Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Probability inequality proof I'm stuck on a homework question and don't even know where to start. Here it goes:
If A and B are two events which are not impossible, prove that $$P(A\land B)\times P(A\lor B)\le P(A)\times P(B)$$
| In general if $d+a=c+b$ and $a\leq b\leq c\leq d$ then:
$$ad=\frac{1}{4}\left(d+a\right)^{2}-\frac{1}{4}\left(d-a\right)^{2}\leq\frac{1}{4}\left(c+b\right)^{2}-\frac{1}{4}\left(c-b\right)^{2}=bc$$
This as a direct consequence of:$$d-a\geq c-b\geq 0$$
This can be applied by taking $a=P(A\cap B)$, $b=P(A)$, $c=P(B)$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/644009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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a question about double integral Let $a,b$ be positive real numbers, and let $R$ be the region in $\Bbb R^2$ bounded by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Calculate the integral
$$
\int\int_R\left(1-\frac{x^2}{a^2}-\frac{y^2}{b^2}\right)^{3/2}dx\,dy
$$
my question is I don't know anything about $R$, the function $\f... | It sounds like you're just a bit confused about notation. $R$ is simply the name of the region. The notation
$$\iint\limits_{R} f(x,y) \, dA$$
simply means that we should integrate over the region $R$. In your case, $R$ is defined to be the region contained inside the ellipse
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Intuition of why $\gcd(a,b) = \gcd(b, a \pmod b)$? Does anyone have a intuition or argument or sketch proof of why $\gcd(a,b) = \gcd(b, a \pmod b)$?
I do have a proof and I understand it, so an intuition would be more helpful.
The proof that I already have:
I show $\gcd(a,b) \mid \gcd(b, a \pmod b)$ and $\gcd(b, a \pm... | Suppose each of $a$ and $b$ is an integer number of miles. Then so is $a\bmod b$.
If a mile is a "common measure" (as Euclid's translators say) of both distances, then a mile is a common measure of what's left when $b$ has been taken from $a$ as many times as possible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/644252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Is a function determined by its integrals over open sets? If $f \in L^1(\mathbb R)$ satisfies
$$
\int_U f = 0
$$
for every open set $U \subset \mathbb R$, then is it true that $f = 0$ a.e.?
| Since $f$ is measurable, the set $A=\{x\in\mathbb{R}\mid f(x)>0\}$ is measurable. Therefore, by regularity of the Lebesgue measure, $m$, for every $\varepsilon>0$ there exists an open set $U$ such that $A\subset U$ and $m(U\setminus A)<\varepsilon$. Let $f^+$ and $f^-$ denote the positive and negative part if $f$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/644347",
"timestamp": "2023-03-29T00:00:00",
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Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$. Question: Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$.
Setting the 2 equations equal w.r.t. $z$, $x^2+y^2+1=2-x^2-y^2 \rightarrow x=\pm\sqrt{\frac 12-y^2}$
Therefore the boundary of $y=\pm\frac {1}{\sqrt2}$.
So to find ... | Your setup is right. Here is the method you could have done to compute the volume.
Assume the density is $f(x,y,z) = 1$, so
$$V = \iiint_D \,dx\,dy\,dz$$
We are given that the solid is bounded by $z = x^2 + y^2 + 1$ and $z = 2 - x^2 - y^2$. As I commented under your question, you need to use cylindrical coordinates ... | {
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How to prove $\left(\frac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\frac{n}{n+1}\right)^n$
Show that:
$$\left(\dfrac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\dfrac{n}{n+1}\right)^n$$
where $n\in \Bbb N^{+}.$
If this inequality can be proved, then we have
$$\lim_{n\to\infty}\sq... | Here is a partial answer :
If a sequence $u=(u_n)_{n\ge1}$ of real numbers converges, then the sequence $\left(\frac{1}{n}\sum_{k=1}^nu_k\right)_{n\ge1}$ converges to the same limit. This is the well known Cesaro's lemma.
It can be proved that the converse is false (consider the sequence $u=((-1)^n)_{n\ge1}$) but becom... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convert the power series solution of $(1+x^2)y''+4xy'+2y=0$ into simple closed-form expression $(a)$Use two power series in $x$ to find the general solution of
$$(1+x^2)y''+4xy'+2y=0$$
and state the set of $x$-values on which each series solution is valid.
$(b)$ Convert the power series solutions in $(a)$ into simple c... | Hint: Your power series are both geometric series.
| {
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"timestamp": "2023-03-29T00:00:00",
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pdf for non-central gamma distribution I have a given gamma distribution as:
$f(x;k,\theta) = \frac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{\frac{-x}{\theta}}$ and a non-centrality parameter $\delta$.
Now, I need to find the pdf of this non-central gamma distribution $f(x;k,\theta,\delta)$?
I have found an expression of this ... | As far as my monte-carlo simulation and closed form expression match, the non-central gamma could be well approximated by Amoroso distribution i.e.,
$f(x;k,\theta,\delta) = \frac{1}{\theta^k\Gamma(\theta)}(x-\delta)^{k-1}e^{\left(\frac{-(x-\delta)}{\theta} \right)}$
where $\delta$ is the location parameter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/644804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given two odd primes, $p\neq q$, prove that there are no primitive roots $\mod(pq)$
Given two odd primes, $p\neq q$, prove that there are no primitive roots $\mod(pq)$
I don't know where to start with this, any help would be appreciated.
| Hints:
*
*$\phi(pq)$ is an even multiple of both $\phi(p)=p-1$ and $\phi(q)=q-1$.
*Check what happens when you raise a residue class to power $\phi(pq)/2$.
| {
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Equation with the variable in the exponent and also in the base Does anyone know how to solve this equation, with the variable in the exponent and also in the base?
$$1.05^{2y}-0.13y-1=0$$
Thank you very much.
| Equations like this can sometimes be "solved" using the Lambert W function, but many do not define that as a solution. Usually you are reduced to numeric rootfinding, which is discussed in any numerical analysis book. This one has a root at $y=0$ and another near $5.5$ as shown by this Alpha plot. Alpha gives this so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/645001",
"timestamp": "2023-03-29T00:00:00",
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Convergence of the integral $\int\limits_{1}^{\infty} \left( \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+3}} \right) \, dx$ Would someone please help me prove that the integral
$$
\int\limits_{1}^{\infty} \left( \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+3}} \right) \, dx
$$
is convergent?
Thank you.
| Use
$$\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+3}}=\frac{\sqrt{x+3}-\sqrt{x}}{\sqrt{x^2+3x}}=\frac{3}{(\sqrt{x+3}+\sqrt{x})\sqrt{x^2+3x}}$$
So, the integrand is positive and $\le \frac{3}{2x\sqrt{x}}$.
Here I use $\sqrt{x^2+3x}>x$ and $\sqrt{x+3}>\sqrt{x}$
So the integral converges by the comparison criterium.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/645070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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exponential equation with different bases We have $3^x-5^\frac{x}{2}=4$ My question is what we can do here ? Can we solved it algebraically or we need to notice that $x=2$ and then show that for $x \neq 2$ there aren't any other solutions?
| Your second approach, viz. showing there are no other solutions would also require some algebra. With $2t = x$, you can write the equation as
$$(4+5)^t = 4 + 5^t$$
This is obvious for $t=1$ i.e $x=2$.
For $t > 1$, we have $(4+5)^t > 4^t+5^t > 4+5^t$
for $0 < t < 1$, let $y = \frac1t > 1$, then $(4+5^t)^y > 4^y+5 > 4+... | {
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Absolute continuity of a nondecreasing function Can anyone give me a hint on how to approach this problem? It's another problem from an old qualifying exam.
Suppose that $f\colon \mathbb R \to \mathbb R$ is nondecreasing, $\int_{\mathbb R} f' = 1$, $f(-\infty) = 0$, and $f(\infty) = 1$. Prove that $f$ is absolutely con... | One approach is as follows:
Since $f$ is non-decreasing, it is differentiable ae. [$m$] and $f(x)-f(y) \ge \int_y^x f'(t) dt$ for all $x>y$.
Use the fact that $1 = \lim_{x \to + \infty} f(x) - \lim_{x \to - \infty} f(x) = \int_{\infty}^\infty f'(t) dt$ to show that,
$f(x)-f(y) = \int_y^x f'(t) dt$ for all $x>y$.
