Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Explain Why This Average of Averages Is Incorrect Five properties are sold at varying prices and number of square feet.
SQ FT PRICE Price/Sq Ft
1635 $630000 $385.32
2045 $675000 $330.07
1900 $685000 $360.53
2045 $700000 $342.30
2305 $715000 $310.20
==== ======== =... | The word "average" here is ambiguous. They are using it to mean the average across properties, while you are using it to mean average across square feet. To see that these are different, use a more exaggerated example. Suppose we have two properties:
SQ FT PRICE Price/Sq Ft
10000 $1000000 $100
100 ... | {
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"timestamp": "2023-03-29T00:00:00",
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$e^\pi - \pi^e < 1$? We have Comparing $\pi^e$ and $e^\pi$ without calculating them but it doesn't give an approximation of the actual difference. Is there a way without calcualting an approximation of them to prove $e^\pi - \pi^e < 1$ ?
| If that can help:
Let $f(x):=e^x-x^e$. This function has a minimum at $x=e$ (double root), and the second order Taylor development is
$$y\approx g(x):=e^{e-1}(x-e)^2.$$
This approximation exceeds $f$, but we still have $g(\pi)<1$.
In blue, $f$, in black, $g$.
| {
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Finding $l^p$ norm of $2\times2$ matrix for $p\notin\{1,2,\infty\}$ Suppose I have the matrix $A=\begin{pmatrix}1 & 5 \\ 5 & 2\end{pmatrix}$ and I plot the image of the unit ball in $l^4$ under $A$.
How can I use this to determine the induced $4$-norm of $A$? I understand that $\|A\|_4=\underset{\|\vec{x}\|_4=1}{max}\... | Consider a point $P$ on the red curve and $P'$ the intersection of the ray $OP$ with the blue curve. Then $\|OP'\|_4=1$ so $\|OP\|=\frac{OP}{OP'}$. Therefore, you need to find $P$ so that $\frac{OP}{OP'}$ is maximum. Alrenatively, you need to find the smallest dilation of the blue curve that contains inside the red cu... | {
"language": "en",
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Function field of integral scheme Let $X$ be an integral scheme. For any affine open $U = \operatorname{Spec}(A)$ the field of fractions $K(A)$ of $A$ is the stalk of $X$ at its generic point. This is called the function field of $X$ and denoted by $K(X)$. Now let $U$ be some open set of $X$, not necessarily affine. Th... | You're correct - $\Bbb P^1_k$ is a counterexample. The open subset $\Bbb P^1_k$ has functions (and thus fraction field of those functions) $k$, while any other nonempty open subset has fraction field of it's regular functions $k(x)$.
As for the wikipedia page, one potential way to fix the inaccuracy of the claim is to... | {
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Why is left ideal called right? For operations between elements of an algebraic structure:
*
*If $a \cdot b = c$, then $a$ is a left divisor of $c$;
*If $a \cdot b = 0$, then $a$ is a left zero divisor;
*If $a \cdot b = e$, then $a$ is a left inverse of $b$;
*...
For operations between an element and an algebra... | One defining property of a right ideal $A\subseteq S$ is that $As\subseteq A$ for any $s\in S$. In words, $A$ is closed under right multiplication.
So it makes sense to call it a right ideal. That is not to say it is senseless to call it a left ideal. Some times when establishing a convention one has to choose between ... | {
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Find a function such that $ \int_{-\pi}^{\pi} f(x)\sin(nx)dx = \frac{(-1)^n}{\sqrt n} $ and $ \int_{-\pi}^{\pi} f(x)\cos(nx)dx = 0 $ As the title states, I must say if the function exists or not.
I'm not sure where to begin...
Is there a general method or approach to finding this type of functions?
All I can think is t... | It tells you what the Fourier series of $f$ should be. So by the definition of the polylogarithm $\operatorname{Li}_s(z)$ and its integral representation,
$$
f(x) = \sum\limits_{n = 1}^\infty {\frac{{( - 1)^n }}{{\sqrt n }}\sin (nx)} = \Im \sum\limits_{n = 1}^\infty {\frac{{( - 1)^n \mathrm{e}^{\mathrm{i}xn} }}{{\sq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\lim\limits_{n \to \infty}\sin(\pi\sqrt{n^2+1})$ I have a question regarding my method of finding a limit for my analysis class.
$\lim\limits_{n \to \infty}\sin(\pi\sqrt{n^2+1})$.
