the curves are the same.
*
*On $ [0,2\pi]:$ $z(t) = e^{it}, z_1(t)= e^{-it}.$
*On $ [0,1]:$ $z(t) = t, z_1(t) = t^2.$
*On $ [0,2\pi]:$ $z(t) = e^{it}, z_1(t)= e^{2it}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
An Elementary Inequality Problem Given $n$ positive real numbers $x_1, x_2,...,x_{n+1}$, suppose: $$\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_{n+1}}=1$$ Prove:$$x_1x_2...x_{n+1}\ge{n^{n+1}}$$
I have tried the substitution $$t_i=\frac{1}{1+x_i}$$
The problem thus becomes:
Given $$t_1+t_2+...+t_{n+1}=1, t_i\gt0$$... | From last, let $\sqrt[n]{t_1t_2...t_{n+1}}=A$ with $AM-GM$:
$$\frac{t_2+t_3+...+t_{n+1}}{n}\geq\sqrt[n]{t_2t_3...t_{n+1}}=\frac{A}{\sqrt[n]{t_1}}$$
so for the first
$$t_2+t_3+...+t_{n+1}\geq\frac{nA}{\sqrt[n]{t_1}}$$
and for the rest take like this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Probability of getting the first red ball before the first blue ball Consider an urn containing $N = N_1 + N_2 + N_3$ balls, more precisely, an urn containing $N_i \gt 0$ balls of colour $i$, $i = 1, 2, 3$. We draw the balls without replacement. Show that the probability of getting a ball of colour $1$ before a ball of... | Disregarding the irrelevant colour-3 balls, an outcome of the experiment gives you a random permutation of the $N_1+N_2$ balls of colours 1 and 2. The event in question is that the first of these balls is colour $1$. Since each of the $N_1 + N_2$ balls is equally likely to be first, and $N_1$ of them are of colour $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Comparison of sum of the minimum distances from the vertices between two triangles
*
*Can I say that the sum of the minimum distances from the vertices in triangle ABC is less than the sum of the minimum distances from the vertices in triangle DEF, if given that the perimeter of triangle ABC is less than the perimete... | No and no.
In the depicted configuration, the triangles $F_1 F_2 A_1$ and $F_1 F_2 A_2$ have the same perimeter, but the lengths of their Steiner nets (i.e. the sum of distances from the vertices for the Fermat point of such triangles), given by $F_2 V_1$ and $F_1 V_2$, is not the same.$^{(*)}$
In general, if $ABC$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to use Triangle inequality to prove $|(x+y)-5| < 0.05$ when $|x-2| < 0.01$ and $|y-3| < 0.04$ It's the first day of calculus, and it's been almost a year since I've been in college algebra, and really stuck on the following homework question:
"Suppose that $| x - 2| < 0.01$ and $| y - 3 | < 0.04$. Use the
Triangle ... | One thing about the problem that was given to you that could confuse you
is that it uses the same symbols $x$ and $y$ that are in the
Triangle Equality (at least in the version of that fact that you've seen).
So rename the variables in the Triangle Inequality.
They're just arbitrary names for anything you could plug in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
If X and Y are two independent standard normal r.v.s, calculate $P(X+Y \in [0,1] \mid X \in [0,1])$ I'm not sure how to solve this. This is my attempt so far.
So if X,Y two independent standard normal r.v.s, we have:
$$\mathbb{P}(X+Y\in [0,1] \mid X \in [0,1])=\frac{\mathbb{P}(\{X+Y\in [0,1]\} \cap \{X \in [0,1]\})}{\m... | Write
$$ \{ X +Y \in [0, 1], X \in [0, 1]\} = \{ (X, Y) \in D_1 \} \cup \{ (X, Y) \in D_2 \}, $$
where
\begin{align*}
D_1 &= \{(x, y) : 0 \leq x \leq 1 \text{ and } 0 \leq y \leq 1 \text{ and } 0 \leq x+y \leq 1 \} \\
D_2 &= \{(x, y) : 0 \leq x \leq 1 \text{ and } -1 \leq y \leq 0 \text{ and } 0 \leq x+y \leq 1 \}.
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Trouble calcualting simple limit I'm not sure if my solution is correct. The limit is:
$\lim_{x\to0}\cot(x)-\frac{1}{x}$. Here is how I tried to solve it:
1. $\lim_{x\to0}\cot(x)-\frac{1}{x}$ = $\lim_{x\to0}\cot(x) - x^{-1}$
2. Since $\cot(0)$ is not valid, apply the de l'Hôpital rule $(\cot (x) )' = -\frac{1}{\sin^... | For the first one, $\tan x \sim x$ for small $x$ by Taylor's yields $\cot x\sim 1/x$ which should make the limit pretty easy.
To apply L'Hôpital's I always convert it to the indeterminate form $\frac{\infty}{\infty}$ to avoid mistakes. Although you are justified in applying L'Hôpital's in this case as $\infty-\infty$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Group acting by isometries on a tree
I've read the above in a book called "Translation equivalence in free groups" by Ilya Kapovich, Gilbert levitt, Paul Schupp, and Vladimir Shpirain.
Why does the infimum can be replaced by a minimum? How can I actually show that?
| Here is a proof basically from Culler and Morgan's paper Group actions on $\mathbb{R}$-trees.
Say $g$ fixes no element of $X$, and consider the arc $[x,gx]$, with midpoint $m$, and the arcs $g[x,gx], g^{-1}[x,gx]$. Note the translated arcs do not contain $m$ since then $m$ would be the midpoints of those arcs, so $g$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $G$ be a finite abelian group and $H$ be a subgroup of $G$. Then there is an epimorphism from $G$ to $H$. Let $G$ be a finite abelian group and $H$ be a subgroup of $G$. Then there is an epimorphism from $G$ to $H$.