Now us... | {
"language": "en",
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Prove $1/x$ is not uniformly continuous $f: (0,+\infty) \to (0,+\infty)$ $f(x) = 1/x$, prove that f is not uniformly continuous.
Firstly, I negated the definition of uniform convergence obtaining:
$\exists \epsilon > 0 $ s.t. $\forall \delta > 0 $ with $|x-y| < \delta$ & $x,y \in (0, + \infty)$ and $|f(x) - f(y)| = \le... | Here a full answer (that i writte too to practice) but take into account that I am just a student so I hope it is correct.
1 - First let recall the definition of a non uniformly continuous function.
It exists at least one $\epsilon_0>0$ such that for every $\delta>0$ that we can choose it will always exists at least $x... | {
"language": "en",
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if $g^k=e$ then $\chi(g)=\sum_j^n \zeta_k$
Let $G$ be a group. Let $g \in G$ and $g^k=e.$ Let $\chi$ be an
$n$-dimensional character of the group $G.$ Let $\zeta_k$ be $k$-th root of unity.
Prove that $\chi(g)$ is equal to sum of a $k$-th roots of unity.
My trying. Consider the cyclic subgroup $C_k \in G,$ Su... | If $g^k = e$ then since the function $\rho: G \to GL_n(\mathbb C)$ that constitutes your representation is a homomorphism you have that $\rho(g^k) = \rho(g)^k = $ I, $ $thus in particular you know that all of the eigenvalues of $\rho(g)$ must be kth roots of unity (as for any $v \in \mathbb C^n$ you have that $\rho(g)^... | {
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What makes a condition unary vs. n-ary (n>1)? For any two disjoint sets $A$ and $B$, a set $W$ is a connection of $A$ with $B$ if
*
*$Z\in W\implies (\exists x\in A)(\exists y\in B)[Z=\{x,y\}]$
*$(\forall x\in A)(\exists !y\in B)[\{x,y\}\in W]$
*$(\forall y\in B)(\exists !x\in A)[\{x,y\}\in W]$
I know that each ... | Say, $P(A,B,W)$ is the first order proposition that expresses that $W$ is a connection between $A$ and $B$, Then consider
$${\rm isConn}(W):= " \exists A\exists B:P(A,B,W)"\,.$$
| {
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What is the difference between a point and a vector? I understand that a vector has direction and magnitude whereas a point doesn't.
However, in the course notes that I am using, it is stated that a point is the same as a vector.
Also, can you do cross product and dot product using two points instead of two vectors? I ... | There is a difference of definition in most sciences, but what I suspect you're asking about is a rather nice one-to-one correspondence between points in real space (say perhaps $\mathbb{R}^n$) and vectors between $(0, 0, 0)$ and those points in the space of $n$-dimensional vectors.
So, for every point $(a, b, c)$ in $... | {
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Prove this is a subspace Let $ W_1, W_2$ be subspace of a Vector Space $V$.
Denote $W_1+W_2$ to be the following set
$$W_1+W_2=\left\{u+v, u\in W_1, v\in W_2\right\}$$
Prove that this is a subspace.
I can prove that the set is non emprty (i.e that it houses the zero vector).
pf: Since $W_1 , W_2$ are subspaces, then t... | If $w_1,w_2 \in W_1+W_2$, then $w_k=u_k+v_k$ for some $u_k \in W_1$ and $v_k \in W_2$. Since $u_1+u_2 \in W_1$ and $v_1+v_2 \in W_2$, we have $w_1+w_2=(u_1+u_2) + (v_1+v_2) \in W_1+W_2$.
Similarly, if $w \in W_1+W_2$, then $w=u+v$ for some $u \in W_1$ and $v \in W_2$, since $\lambda u \in W_1$ and $\lambda v \in W_2$, ... | {
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Are all $\mathbb{Z}/(6)$-modules injective? I'm trying to show that every $\mathbb{Z}/(6)$ module is injective. My strategy is to use Baer's Criterion.
The only nontrivial ideals of $\mathbb{Z}/(6)$ are $(2)=(4)=\{0,2,4\}$ and $(3)=\{0,3\}$. Suppose $g:(2)\to Q$ is a morphism into any $\mathbb{Z}/(6)$ module $Q$. I try... | *
*Every module over a field (i.e. vector space) is injective.
*If $C_1,C_2$ are categories with injective objects $I_1 \in C_1$, $I_2 \in C_2$, then $(I_1,I_2)$ is an injective object of $C_1 \times C_2$.
*If $R_1,R_2$ are rings, there is an equivalence of categories $\mathsf{Mod}(R_1 \times R_2) \simeq\mathsf{Mod}... | {
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Prove $\int \limits_0^b x^3 = \frac{b^4}{4} $ by considering partitions $[0, b]$ in $n$ equal subinvtervals. I was given this question as an exercise in real analysis class. Here is what I came up with. Any help is appreciated!
Prove $\int \limits_0^b x^3$ = $\frac{b^4}{4} $ by considering partitions [0, b] in $n$ equ... | Say that you want to use $n$ subintervals. So you want to integrate $x^3$ over the range b(i-1)/n and b i/n, index $i$ running fron $0$ to $n$.
The result of this integration for this small range is simply given by
b^4 (-1 + 4 i - 6 i^2 + 4 i^3) / (4 n^4)
You must now add up all these terms from $i=0$ to $i=n$ ... | {
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Is the following set is compact Consider the set of all $n \times n$ matrices with determinant equal to one in the space of $\mathbb R^{n\times n}$.
My idea is compact because determinant function is continous ant it is bijective from the given set to $\mathbb R$ and $\mathbb R$ is Hausdorff, so image of compact set ... | In $\mathbb R^m$, with the usual metric topology, a set is compact iff it is closed and bounded. The set you describe is closed (since it is the inverse image of a closed set under a continuous function) but it fails to be bounded if $n>1$ (and the case $n=1$ is trivial, do you see why?). Try to find matrices with arbi... | {
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Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be multiplicative. Is it exponential? For function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f\left(x+y\right)=f\left(x\right)f\left(y\right)$
and is not the zero-function I can prove that $f\left(1\right)>0$
and $f\left(x\right)=f\left(1\right)^{x}$ for each $x\in\math... | Continuity (or continuity in some point or measurability) is required. See Cauchy's functional equation. Your problem is reducible to this.
| {
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MATLAB: Approximate tomorrow's temperature with 2nd, 3rd and 4th polynomial using the Least Squares method. The following is Exercise 3 of a Numerical Analysis task I have to do as part of my university course on the subject.
Find an approximation of tomorrow's temperature based on the last 23
values of hourly tempe... | Let $\mathbf{t}\in\mathbb{R}^{23}$ be the last 23 samples you have. To fit these to an $N^\mathrm{th}$ order polynomial in terms of the hour, i.e., $t = \sum_i p_i h^i$, where $\mathbf{p}\in\mathbb{R}^{N}$ is the vector of coefficients and $h\in\mathbb{R}$ is the time in hours, first set up the system using the measur... | {
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How do I convince my students that the choice of variable of integration is irrelevant? I will be TA this semester for the second course on Calculus, which contains the definite integral.
I have thought this since the time I took this course, so how do I convince my students that for a definite integral
$$\int_a^b f(x... | Maybe it helps to investigate the wording: integration variable is just a fancy name for what we used to call placeholder in elementary school when we solved
3 + _ = 5
and used an underscore or an empty box as the placeholder. Isn't it obvious then that the symbol (or variable name) cannot have an effect on the solut... | {
"language": "en",
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Showing that a map defined using the dual is a bounded linear operator from X' into X' I have trouble answering the second part of the following exercise. Any help would be appreciated!
Let $(X, \| \cdot \|)$ be a reflexive Banach space. Let $\{ T_n \}_{n = 1}^\infty$ be a sequence of bounded linear operators from $X$ ... | After you have shown that $A := \sup\limits_n \lVert T_n'\rVert < \infty$, you have a known bound on $S(f)(x)$ for every $x\in X$ and $f\in X'$, namely
$$\lvert S(f)(x)\rvert = \lim_{n\to\infty} \lvert T_n'(f)(x)\rvert \leqslant \limsup_{n\to\infty} \lVert T_n'(f)\rVert\cdot \lVert x\rVert \leqslant \limsup_{n\to\infty... | {
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Equations of planes and lines in 3-space I'm reading Strang's book "Linear Algebra and it's applications" and he writes in the first chapter that an equation involving two variables in still a plane in 3-space.