The method I used was I noticed that $\lim\limits_{n \to \infty}n = \lim\limits_{n \to \infty}(n^2+1)$ I think, so then that means $\lim\l... | $|\sin (π(n^2+1)^{1/2}-nπ)+nπ)|=$
$|\sin(π(n^2+1)^{1/2})\cos nπ+$
$ \cos (π(n^2+1)^{1/2}-nπ)\sin nπ)|=$
$|\sin (π(n^2+1)^{1/2}-nπ)\ cos (nπ)|=$
$|\sin (π(n^2+1)^{1/2}-nπ)||\cos (nπ)|$
$=|\sin (π(n^2+1)^{1/2}-nπ)|\cdot 1.$
$f(n)=π(n^2+1)^{1/2}-nπ=$
$π\dfrac{1}{(n^2+1)^{1/2}+n};$
$\lim_{n \rightarrow \infty}f(n)=0$;
Fina... | {
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Graph Laplacian appellation I was trying to figure out why the laplacian matrix of a graph
$$L=D-A$$
is named so. For this, I draw a 2D grid and tried to find the laplacian at some point $f_{x,y}$
I can write down the following:
$$
\Delta f(x, y) \approx \frac{f(x-h, y)+f(x+h, y)+f(x, y-h)+f(x, y+h)-4 f(x, y)}{h^{2}}
... | In terms of "sign": there's a long debate between geometers and analysts about whether the Laplacian should be the trace of the Hessian or its negative. From the functional analytic point of view (which will be useful also for the graph Laplacian), defining the Laplacian as the negative trace of the Hessian has the adv... | {
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Nonlinear System Stability using Lyapunov I am required to find the stability of this nonlinear system and for which values of $k$ is the system stable.
$\dot x=x.(x^2-1-k)$
I am trying to use quadratic Lyapunov function, and used the function $g(x)=x^2/\sqrt {k}$ to constraint $k$.
I am confused if I am doing it the r... | If you want to investigate the local stability of an equilibrium point of a nonlinear system it is usually easier to linearize. If all eigenvalues of linearized system have a negative real part then that equilibrium point is (asymptotically and exponentially) stable and if any of the eigenvalues has a positive real par... | {
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If $X$ has CDF $F$, find the CDF of $Y=X^2$
Let $X$ be a random variable with cumulative distribution function $F$. Let $Y=X^2$. Find the cumulative distribution function $F_Y$ of $Y$ in terms of $F$.
First we observe that $0\leq X^2$, and hence $0\leq Y$. So if $t<0$ then
$$F_Y(t)=P(X^2\leq t)=0.$$
Now if $0\leq t$... | What you have done is not correct. You are assuming that $X \geq 0$ which is not given.
Let $t \geq 0$. Note that for $X^{2} \leq t$ iff $-\sqrt t \leq X \leq \sqrt t$. So $P(X^{2} \leq t)=P(X \leq \sqrt t)-P(X<-\sqrt t)$. This can be written as $F(\sqrt t)-F((-\sqrt t)-)$ where $F(x-)=sup_{y<x} F(y)$, the left hand l... | {
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Proof by induction: $2^{2n}-1$ is a multiple of $3$ I'm learning proofs by induction and I'm a little confused on how they work exactly. This is what I have.
Theorem: $\forall n\in\mathbb N_0$, $2^{2n}-1$ is a multiple of 3.
old proof with mistakes:
Base: $n=1$
$2^{2(1)}-1 = 4-1 = 3$
$3 = 3m, m\in\mathbb N$
$3$ is a ... | You have applied the induction hypothesis wrongly. We have $4 (2^{2n}) -1=4 (2^{2n}-1)+3=4(3m)+3=3(4m+1)$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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100 persons 100 sweets problem
There are $100$ persons, including men,women and children.
Then there are $100$ sweets.
*
*Each man will get $10$ sweets
*Each woman will get $5$ sweets
*Each child will get $.5$ (i.e half) of the sweet
At the end of sharing every 100 person should get sweets, and t... | The part you are ignoring is that this is a diophantine equation -- that is, all of the variables must be non-negative integers. That will eliminate a large number of solutions to the two equations you listed.
Let's do it intuitionistically to start. We will imagine that there were $100$ children. That obviously doe... | {
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Isolate Variable in Fraction I can approximate u with a calculator by guessing or using excel but I want to isolate it.
$100 = \dfrac{1 + \dfrac{1}{(1+u)^6}}{u}$
Can not seem to do it by hand myself. Is it possible using only simple algebra?
| This looks very much as a finance problem.