(That is, an epimorphism from a finite abelian group to its subgroup.)
| By Exercise II.5.8 in Hungerford's Algebra,
$G\cong \prod_{i=1}^{n}P_i$ is the direct product of its Sylow $p$-subgroups.
By Exerise 24.56 in Gallian's Contermporary Abstract Algebra,
$H\cong \prod_{i=1}^{n}(H\cap P_i)$ is the direct product of the Sylow $p$-subgroups of $H$.
So it is sufficient to consider the case $G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Algebraic proof of the chain rule? I would like to prove the chain rule: given $f$ and $g$ polynomial functions, $h = f \circ g$, and $a \in \mathbb{R}$, that $h'(a) = f'(g(a)) \cdot g'(a)$. However, I would like to do so without using the limit definition of the derivative or any sort of differentiation rules.
So far... | This is not a full solution, but it does describe another approach to defining the derivative of a polynomial that does not involve limits, and that is related the division algorithm.
Let $P(x)$ be any polynomial. Choose some real number $a$. Then if we divide $P(x)$ by $x-a$ we get a quotient, $Q(x)$, and a constant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Find $y'$ given $y\,\sin\,x^3=x\,\sin\,y^3$? The problem is
$$y\,\sin\,x^3=x\,\sin\,y^3$$
Find the $y'$
The answer is
Can some explain how to do this, please help.
| I have found that it helps students to understand implicit differentiation if first they think of both $x$ and $y$ as functions of some third variable such as $t$ and take the derivative of both sides with respect to $t$, being careful to use the product rule, chain rule, etc when needed. Then as a final step, multiply... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Coincidence of standard derivative and weak derivative Let $f:\mathbb{R}^n \to \mathbb{R}$ be in $W^{1,p}(\mathbb{R}^n)$ and differentiable (in the classical sense) almost everywhere.
Is it true that the standard derivative and the weak derivative conicide?
When $p>n$ this is a corollary from theorem 4.9 ("LECTURES ON... | This follows from the ACL characterization of Sobolev spaces, see https://en.wikipedia.org/wiki/Sobolev_space#Absolutely_continuous_on_lines_.28ACL.29_characterization_of_Sobolev_functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to express the cardinality of $∏_{1≤i≤n} A_i$ in terms of cardinalities $|A_1|, |A_2|, . . . , |A_n|$ I was given the problem:
For finite sets $A_1, A_2,\dotsc , A_n$ define their Cartesian product $\prod_{i=1}^n A_i$ as the
set of all $n$-sequences $(x_1, x_2,\dotsc, x_n)$, where $x_i \in A_i$ for every $i = 1, 2,... | We know that $$|A\times B|=|A|\times|B|\qquad(1)$$.
We want to show $$\left|\prod_{i=1}^nA_i\right|=\prod_{i=1}^n|A_i|$$ is true for any natural number $n$, where $\prod_{i=1}^n|A_i|=|A_1|\times|A_2|\times\dotsc\times|A_n|$. So, we have use induction.
The base case $n=1$ ($|A_1| = |A_1|$) is trivial. Now suppose induc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $ \sin\theta + \cos\theta = \frac 1 2$, what does $\tan\theta + \cot\theta$ equal? A SAT II question asks:
If $ \sin\theta + \cos\theta = \dfrac 1 2$, what does $\tan\theta + \cot\theta$ equal?
Which identity would I need to solve this?
| Hint
$$\sin\theta+\cos\theta=\frac{1}{2} \implies \left( \sin\theta+\cos\theta \right)^2 = \frac{1}{4} \iff \color{blue}{\cos\theta\sin\theta} = \cdots$$
and
$$\tan\theta+\cot\theta = \frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta} = \frac{\cos^2\theta+\sin^2\theta}{\cos\theta\sin\theta}= \frac{1}{\color{bl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Prove that $f$ is increasing if and only if a given inequality holds Let $f: [0, \infty) \to \mathbb{R}$ be a continuous function. Prove that $f$ is increasing if and only if:
$$\int_a^b f(x) dx \leq bf(b) - af(a), \, \forall \, \, 0 \leq a \leq b.$$
I have no difficulties in proving that if $f$ is increasing then the ... | Note that if $f(a) > f(b) $ then by continuity we can choose $c\in(a, b] $ such that $f(x) > f(c) $ for all $x\in [a, c) $ and hence $$\int_{a} ^{c} f(t) \, dt>(c-a) f(c) $$ And given condition on $f$ implies that integral above is not greater than $cf(c) - af(a) $. It now follows that $f(a) <f(c) $ which is contrary t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
If $f(1)$ and $f(i)$ real, then find minimum value of $|a|+|b|$ A function $f$ is defined by $$f(z)=(4+i)z^2+a z+b$$ $(i=\sqrt{-1})$for all complex number $z$ where $a$ and $b$ are complex numbers. If $f(1)$ and $f(i)$ are both purely real, then find minimum value of $|a|+|b|$
Now
$f(1)=4+i+a+b$ which means imaginary ... | If $a=a_1+a_2i$ and $b=b_1+b_2i$ then you have that $b_2+a_2=-1, b_2+a_1=1$. So you can pick any $b_1$, which, since you are seeking a minimum, means you can choose $b_1=0$. You get $b=b_2i$ and $a_2=-(b_2+1)$ and $a_1=1-b_2$. So you are trying to minimize:
$$\sqrt{a_1^2+a_2^2}+\sqrt{b_1^2+b_2^2}=\sqrt{(-1-b_2)^2+(1-b_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
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