"The second plane is 4u - 6v = -2. It is drawn vertically, because w can take any value. The coefficient of w... | A linear equation reduces the dimension of the ambient space by 1. You can think of it as restricting one variable, as a function of all the others.
Hence, a linear equation in 2 dimensions is a one-dimensional space, or a line. A linear equation in 3 dimensions is a 2-dimensional space, or a plane. And so on.
Foll... | {
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Hyperplane sections on projective surfaces I am studying Beauville's book "Complex Algebraic Surfaces".
At page 2 he defines the intersection form (.) on the Picard group of a surface.
For $L, L^\prime \in Pic(S)$
$$(L.L^\prime)=\chi(\mathcal{O}_S)-\chi(L^{-1})-\chi(L^{\prime-1})+\chi(L^{-1}\otimes L^{\prime-1})$$
Why ... | This self-intersection is exactly the degree of $S$.
Concretely, choose $H$ and $H'$ in general position, then $S \cap H$ and $S \cap H'$ are two curves on $S$, and they intersect in some number of points. This
already shows that the intersection is non-negative. The fact that it is positive
is a general fact abou... | {
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} |
Get the rotation matrix from two vectors Given $v=(2,3,4)^t$ and $w=(5,2,0)^t$, I want to calculate the rotation matrix (in the normal coordinate system given by orthonormal vectors $i,j$ and $k$) that projects $v$ to $w$ and to find out which is the rotation axis.
I Started by calculating the vector product $v \times ... | Try writing the matrix $A_{B_2}$ in terms of the basis $B_2 = \{\hat v, \hat w, \widehat{v \times w}\}$. This is very similar to the matrix for the plane that you've already written.
Then, think about if you know a way to change from the standard basis $B_1 = \{\hat i, \hat j, \hat k\}$ to this basis. That is, let $P... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Correct way to get average price probably a basic question for a lot of you guys, but it was a subject of a friendly debate at my work earlier - needless to say none of the involved were accountants. In short, we are thinking about which is the proper way to get the average price of a sold item. Here is a simple exampl... | The latter is better, because it actually tells you that what you have earned is the same as if you've sold all the items at this average price.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/646784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question about degree of a polynomial Let $R$ be a commutative ring with identity $1 \in R$, let $R[x]$ be the ring of polynomials with coefficients in $R$, and let the polynomial $f(x)$ be invertible in $R[x]$. If $R$ is an integral domain, show that $\text{deg}(f(x))=0$
| Go for a contradiction. Assume that $\deg(f)\geq 1$ Write out the polynomial, which has nonzero leading coefficient $a_n$
$$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$
Write the inverse function with nonzero lead coefficient
$$f^{-1}(x)=b_mx^m+b_{m-1}x^{m-1}+...+b_1x+b_0$$
$f(x)f^{-1}(x)$ has the lead term $a_nb_mx^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why $\frac{|1-z|}{1-|z|}\le K$ corresponds to the region defined by the Stolz angle? In his presentation of Abel's theorem, Ahlfors mentions that for a fixed positive number $K$, the region defined by \begin{equation}
\frac{|1-z|}{1-|z|}\le K
\end{equation} corresponds to the region inside the unit circle and in a cert... | It's not an equality. The Stolz angle with opening $\alpha > 0$ and radius $r$ is
$$S(\alpha,r) = \{1 - \rho e^{i\varphi} : 0 < \rho < r,\; \lvert\varphi\rvert < \alpha\},$$
a circular sector that for $\alpha < \pi/2$ and small enough $r$ (depending on $\alpha$) is contained in the unit disk. Its boundary consists of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/647025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How prove exists a sequence $\{a_{n}\}$ of real numbers such that $\sum_{n=1}^{\infty}a^2_{n}<\infty,\sum_{n=1}^{\infty}|a_{n}b_{n}|=\infty$ Suppose that the series $\displaystyle\sum_{n=1}^{\infty}b^2_{n}$ of postive numbers diverges. Prove that
there exists a sequence $\{a_{n}\}$ of real numbers such that
$$
\sum_{n... | Define
$$
s_n=\sum_{k=1}^nb_k^2
$$
and
$$
a_n=\frac{b_n}{\sqrt{s_ns_{n-1}}}
$$
Without loss of generality, assume $b_1\ne0$.
Since $u-1\ge\log(u)$ for $u\gt0$,
$$
\begin{align}
\sum_{k=2}^n a_kb_k
&=\sum_{k=2}^n\frac{s_k-s_{k-1}}{\sqrt{s_ks_{k-1}}}\\
&=\sum_{k=2}^n\sqrt{\frac{s_k}{s_{k-1}}}-\sqrt{\frac{s_{k-1}}{s_k}}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/647085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Real- Valued Random Variable This is from Ross Ihaka's notes about Time Series Analysis.
Any random variable which has probability 1 of being zero will have $\langle X,X \rangle = 0$, which violates the requirement that this only happen when $X=0$.
*
*How can a real-valued random variable has probability 1 of be... | A random variable is a (measurable) function from a sample space to the real numbers. There is nothing wrong with mapping (almost) everything to value 0. This is still a random variable.
When $X=0$ with probability $1$, $\langle X,X\rangle$ equals to 0.
Here is an example, let the sample space be the interval $[0,1]$ w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basic differentiation: second derivative I'm currently teaching myself some differential equations by watching the MIT OCW series on the topic. In This video, at 21:50mins, the lecturer calculates the following derivatives:
1st $y'=x^2-y^2$
2nd $y''=2x-2yy'$
My simple question is, how he came to the second one. Is this... | What's going on in the video, and in your posted problem, is what we refer to as implicit differentiation.
We view $y$ as a function of $x$, and thus, need to use the chain rule: $$y'(x) = x^2 - [y(x)]^2 \implies y''(x) = 2x - 2y(x)y'(x)$$ The author simply omits the parenthetical argument $(x)$: $$y'' = 2x - 2yy''$$
... | {
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Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ I can't crack this one.
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that $a,b,c,z \in \mathbb{Z}$ and $c\neq 0)$
| This question is closely related to this one. There is a small difference, but apparently both solutions provided there fit in here too, after a very small modification:
Assume $c\neq0$.
Let $p_1,p_2,\ldots,p_i$ be the common prime divisors of $c$ and $b$, with their respective powers $e_1,\ldots,e_i$ in the prime fact... | {
"language": "en",
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Graph Theory. Prove that $\sum_{v}^{} \frac{1}{1+d(v)} \ge \frac{n^2}{2e+n} $ Let e denote the number of edges and n the number of vertices. We can assume that the graph G is simple. Prove that
$\sum_{v}^{} \frac{1}{1+d(v)} \ge \frac{n^2}{2e+n} $
Any help/hints would be very much appreciated!
| Recast the right-hand side as follows:
$$\frac{n^2}{2e+n} = \sum_v\frac{n}{2e+n} = \sum_v \frac{1}{1+ \frac{2e}{n}}.$$
From the handshaking lemma, it follows that $\frac{2e}{n} = \overline{\delta}$, the average degree. Instead of working with the above quantity, we work with the harmonic mean, which is incidentally equ... | {
"language": "en",
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Proving that $(a_n) $ defined by induction to be: > $a_1=2.2$, $a_{n+1}=5-\frac6{a_n}$ is converging and finding the limit
Let there be the sequence $(a_n) $ defined by induction to be:
$a_1=2.2$, $a_{n+1}=5-\frac6{a_n}$ $\forall n\ge 1$
Prove that the sequence is converging and calculate it's limit.
So what needs ... | Ok. I don't know how you resolved this in class but here's one way.
Let $f$ be defined by $$f(x) = 5-\frac{6}{x}$$
So we have, $a_1 = \frac{11}{5}$ and $$\forall n \geqslant 1, \qquad a_{n+1} = f(a_n)$$
Study of $f$
$f$ is defined for $x\neq 0$. We are looking for stable intervals by $f$ (interval $I$ such that $f(I) \... | {
"language": "en",
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In $\ell^p$, if an operator commutes with left shift, it is continuous? Our professor put this one in our exam, taking it out along the way though because it seemed too tricky. Still we wasted nearly an hour on it and can't stop thinking about a solution.