Let us rewrite it as
$$\frac{u}{1+\frac{1}{(1+u)^6}}=a$$
Develop the lhs as a Taylor series to get
$$\text{lhs}=\frac{1}{2}u+\frac{3 }{2}u^2-\frac{3 }{4}u^3-4 u^4+\frac{51 }{8}u^5+O\left(u^6\right)$$ and use series reversion to get
$$u=2 a-12 a^2+156 a^3-2392 a^4+40560 a^5+O\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Assume $\lambda_\min(A_kA_k^{\rm T})>\varepsilon$ and show that $A_k^{\rm T}(A_kA_k^{\rm T})^{-1}$ is bounded For all $k\in\mathbb{N}$, let $A_k\in\mathbb{R}^{n\times m}$, where $n\leq m$, and assume that there exists $\varepsilon>0$ such that for all $k\in\mathbb{N}$, $\lambda_\min(A_kA_k^{\rm T})>\varepsilon$, where ... | Let $A= U \Sigma V^T$ where $\Sigma$ has the same form as $A$. Then
$A^T (A A^T)^{-1} = V \Sigma^T (\Sigma \Sigma^T)^{-1}U^T$.
We have $\Sigma^T (\Sigma \Sigma^T)^{-1} = \begin{bmatrix} \operatorname{diag}({1 \over \sigma_1},\cdots, {1 \over \sigma_n}) \\ 0 \end{bmatrix}$. Hence $\|A^T (A A^T)^{-1}\| = {1 \over \sigma_... | {
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Exponential families for normal distribution $f_Y (y; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp(-\frac{(y-\mu)^2}{2\sigma^2 })$ is the normal distribution pdf
I am trying to get this in the form $$Y∼f_Y (y;θ,ϕ)= \exp\left[\frac{yθ-b(θ)}{a(ϕ)}+ c(y,ϕ)\right]$$
my notes did this in one step
$$f_Y (y; \mu, \si... | If your starting point is that $f_Y (y; \mu, \sigma^2) = f_Y (y; \theta, \phi)$ then you are requiring that $\mu = \theta$ and $\sigma^2 = \phi$ from the outset, which means you do have to write that first term as $$\frac{y \mu - \mu^2 /2 }{\sigma^2}$$ so that $\theta$ matches up with $\mu$.
In other words, you're n... | {
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Conics consisting of two points/lines makes them rank 2 While studying conics, I came across this concept and example:
Degenerate conics. If the matrix $C$ is not of full rank, then the conic is termed degenerate. Degenerate point conics include two lines (rank 2), and a repeated line (rank 1).
Example. The conic
$$C ... | For the two-point/line degenerate conics, the explanation is already there in the text: “The null vector is $\mathbf x=\mathbf l\times\mathbf m$” [emphasis mine]. We can drill down into this statement a bit, though.
What is the dimension of the null space of $\mathbf l\mathbf m^T+\mathbf m\mathbf l^T$? Well, $$(\math... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to use Chinese Remainder Theorem A cubic polynomial $f(x)=ax^3+bx^2+cx+d$ gives remainders $-3x+3$ and $4x-1$ when divided by $x^2+x+1$ and $x^2+2x-4$. Find the value of $a,b,c,d$.
I know it’s easy but i wanna use Chinese Remainder Theorem(and Euclidean Algorithm) to solve it.
A hint or a detailed answer would be ... | $$ \left( x^{2} + x + 1 \right) \left( \frac{ - x - 7 }{ 31 } \right) - \left( x^{2} + 2 x - 4 \right) \left( \frac{ - x - 6 }{ 31 } \right) = \left( -1 \right) $$
is all you need. Cleaning up,
$$ (x+7) \left( x^{2} + x + 1 \right) - (x+6) \left( x^{2} + 2 x - 4 \right) = 31 $$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Estimating the error in the alternating series So I have the following question here:
Find the Macluarin series of $\displaystyle F(x) = \int_{0}^{x} (1+t^2)\cos(t^2)dt$. Use this series to Evaluate $F(\frac{\pi}{2})$ with an error less than $0.001$.
Now, I know the basic idea. The Maclaurin series of $\displaystyle ... | Since they are each alternating series and eventually the terms are decreasing you can use the alternating series rule on each one. If you make your error criteria on each series to be half the desired error, then the overall error when you combine the two will be what you want.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What if we don't accept ex falso quodlibet? What happens in a logical system if we do not state ex falso quodlibet?
$\bot\rightarrow P$
| See Paraconsistent Logic for a family of logics that reject Ex Falso (aka: Principle of Explosion).