What we have: The left shift $L : \ell^p \to \ell^p$
$$L(x_1,x_2... | The statement is false, as I discovered here. Since not everybody has access to the paper, let me provide a summary of the argument:
Let $R=\mathbb C[t]$, $L$ the left shift operator, and view $\ell^p$ as an $R$-module by defining $t\cdot x=Lx$. Let $X=\sum \ker L^i \subset \ell^p$ be the subspace of eventually-zero ... | {
"language": "en",
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"source": "stackexchange",
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Bipartite proof Let $G$ be a graph of order $5$ or more. Prove that at most one of $G$ and "$G$ complement" is bipartite.
I'm lost as to what needs to be done. I know that A nontrivial graph $G$ is bipartite if and only if $G$ contains no odd cycles.
| Let me give you a hint towards a much, much more elementary solution:
Hint: It is enough to show that if $G$ is bipartite, then $\bar{G}$ (the complement of $G$) is not bipartite.
To that end, suppose that $V(G)=A\cup B$ is a bipartition of $G$. Then in $E(G)$, there are no edges inside $A$, no edges inside $B$, and t... | {
"language": "en",
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Evaluate the contour integral $\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$ Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate
$$\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$$
I know that
\begin{equation}
\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz = \frac{1}{z^4}\left(z-\frac{z^3}{3!... | Cauchy's integral formula is
$$f^{(n)}(z) = \frac{n!}{2\pi i} \int_\gamma \frac{f(\zeta)}{(\zeta-z)^{n+1}}\,d\zeta,$$
where $\gamma$ is a closed path winding once around $z$, and enclosing no singularity of $f$.
Thus in your example, $n = 3$, and you need the third derivative,
$$\int_{\gamma(0,1)} \frac{\sin z}{z^4}\,d... | {
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Homogenous polynomial and partial derivatives I'm struggling to understand this part in a book I'm reading:
Let $F$ be a projective curve of degree $d$ with $P\in F$. Wlog,
suppose $P=(a:b:1)$. Let's look the affine chart $(a,b)\mapsto
(a,b,1)$.
Let $f$ be the deshomogenization of $F$, we can write $f$ in this way:... | Question: "I know this should be a silly question, but I'm a beginner in this subject and I really need help, if anyone could help me I would be grateful. Thanks"
Answer: When trying to prove a formula you should verify the formula in an explicit "elementary " example first, then try to generalize.
Example: Let $F:=x^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer. I am confused as to how to solve this question.
For the Base case $n=1$, $(2^{2(1)} - 1)\,/\, 3 = 1$, base case holds
My induction hypothesis is:
Assume $2^{2k} -1$ is divisible by $3$ when $k$ is a positive integer
So, $2^{2k} -... | Hint: If $2^{2k} - 1$ is divisible by $3$, then write
\begin{align*}
2^{2(k + 1)} - 1 &= 2^{2k + 2} - 1 \\
&= 4 \cdot 2^{2k} - 1 \\
&= 4 \cdot \Big(2^{2k} - 1\Big) + 3
\end{align*}
Do you see how to finish it up?
This technique is motivated by attempting to shoehorn in the term $2^{2k} - 1$, since that's the only piec... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are there any nontrivial ways to factor n-cycles into a product of cycles? I was reading a proof here about the simplicity of $A_n (n \ge 5)$. It states (and proves) a lemma about 3-cycles:
A 3-cycle $(a, b, c)$ may be written as $(a, b, c) = (1, 2, a)^{-1}(1, 2, c)(1, 2, b)^{-1}(1,2, a)$ (here, multiplication is right... | I believe you need another way of looking on conjugation:
If the group $S_n$ is realized as acting on $n$ points, then conjugating by some permutation is just renumerating points. (i.e. if you have a cycle $\sigma=(123)$ and a renumeration $\tau=(12)$ then the new cycle will be $\sigma=(213)$ and that is exactly the co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An inequality concerning triangle inequality. $a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3 \ge 0$, and given that $a_i+b_i \ge c_i$ for $i = 1,2,3$. I'd like the following inequality to hold, but can't find a proof, so I'd appreciate some help. $$\sqrt{a_1^2 + a_2^2 + a_3^2} + \sqrt{b_1^2 + b_2^2 + b_3^2} \ge \sqrt{c_1^2 + c_2... | Good news. It does hold.
Note that applying Minkowski's inequality to the LHS, we have:
$$\sqrt{a_1^2 + a_2^2 + a_3^2} +\sqrt{b_1^2 + b_2^2 + b_3^2} \ge \sqrt{(a_1 +b_1)^2+(a_2 +b_2)^2 +(a_3 +b_3)^2}$$
Now all we need to do is show that
$$\sqrt{(a_1 +b_1)^2+(a_2 +b_2)^2 +(a_3 +b_3)^2} \ge \sqrt{c_1^2+c_2^2+c_3^2}$$
Wh... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Way to find volume of the solid A solid has a square base of side $s$ . The upper edge is parallel to the base and has length $2s$. All other edges have length $s$ . What is the volume of the solid ?
NB : The volume of the tetrahedron with all sides length l is $ V = \dfrac{\sqrt2}{12}l^3$
| Converting comments to answer.
Especially given your "NB", it seems like you're describing a solid formed by slicing a regular tetrahedron by a plane parallel to (and half-way between) two opposite edges. In that case, the volume of the solid is half the volume of a regular tetrahedron of side $2s$.
Imagine a regular ... | {
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Show that $-3$ is a primitive root modulo $p=2q+1$ This was a question from an exam:
Let $q \ge 5$ be a prime number and assume that $p=2q+1$ is also prime. Prove that $-3$ is a primitive root in $\mathbb{Z}_p$.
I guess the solution goes something like this:
Let $k$ be the order multiplicative order of $-3$ modulo p.... | Hint: If $(-3)^q \equiv 1 (mod p)$ then $-3$ is a quadratic residue modulo $p$ (because if $\xi$ is a primitive root and $(\xi^k)^q \equiv 1$ then $k$ is even and $\xi^k$ is a quadratic resieue)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/648616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Should $f(x) \equiv 0$ if $0\le f'(x)\le f(x)$ and $f(0)=0$? Assume $f(x)$ is a real-function defined on $[0,+\infty)$ and satisfies the followings:
*
*$f'(x) \geq 0$
*$f(0)=0$
*$f'(x) \leq f(x)$
Should we always have $f(x) \equiv 0$ ? Thanks for any solution.
| As $f(0)=0$ and for all $x\geq0$, $f'(x)\geq0$, we have $f(x)\geq0, \forall x\geq0$.
Define
$$g(x)=\mathrm e^{-x}f(x)$$
and compute
$$g'(x)=\mathrm e^{-x}\left(f'(x)-f(x)\right)\leq 0$$
as $g(0)=0$ we have $g(x)\leq 0$ for all $x\geq0$.
Therefore we have
$$f(x)=\mathrm e^xg(x)\leq 0,\quad \forall x\geq0,$$
which prove... | {
"language": "en",
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Why is $\vec a\downarrow\vec c=\vec a\downarrow(\vec b\downarrow\vec c)$? I know how to draw a driagram to show that it's true, but I can't really explain it mathematically / algebraically. This is about projection vectors, if the notation is unclear.
EDIT:
This is the notation I've learned, but after having read about... | The projection can be written as $$\vec a \downarrow \vec c=\frac{\vec c\cdot \vec a}{\vec c \cdot \vec c}\vec c$$
What do you get if you let $\vec c=\vec b \downarrow \vec c$?
| {
"language": "en",
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Financials Maths- Credited Interest You invest £1,000 in an account for 5 years at 9% pa nominal. How much will you get at the end of the 5 years if the interest is is credited:
a) annually; b) 6 monthly; c) 3 monthly; d)monthly?
Approximate how much you would get if interest was credited daily. Which method... | Hint: You are correct for the annual calculation. If the interest is credited every half year, you get half the amount of interest ($4.5\%$) each time, so it would be $1000(1.045)^{10}$ You should find that shorter crediting periods are better.
| {
"language": "en",
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Extending a holomorphic function defined on a disc Suppose $f$ is a non-vanishing continous function on $\overline{D(0,1)} $ and holomorphic on ${D(0,1)} $ such that $$|f(z) | = 1$$ whenever $$|z | = 1$$
Then I have to prove that f is constant.
We can extend $f$ to all $\mathbb{C}$ by setting $$f(z) = \frac{1}{\overlin... | To prove that $f$ is identically constant, you'd better employ the maximum principle, according to which the maximum of $|f|$ on $\overline{D}$ equals 1. But $f$ is non-vanishing on $\overline{D}$, hence $\frac{1}{f}$ is holomorphic on $D$ while $\bigl|\frac{1}{f}\bigr|=1$ on $\partial{D}$. By the same maximum princip... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Bijection from a set of functions to a Cartesian product of sets Let S be an arbitrary set. Let $F=\{f:\{0,1\}\to S\}$ be the set of functions from $\{0,1\}$ to S. Construct a bijection $F→S \times S$.
I think I would define the function $a(f)=(f(0),f(1))$ because we know that both $f(0)$ and $f(1)$ are in $S$, but I ... | Consider the following mapping $\phi:F\to S\times S$: $$\phi(f)=(f(0),f(1))\quad\forall f\in F.$$ It's injective, since if $\phi(f)=\phi(g)$ for $f,g\in F$, then $(f(0),f(1))=((g(0),g(1))$. This means that $f(0)=g(0)$ and $f(1)=g(1)$, so that $f=g$ identically. It's also surjective, since if $(s,t)\in S\times S$, then ... | {
"language": "en",
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Irrationals: A Group? I understand that the set of irrational numbers with multiplication does not form a group (clearly, $\sqrt{2}\sqrt{2}=2$, so the set is not closed). But is there a proof or a counter-example that the irrationals with addition form (or do not form) a group?
Thank you!
Edit: In particular, I am wond... | To speak to the spirit of your question a bit: the rational numbers are a so-called normal subgroup of the reals (since the reals form an abelian group, and all subgroups of an abelian group are normal), so we can talk about the quotient group of the reals by the rationals, $\mathbb{R}\ /\ \mathbb{Q}$. Each element of... | {
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Set Theory and Equality Let $A$ and $X$ be sets. Show that $X\setminus(X\setminus A)\subseteq A$, and that equality holds if and only if $A\subseteq X$.
I understand why this holds but am not sure how to 'show' this. Any advice would be appreciated.
| Start with the definition/alternate expression for "setminus": $A\setminus B=A\cap B^c$, where $B^c$ is the complement of $B$:
$$X\setminus(X\setminus A)=X\cap (X\cap A^c)^c = X\cap (X^c \cup A) = (X\cap X^c) \cup (X \cap A)= X\cap A$$
Does that help?
| {
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Find greatest value of $y(x) = (0.9^x)(300x + 650)$ Question and attempt
$y(x) = (0.9^x)(300x + 650)$
Estimate at what x value that y reaches its maximum value
The only way I could think of would be to use derivatives, so I tried it:
$y'(x) = (0.9^x)' \times (300x + 650) + 0.9^x \times (300x + 650)'$
$=$ ... | You were asked to give an approximation, not an exact answer. If you take the derivative and compare it to zero, you get the rather difficult expression:
$$(300x+650)[\ln(0.9)×0.9^x]+0.9^x×300=0$$
We know $\ln (1+x) \sim x$ for small $x$. So, we get:
$$(300-30x-65)0.9^x=0$$
This will only be zero when $235=30x$, or whe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/649371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Help with recurrence relation It 's been a long time since I touched this kind of math , so please help me to solve the relation step by steps :
$V_k = (1+i)*V_{k-1}+P$
I know the answer is $V_k = (P/i)*((1+i)^k-1) $
Thanks in advance.
| If you want a methodical way to find the answer that you don't know yet, you could use generating functions. Defining $F(x) = \sum_{k=0}^\infty V_k x^k$, and assuming the initial condition is $V_0=0$, the recursion gives
\begin{align*}
F(x) &= V_0 x^0 + \sum_{k=1}^\infty \big( (1+i)V_{k-1} + P \big) x^k \\
&= 0 + (1+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Difference between a "topology" and a "space"? What do we mean when we talk about a topological space or a metric space? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space? What is it that the word means, and if there are multiple meanings... | I think of a "space" as the conceptually smallest place in which a given abstraction makes sense. For example, in a metric space, we have distilled the notion of distance. In a topological space, we are in the minimal setting for continuity.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Show $\forall x \exists y F(x,y)$ does not imply $\exists y \forall x F(x,y)$ Show:
$\exists y \forall x R(x,y) \rightarrow \forall x \exists y R(x,y)$
$\forall x \exists y F(x,y)$ does not imply $\exists y \forall x F(x,y)$
How do proofs of this nature usually work? When I try to prove the first one by saying:
$\mathc... | Consider the sentences $\forall x\exists y\ x=y$ and $\exists y\forall x\ x=y$ in a domain with two elements.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that Spec(α) and Spec(ß) partition positive integers iff α and ß are irrational and 1/α + 1/ß = 1 From Concrete Math, problem 3.13 asks:
"Let α and ß be positive real numbers. Prove that Spec(α) and Spec(ß) partition positive integers if and only if α and ß are irrational and 1/α + 1/ß = 1"
The solution claims:... | The hint is in the phrase "for large $n$".
As you proved, we must have that for all $n$,
$$n(1/\beta + 1/\alpha) + 1/\alpha + 1/\beta - \{(n+1)/\beta\} - \{(n+1)/\alpha\} = n.$$
Let us denote $1/\beta + 1/\alpha$ by $c$, to rewrite this as
$$nc + c - \{(n+1)/\beta\} - \{(n+1)/\alpha\} = n.$$
As $0 \le \{x\} < 1$ for a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding limit of a sequence in product form \begin{equation} \prod_{n=2}^{\infty} \left (1-\frac{2}{n(n+1)} \right )^2 \end{equation}
I need to find limit for the following product..answer is $\frac{1}{9}$.
I have tried cancelling out but can't figure out.
Its a monotonically decreasing sequence so will converge to it... | The product is equal to $\left(\frac{2-1}{2+1}\frac{n+2}{n}\right)^2$
and the limit is 1/9.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/649789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
open map equivalent definition $f : (X,\tau_X) \to (Y,\tau_Y) $ continuous and surjective. I need to prove that
$f$ is open $\Longleftrightarrow \forall U\in\tau_X, f^{-1}(f(U))\in \tau_X$
Proof:
$\implies)$ By definition, $f$ is open if $\forall U\in\tau_X, f(U)\in\tau_Y$. As f is continuous, $f^{-1}(f(U))\in \tau_X$... | This isn't true. Let $X$ denote the real line with the discrete topology, let $Y$ denote the real line with the usual topology, and let $f : X \to Y$ be the identity map. Clearly $f$ is a continuous surjection, and is not an open mapping. However $f^{-1} [ f [ U ] ] = U$ for all (open) $U \subseteq X$.
| {
"language": "en",
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Can a function have overlapping range? Consider the function $f : \mathbb{R}\rightarrow\mathbb{R}$ defined by
$$f(x)=\left\{\begin{align}x^2 - 2 & \text{if}\,x > 0,\\
x - 1 & \text{if}\, x \le 0.\end{align}\right.$$
Find a right inverse of $f$.
The answer that I came up to this question was this:
$$f(g(x)) = x$$
$$g(x)... | You are right in that your $g$ is not a function.
In the overlapping range, you need to decide which one of the two branches to choose.
Solution:
For example, you can choose
$$g(x)=\left\{\begin{align}\sqrt{x+2} & \text{if}\,x>-2\\
x+1 & \text{if}\,x\le -2\end{align}\right.$$
Now suppose that $x>-2$. Then $\sqrt{x+2}>0... | {
"language": "en",
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Picking a correct answer in a multiple choice test answering randomly I was reading a book about techniques to pass a multiple choice test and I found a passage that seems strange to me.