See also: Walter Carnielli & Marcelo Esteban Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation (Springer, 2016),
as well as: Holger Andreas & Peter Verdée (editors), Logical Studies of Paraconsisten... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the level set $\{f=1\}$ of the Minkowski functional of $C$ equal to the boundary of $C$? Let $C$ be a convex and compact subset of $\mathbb{R}^d$. Assume that $\boldsymbol{0}$ belongs to the interior of $C$.
The Minkowski functional of $C$ is
\begin{align*}
f \colon \mathbb{R}^d & \to [0,+\infty)\\
\boldsymbol{x} &\... | Your gut feeling seems correct to me.
Let $\varepsilon>0$ such that $B(0,\varepsilon)\subseteq C$ and assume $x\in \tau C$ for some $\tau\in (0,1)$. Then, $\frac{1}{\tau}x\in C$ and accordingly, we get that
$$
B(x,(1-\tau)\varepsilon)=\tau \left\{\frac{1}{\tau}x\right\}+(1-\tau)B(0,\varepsilon)\subseteq C,
$$
which pro... | {
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Show that $ ‖ A^2 ‖_2 = ‖ A ‖^2_2$ for a symmetric matrix $A$ I want to show that $ ‖ A^2 ‖_2 = ‖ A ‖^2_2$ for a symmetric matrix $A$. So far, I got $$‖ A^2 ‖_2 = \underset{x\neq0}{\max} \frac{‖ A^2x ‖}{‖ x ‖} = \underset{x\neq0, ‖ x ‖ = 1}{\max} ‖ A^2x ‖$$ $$ ‖ A ‖^2_2 = (\underset{x\neq0}{\max} \frac{‖ Ax ‖}{‖ x ‖})^... | You should use the hypothesis.
Hint: $\|Ax\|^2 =x^TA^2x\le \|A^2x\|\le\|A^2\|$ if $\|x\|\le 1$,
and for any two matrices $A,B$ we have $\|AB\|\le \|A \|\, \|B\|$.
| {
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"timestamp": "2023-03-29T00:00:00",
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The supremum of the $n$th derivative of a holomorphic function is bounded by the $L^1$ norm Let $U\subset \mathbb{C}$ be open, and $A\subset U$ compact. Suppose $n\geq 0$ where $n\in \mathbb{Z}$. Prove that there exists a constant $k$ (allowed to depend on $n$, $A$, and $U$) such that for any function $f$ which holomor... | Let $4d>0$ be the distance from $A$ to $\partial U$. By compacity, there are finitely many closed discs of radius $d$ centered at the points of $A$ that cover it, so if we prove the relation required for the part of $A$ in each such disc we are done by taking for $k$ the maximum of all the $k's$ obtained at each disc a... | {
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Solving $\cos^2 x+\cos^2 2x +\cos^2 3x=1$ I picked this up from the IMO 1962.
Solve for $x$:
$$\cos^2 x+\cos^2 2x +\cos^2 3x=1$$
At first I thought it was trivial to bring this in an IMO, but I realized approaching it directly brings a power of 6 which is not all too friendly.
Is there a sneaky way to solve this?... | Let $t:=\cos^2x$.
We have
$$t+(2t-1)^2+(4t-3)^2t=1$$
or
$$16t^3-20t^2+6t=0.$$
The roots are $0,\dfrac12,\dfrac34$, nothing really difficult.
| {
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"timestamp": "2023-03-29T00:00:00",
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Index of a number field and its subfields Let $F$ and $L$ be number fields, let $FL$ be their compositum, and let the discriminants of these two fields be coprime.
Given one of the extensions $F / \mathbb{Q}$ and $L / \mathbb{Q}$ is Galois I want to
show $[F L: \mathbb{Q}]=[F: \mathbb{Q}][L: \mathbb{Q}].$ I tried apply... | I don't think there is any elementary solution because it doesn't hold when replacing $\Bbb{Q}$ by another number field $K$ as there might be some unramified extension $F/K$ (so that $Disc_{F/K}(F)=O_K$) and letting $L=F$ it fails.
A solution is that $[F L: \mathbb{Q}]=[F: \mathbb{Q}][L: \mathbb{Q}]$ is equivalent to ... | {
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How do they check really large primes? Currently, the largest prime know is a mersenne, $2^{82,589,933} − 1$. That’s an $82,589,933$-bit number if I am correct. Considering that RSA codes of as low as 1024 bits can be considered safe, how was this number factored to check if it is prime? I can kind of answer that quest... | After the famous 'primes is in P' paper, there are polynomial time algorithms for testing whether a number is prime. According to wikipedia the run time of these is $O(\log(n)^6)$ and while this is massively faster than actually factoring a number like $
2^{82,589,933}−1$ the 6th power is still too big to make this fea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding the equation of a hyperbola
Find the equation of the hyperbola with center on $2y+x-1=0$, with an asymptote $y+2x-5=0$, and a focus $(1,0)$.