Every question has 5 possible answers, you get 1 point for each correct answer, 0 for each not given answer and -.25 for each wrong a... | No, it's technically incorrect as you can't say "one WILL be correct." It should read something like "you are expected to get one fifth of the questions right"
However, if I may deduce where the author is going with this, he will probably say something of the sort "So, you will get one question right (+1) and four inco... | {
"language": "en",
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For a non-constant entire function which property is possible? Let $f$ be a non-constant entire function.Which of the following properties is possible for each $z \in \mathbb{C}$
$(1) \ \ \mathrm{Re} f(z) =\mathrm{Im} f(z)$
$(2) \ \ |f(z)|<1$
$(3)\ \ \mathrm{Im} f(z)<0$
$(4)\ \ f(z) \neq 0 $
I tried for $(2)$ and $(3... | Hints: For $1,$ note that $f$ can't be $0$ everywhere, nor can its derivative. Hence, there is some non-empty open set that $f$ maps to an open set. (Why?) Can the line $\operatorname{Re}(w)=\operatorname{Im}(w)$ contain any non-empty open set?
For $4,$ try to think of an example of a non-constant entire function that ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Small question on relative holology if $Y\subset X$ , what is $\ker \delta$ such that $\delta: H_k(X,Y)\rightarrow H_k(Y)$ ?
is it $\ker \delta = H_k(X,Y)$ ?
$\delta$ is the usual connecting homomorphism from the long exact sequence of relative pairs
Please
Thank you.
| There's not really much more I can say than is on Wikipedia or can be found in any intro to algebraic topology text. The map $\delta$ which appears in the long exact sequence of a relative pair $(X,Y)$ as $$\cdots \to H_n(Y) \to H_n(X) \stackrel{f_*}{\to} H_n (X,Y) \stackrel{\delta}{\to} H_{n-1}(Y) \to \cdots$$ is the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which methods different than the natural one can one devise to confirm that the limit is $\;2/\pi\;$? Good evening,
I have found this exercise (https://math.stackexchange.com/questions/633509/which-methods-different-than-the-natural-lim-n-to-infty-frac-cos1-cos)
What is the limit of:
$$\lim_{n\to\infty}\dfrac{|\cos{1}|... | Hint: If one replaces $n$ with $n\bmod {2\pi}$, the quotient tuens into something like a "distorted" Riemann sum. How distorted can it be? What does it mean of $n_1\bmod {2\pi}$ and $n_2\bmod{2\pi}$ are "unusually" close to each other?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Group with an even number of elements. If $G$ is a group such that $|G|=2n$. Prove that there's an odd number of elements of order 2, and then there's an element which is its own inverse, besides of the identity.
If we consider all the elements of $G$ that have order different than 2, we have two cases: 1) the elements... | Your first try is correct. By your proof, you know the number of elements of order 2 is odd, hence can not be zero. This means there exists at least one, say $g$, which is just the element you wanted in the second part.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convex function in its interior Let $f$ be a convex function on an open subset of $R^{n}$. How to prove $f$ is continuous in the interior of its domain.
For $n=1$, let $f$ be convex on the set $(a,b)$ with $a<s<t<u<b$
Then using the inequality $\frac{f(t)-f(s)}{t-s} \leq \frac{f(u)-f(s)}{u-s} \leq \frac{f(u)-f(t)}{... | You can see the solution in the following lecture note
enter link description here
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How find this ODE solution $\frac{f(x)}{f(a)}=\left(\frac{x}{a}\right)^2e^{(x-a)\left(\frac{f'(x)}{f(x)}-\frac{2}{x}\right)}$ let $a>0$ is constant number,and the function $f(x)$ such follow ODE
$$\dfrac{f(x)}{f(a)}=\left(\dfrac{x}{a}\right)^2e^{(x-a)\left(\dfrac{f'(x)}{f(x)}-\dfrac{2}{x}\right)}$$
Find the $f(x)?$
T... | You could turn it into a first order linear differential equation by substituting $\ln(f)=\phi$, since $\phi'=\ln(f)'=\frac{f'}{f}$ appears on the right side.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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what is the minimum number of points in affine plane. what is minimum number of points in affine plane,
By the way: Here are the $\textbf{Three Axioms}$ for affine plane.
*
*Given two distinct points $\textbf{P}$ and $\textbf{Q}$, there is only one line passing through them
*Given a point $\textbf{P}$ and a line $... | HINT: If $K$ is any field, the vector space $V=K^2$ has always the structure of affine plane.
Now take for $K$ the smallest field you know.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Check whether the function is continuous at 0 - what went wrong? I have to check the whether the following function is continuous:
$$ \
f:\mathbb{R}\rightarrow \mathbb{R},~f(x)=\left\{
\begin{array}{lll}
e^{1/x} &\text{if} & x < 0,
\\
0 & \text{if}& x \ge 0.
\end{array}\righ... | If $x_n \to 0$ and $x_n<0$ then $\frac{1}{x_n} \to -\infty$ and $e^{1/x_n} \to 0$.
So the left limit is 0. The right limit at 0 is evidently 0. Therefore the function iscontinuous at 0.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Prove that this sequence converge. I am obliged to prove that this sequence:
$\large {a_n=(1+\frac{1}{3})(1+\frac{1}{9})(1+\frac{1}{27})...(1+\frac{1}{3^n})}$
is convergent sequence.
I mean I was thinking about this and I know that $\large\lim_{n \to \infty} (1+\frac{1}{3^n})=1 $
From this I know that it will be probab... | Take $\log$:
$$\log\lim_{n\to\infty} a_n = \lim_{n\to\infty}\log a_n = \lim_{n\to\infty}\sum_{k=1}^n\log(1+{1\over 2^k}) = \sum_{k=1}^\infty\log(1+{1\over 2^k});$$
now, as
$$\lim_{k\to\infty}{\log(1+{1\over 2^k})\over{1\over 2^k}}=1,$$
the behavior of $\sum\log(1+{1\over 2^k})$ is the same that the behavior of $\sum {1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this expression a quadratic form I have an matrix expression that basically is of the form:
\begin{equation}
tr(B X BX )
\end{equation}
Where $B$ and $X$ and nonsquare matrices. $B$ is $p \times n$, $X$ is $n \times p$.
It seems to me this trace expression is a quadratic form because I got it as part of a longer ... | You are almost there. Notice that the elements of $\mathrm{vec}(X^T)$ are just a reordering of $\mathrm{vec}(X)$. The two are related by a permutation matrix that is sometimes known as a stride permutation. If $X\in \mathbb{R}^{m\times n}$ then the stride permutation matrix $L_m^{mn}$ satisfies the equation $L_m^{mn... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Continuity of sum of functions Suppose we have that $f_n:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and $f_n\geq0$ for all $n\in\mathbb{N}$. Assume that $f_n(x)\leq1$ for all $x\in[-n,n]$. My question is: Why is the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $f=\sum_{n\in\mathbb{N}}{\frac{f_n}{2^n}}$ c... | Appy M-test the the tail of the series $\sum_{k=n}^\infty{f_k\over 2^k}$ in $[-n,n]$. $f =$ finite sum of continuous functions in $\Bbb R$ + continuous tail in $[-n,n]$ is continuous in $[-n,n]$. So, $f$ continuous in $\Bbb R$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how to find $ \lim\limits_{x \to \infty} \left(\sqrt{x^2 +1} +\sqrt{4x^2 + 1} - \sqrt{9x^2 + 1}\right)$ How can I find this?
$ \lim\limits_{x \to \infty} \left(\sqrt{x^2 +1} +\sqrt{4x^2 + 1} - \sqrt{9x^2 + 1}\right)$
| Since for any $A>0$
$$\sqrt{A^2 x^2+1}-A|x| = \frac{1}{A|x|+\sqrt{A^2 x^2+1}}<\frac{1}{2A|x|}$$
holds, we have:
$$\left|\sqrt{x^2+1}+\sqrt{4x^2+1}-\sqrt{9x^2+1}\right|=\left|\sqrt{x^2+1}-|x|+\sqrt{4x^2+1}-2|x|-\sqrt{9x^2+1}+3|x|\right|\leq \left|\sqrt{x^2+1}-|x|\right|+\left|\sqrt{4x^2+1}-2|x|\right|+\left|\sqrt{9x^2+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/651148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Continuous curve, traps itself outside the unit circle. Lets say i have an injective continuous curve $\sigma$ in $\mathbb{C}$, indexed on $[0,\infty)$ and converging to $\infty$. If $\vert \sigma(0)\vert>0$ , is it possible that it can trap itself outside the unit circle? By that i mean, that there doesn't exist an ex... | Let's compactify $\mathbb C$ to $S^2=\mathbb C\cup\{\infty\}$, and add the point at $\infty$ to the image of $\sigma$. Your question amounts to whether the complement of a simple arc $\gamma$ in $S^2$ is path-connected. (A simple arc is a homeomorphic image of $[0,1]$). This is equivalent to asking whether the compleme... | {
"language": "en",
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Positive elements of a $C^*$ (MURPHY, ex 2-2). I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise:
Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra.
(a) If $a,b$ are positive elements of $A$, show that $\sigma(ab)\subseteq\mathbb{R}^+=\left\{x\in\mat... | For (a), you have to use that $\sigma(xy)\cup\{0\}=\sigma(yx)\cup\{0\}$ for any two operators $x,y$. Then
$$
\sigma(ab)=\sigma((ab^{1/2})b^{1/2})\subset\sigma(b^{1/2}ab^{1/2})\cup\{0\}\subset\mathbb R^+.
$$
The point is that $b^{1/2}ab^{1/2}$ is positive.
For (c),
\begin{align}
\max\sigma(a^*a)&=\|a^*a\|=\|a\|^2=1=\... | {
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Nonpiecewise Function Defined at a Point but Not Continuous There I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, polynomials are continuous on $(-\infty, \infty)$, an... | There's always $[x]$, and $\chi_S(x)$ for any proper subset $S$ of $\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/651381",
"timestamp": "2023-03-29T00:00:00",
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Rank of a subgroup of $\mathbb{Z}^3$ given a generating set $(2,-2,0)$, $(0,4,-4)$, and $(5,0,-5)$? Is there some standard approach to finding the rank of a subgroup given a generating set?
In particular, I'm considering the subgroup of $\mathbb{Z}^3$ generated by $v_1=(2,-2,0)$, $v_2=(0,4,-4)$, and $v_3=(5,0,-5)$. Thi... | The submodule generated by $v_i$ is contained in the rank $2$ submodule of $\mathbb Z^3$ consisting of those tuples whose entries add up to zero.
| {
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"url": "https://math.stackexchange.com/questions/651467",
"timestamp": "2023-03-29T00:00:00",
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What is an isomorphism? I'm familiar with the concepts of group isomorphism, ring isomorphism, and graph isomorphism, but it's never been presented to me what an isomorphism is in general: given any X, what is an X isomorphism?
Informally, I understand isomorphism as "preservation of structure", where "preservation" is... | As the comment suggests, looking up the word "morphism" in the context of "Categories" and Objects will give you a more rigorous and general idea about isomorphisms. But I always like to think of isomorphism as something that allows you to make copies of a given "object", be it rings or groups, or Topological spaces or... | {
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Problem of continuous real valued function Which ofthe following statements are true?
a.If $f:\mathbb R\to\mathbb R$ is injective and continuous, then it is strictly monotonic.
b.If $f\in C[0,2]$ is such that $f(0)=f(2)$,then there exists $x_1,x_2$ in [0,2] such that $x_1-x_2=1$ and $f(x_1)=f(x_2).$
c.Let $f$ and $g$ ... | for b> $g(x)=f(x+1)-f(x)$ then $\exists c \in [0,1]$ s.t. $g(c)=0 \implies f(c)=f(c+1)$ .
for c> if $f(x) \neq g(x) $ then $r(x)=f(x)-g(x) \neq0 $ for all $x$.
$0 \neq r(f(x))+r(g(x))=f(f(x))-g(f(x))+f(g(x))-g(g(x))=f(f(x))-g(g(x))$ for all $x$ which is a contradiction so $f(x_1)=g(x_1)$ for some $x_1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does $a$ mean in Taylor series formula? I'm trying to code the Taylor summation in MATLAB, being Taylor's formula the following:
I've also seen $a$ denoted as $x_0$ in distinct bibliography.
Problem is that I'm not sure how should I evaluate or assign for $a$.
At lecture I studied the exponential Taylor's repre... | $a$ is the point for which you calculate the derivatives that you plug in the expansion, along with the displacement from $a$ raised to the correct power, as you seem to understand yourself
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/651776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the tight closure tightly closed? Let $R$ be a commutative noetherian ring containing a field of characteristic $p\gt0.$
For an ideal $I\subset R,$ the tight closure $I^*$ is defined as $$\{f\in R\mid \exists t\in R, t\not\in\mathfrak p, \forall \mathfrak p \text{ minimal prime of }R,\ tf^q\in I^{[q]} \text{ fo... | Take $x_1,\dots,x_n$ a system of generators for $I^*$. By definition there exists $c_i\in R^0$ such that $c_ix_i^q\in I^{[q]}$ for $q\gg0$. Set $c=c_1\cdots c_n$. Then $c(I^*)^{[q]}\subset I^{[q]}$ for $q\gg0$.
Now let $x\in (I^*)^*$. There exists $c'\in R^0$ such that $c'x^q\in (I^*)^{[q]}$ for $q\gg0$. Then $cc'x^q\... | {
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"answer_id": 0
} |
finding a solution for $m$ given $(1+i)z^2-2mz+m-2=0$ Given the equation: $(1+i)z^2-2mz+m-2=0$, while $z$ is complex and $m$ is a parameter.
For which values of $m$ the equation has one solution?
So my idea was to use: $b^2-4ac=0$ for $ax^2+bx+c=0$
But it leads to difficult computation which i could not solve.
Is there... | Your idea makes sense. The equation has one solution if and only if the discriminant $b^2-4ac$ is zero.
$$b^2-4ac=(-2m)^2-4(m-2)(1+i)=4m^2-(4+4i)m+(8+8i)$$
This is a quadratic equation in $m$ which has the two solutions $2i$ and $1-i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/651964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Graph Theory Vertex Problem Let $G$ be a graph of order $8$ with $V(G)=\{v_1, v_2,...,v_8\}$ such that deg $v_i=i$ for $1 \leq i \leq 7$. What is deg $v_8$.
Any help or hints would be greatly appreciated.
| You must join $v_7$ with all other vertices.
After that, $v_1$ is already "fed up". What can you now say about $v_6$? What happens then to $v_2$? And so on.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/652065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Solve the equation $|x-7|-3|2x+1|=0$ This equation is very unfamiliar with me, I never seen things like that because I always solved equations of the form $|\text{something}|=\text{things}$ but never seen equations that look like $|\text{something}|=|\text{things}|$. So if I learn how to solve it I will be able to solv... | |x-7|=|6x+3|
then either
x-7=6x+3
or
x-7=-6x-3
solve these two equations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/652167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 9,
"answer_id": 5
} |
Derivative of $(\ln x)^{\ln x}$ How can I differentiate the following function? $$f(x)=(\ln x)^{\ln x}.$$ Is it a composition of functions? And if so, which functions?
Thank you.
| Note that $$a^b = {\left(e^{\ln a}\right)}^b = e^{b\ln a}$$
so $$\left(\ln x\right)^{\ln x} = {\left(e^{\ln\ln x}\right)}^{\ln x}\\
= e^{\ln x\cdot\ln\ln x} = x^{\ln \ln x}$$ either of which which you should be able to do with methods you already know.
We can apply this technique generally to calculate the derivative o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find the value of the expression The expression $ax^2 + bx + 1$ takes the values $1$ and $4$ when $x$ takes the values $2$ and $3$ respectively. How can we find the value of the expression when $x$ takes the value of $4$?
| Hint $\,\ f(3)=2^2,\ f(2) = 1^2,\ f(0) = (-1)^2\ $ so $\,f-(x-1)^2\!=0,\,$ having $3$ roots $\,x=3,2,0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/652269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Gaussian random variable in $\mathbb{R}^n$ question Let $X=(X_1,...,X_n)$ is a Gaussian random variable in $\mathbb{R}^n$ with mean $\mu$ and covariance matrix $V$.
I want to show that we can write $X_2$ in the form $X_2 = aX_1 + Z$, where $Z$ is independent of $X_1$, and I want to find the distribution of $Z$.
Any he... | Assuming that $X_2 = aX_1 + Z$ holds (with $Z$ and $X_1$ independent), you can find $a$ in terms of particular entries of the covariance matrix. Once you have $a$, you know that $Z=X_2 - aX_1$, so you can find the distribution of $Z$, and check that it is independent of $X_1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/652356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$
I am trying to show that $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$ where $n,m$ are positive integers.
I showed that if $\gcd(n,m)$ is not $1$, then $X^n-Y^m$ is reducible. How to show the other direction. Please help.
| Assume $f(X,Y) =X^n-Y^m=g(X,Y)h(X,Y)$. Then $f(Z^m,Z^n)=0$ implies that one of $g(Z^m,Z^n)$ or $h(Z^m,Z^n)$ is the zero polynomial. Suppose that $g(Z^m,Z^n)=0$. That means that for all $k$, the monomials cancel, i.e. if $$g(X,Y)=\sum a_{i,j}X^iY^j $$
then $$\sum_{mi+nj=k}a_{i,j}=0.$$
Can we ever have $mi+nj=mi'+nj'$? T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 3
} |
How to evaluate the integral $\int_1^n\frac{1}{(\ln x)^{\ln x}}dx$ I'm stuck with this integral to test the convergence of a series. $$\int_1^n\frac{1}{(\ln x)^{\ln x}}dx.$$ Could you give me a couple of hints to compute this integral please? Is it a simple integral or I need to know something special to solve it?
| \begin{array}{l}
\int {\left( {{{\left( {\ln x} \right)}^{ - \ln x}}} \right)} \;dx\\
= \int {\left( {{{\left( {\ln x} \right)}^{ - \ln x}}\cdot1} \right)} \;dx\\
= {\left( {\ln x} \right)^{ - \ln x}}\int 1 \;dx - \int {\left( { - \frac{{{{\left( {\ln x} \right)}^{ - \ln x}}}}{x}\;\cdot\int 1 \;dx} \right)} \;dx\\
=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Elliptic curves in projective form question Let
$K$
be any field with Char
$K
\neq 2, 3$,
and let
$\varepsilon
:
F
(
X_0
;X_1
;X_2
) =
X_1^2
X_2-
(
X_0^3
+AX_0
X_2^2
+
BX_2^3
)$
;
with
$A, B
\in
K$,
be an elliptic curve. Let
$P$
be a point on
$\varepsilon$.
(a).
Show that $3P = \underline{o}$, where $\underline{o}$ is... | (a) Note that $3P=O$ iff $2P=-P$. To compute $2P$ you have to intersect the tangent line $t$ in $P$ with $\varepsilon$. The line $t$ will meet $\varepsilon$ in two points, say $P$ and $Q$, because it already meet $\varepsilon$ at $P$ with multiplicity $2$. In any case, we know that $2P=-Q$. Therefore, if $2P=-P$ then i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
conditional expectation? I'm trying to solve an expected value problem where a biased coin is flipped until a run of five heads is achieved. I need to compute the $E(X)$ where $X$ is the number of tails expected before the run of five heads.
Would this require conditional expectation, since $E(X)$ is dependent on $P(Y)... | The following is a conditional expectation argument. We first deal with an unbiased coin, and then a biased coin. Let $e$ be the required expectation.
Unbiased Coin: If the first toss is a tail (probability $\frac{1}{2}$) then the expected number of tails is $1+e$.
If first toss is a head and the second is a tail (pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Fast calculation for $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$ I want to show that $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$, but is there a faster method than finding the contour and doing all computations?
Otherwise my idea is to do the substitution $x=e^t$, integral than changes to $\int _{-\infty}^{\infty}\frac... | Here is a general approach. Consider the integal
$$ F(s) = \int_{0}^{\infty}\frac{x^{s-1}}{1+x^2}=\frac{\pi}{2\sin(s\pi/2)},\quad 0<Re(s)<2. $$
which is the Mellin transform of the function $\frac{1}{1+x^2}$. Now, our integral can be evaluated as
$$ I = \lim_{s\to 1} F'(s) = 0.$$
Note:
1) To evaluate the above integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Prove $\lim_{x \to \infty} f(x) = L \iff \lim_{x \to 0} g(x) = L$ Let $f(x) = g(1/x)$ for $x>0$. Prove: $\lim_{x \to \infty} f(x) = L \iff \lim_{x \to 0} g(x) = L$ for some $L \in \mathbb{R}$.
I assume I am supposed to use l'Hopital's rule in some way (considering that is what section we are in). I've tried looking at... | Here is the right direction:
Given $\lim_{x \rightarrow \infty} f(x) = L$ we know $\exists N : x> N \Rightarrow |f(x) -L| < \epsilon$.
Set $x' = \frac 1 x$ and $\delta = \frac 1 N$. Then $x>N \Leftarrow\Rightarrow x' < \frac 1 N = \delta$.
Thus $x' < \delta \Rightarrow x> N \Rightarrow |f(x) -L| = |f(\frac 1 {x'}) -L| ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Questions concerning the differential operator Consider the differential equation:-
$a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$.
I'm defining $w = Lw'$ and $x=Lx'$, where $L$ is a constant.
I'm trying to obtain $\... | Here is a start. First make the change of the dependent variable $w=Lz$ (I used z instead of w' to avoid confusion with derivative)
$$ w=Lz \implies D^n w= L D^n z,\quad D=\frac{d}{dx}, $$
so, the differential equation becomes
$$ a \phi(x) + L(bD^3 - cD)z =0 \longrightarrow (1).$$
Now, we use the other change of va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/652907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? A little unclear about this...
| A concrete example:
$$u_1=(1,0,0),u_2=(2,0,0),u_3=(0,1,0),u_4=(0,0,1)$$
$\{u_1,u_2,u_3,u_4\}$ clearlly spans $\mathbb{R}^3$.
On the other hand $u_4 \notin span\{u_1,u_2,u_3\}$, and therefore $\{u_1,u_2,u_3\}$ does not span $\mathbb{R}^3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/652999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
} |
Is the ring $\mathbb{C}[t^2,t^3]$ integrally closed? I am trying to understand if the ring $\mathbb{C}[t^2,t^3]$ in integrally closed (into its field of fractions), but I have no idea about how to proceed. All I have tried until now has failed. Any hints/ideas?
| The field of fractions is $\mathbb{C}(t)$. The element $t$ is integral because it is a root of $x^2-t^2$. But $t \notin \mathbb{C}[t^2,t^3]$. In fact, the integral closure is $\mathbb{C}[t]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/653087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is $\nabla u \in L^{\infty}$ if $u$ is bounded $C^{0}$? I would like to prove something of the form $|A_{1}(u)| \leq c \lVert u \rVert_{L^{\infty}}$ and $|A_{2}(u)| \leq c \lVert \nabla u \rVert_{L^{\infty}}$ for some operators $A_{1},\ A_{2}$ and arbitrary constant $c$. I am working on domain $\Omega \subset \mathbb{R... | Let $\Omega = \overline{B_1(0)}$ and $u(x) := \sqrt{1-\Vert x\Vert^2}$.
Then $u\in C^0(\Omega) \cap L^\infty(\Omega) \cap C^\infty(\Omega^\circ)$, but
$\nabla u \notin L^\infty(\Omega)$, thus this $u$ serves as a counter-example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/653152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$H_1\times H_2 Let $H_1, H_2, G_1, G_2$ be groups. Clearly if $H_1<G_1$ and $H_2<G_2$, then $H_1\times H_2<G_1\times G_2$.
I'm wondering if the converse statement is true. I'm quite sure it's not. Can you find a counterexample?
| It depends even on the very construction of products and the notion of natural inclusion.
For example is $A\times (B\times C)=(A\times B)\times C$ or merely $A\times (B\times C)\cong(A\times B)\times C$? Is $A\times B<A\times (B\times C)$? If yes, this easily leads to a counterexample.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/653220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Absolute continuity preserves measurability In studying absolutely continuous function, I knew that if $f:[a,b]\to\mathbb{R}$ is absolutely continuous, then $f(N)$ has measure zero if $N$ is, and $f(E)$ is measurable if $E$ is.
Suppose continuous function $f:[a,b]\to\mathbb{R}$ is such that if $E$ is measurable, then $... | Suppose $N\subset[a,b]$ is of measure $0$, but $f(N)$ is not. Then, as $f(N)$ is measurable, it contains a non-measurable set $B$. One can then find a subset $A$ of $N$ with $f(A)=B$. As $A$ is measurable with measure $0$, we have a contradiction.
In fact, a continuous function on $[a,b]$ maps every measurable set onto... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/653333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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