Can anyone help me out with this problem?
| First, verify that the given focus lies on the line $x+2y-1=0$, which means that this line is the transverse axis of the hyperbola. Then reflect the given asymptote in this line to get an equation of other asymptote in the form $px+qy+r=0$. An equation of the hyperbola is then $(2x+y-5)(px+qy+r)=k$. Finally, choose $k$... | {
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Upper bound of tail of probability distribution assuming only finite first moment. Let $X$ be a random variable with $\mathbb E{X}<\infty$. By Markov's inequality we have
$$
\mathbb P(X > n)\le \mathbb E(X)/n = O(1/n).
$$
I sort of remembering I saw somewhere $\mathbb E{X}<\infty$ actually implies that
$$
\mathbb P(X >... | Suppose $X$ is non-negative and $EX<\infty$. Then $xP(X>x)\to 0$ as $x\to \infty$. Indeed
first note that $$xI(X>x)\leq XI(X>x)$$
where $I$ is the indicator function. Taking expectations yields that
$$
xP(X>x)\leq EXI(X>x)\to 0
$$
by the dominated convergence theorem.
| {
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} |
Isosceles triangle and altitude
$\triangle ABC$ is isosceles with altitude $CH$ and $\angle ACB=120 ^\circ$. $M$ lies on $AB$ such that $AM:MB=1:2$. I should show that $CM$ is the angle bisector of $\angle ACH$.
We can try to show that $\dfrac{AM}{MH}=\dfrac{AC}{CH}$. We have $\dfrac{AC}{CH}=\dfrac{2}{1}$ because $\... | Note that
$$\sin \angle CMH = \frac{MH}{CM}
=\frac{\frac14MB}{MB\sin30}=\frac12$$
Then, $\angle CMH=30$. Thus, $CM$ is the angle bisector.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3563573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
basic number theory - polynomial congruence
Problem: Determine whether $x^{2} \equiv 5$ mod $120$ has solution. If
so, how many?
NOTE: This is a specific question, but is there a method for answering this question given any set of numbers?
Thoughts: Not exactly sure. I want to rearrange the terms to say that this ... | The answer is zero solutions.
$x^{2}\equiv{5}\mod{120} \rightarrow x^{2}=120y+5 \rightarrow x^{2}\equiv{5}\mod{8}$
$x^{2}\equiv0,1,4\mod{8}$
Can also be $x^{2}\equiv2\mod{3}$ while quadratic numbers are either $0$ or $1\mod{3}$. Credits: lulu
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3563734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove an equation involving Gamma Function Just started learning Gamma function, and we were asked to prove following equation for all positive integer $n$ and non-integer $m$.
$$0 = \sum^n_{i = 0}\frac{n-m-2i}{i!(n-i)!\Gamma (i+m+1) \Gamma (n-m-i+1)}$$
I tried when $n = 1$ and 2. I feel like it's related to an expansi... | $\binom{\alpha}{\beta}:=\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}$ is well-defined (at least) for $\alpha\notin\mathbb{Z}_{<0}$ (assuming $1/\Gamma(\beta):=0$ for $\beta\in\mathbb{Z}_{\leq 0}$). Then $\binom{\alpha}{\beta}+\binom{\alpha}{\beta+1}=\binom{\alpha+1}{\beta+1}$ holds (like in the case o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3563898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Number theory: how to prove $\gcd(a,b,c)=\prod p^{\min(a_p,b_p,c_p)}$ $a,b,c\in\mathbb Z\setminus\{0\}$
$\gcd(a,b,c):=\gcd(\gcd(a,b),c)$
How to show that $\gcd(a,b,c)=\prod p^{\min\{a_p,b_p,c_p\}}$?
My idea is to use $\gcd(a,b)=\prod p^{\min\{a_p,b_p\}}$, but I am stuck on it. Any hints?
Thanks!
| The gcd of given $n$ natural numbers $a_1, a_2, \cdots ,a_n$ is defined as the largest divisor common to $a_1, a_2, \cdots ,a_n$.
If we write the prime factorization of each of the numbers $a_1, a_2, \cdots,a_n$, the gcd will be the products of some power of all the primes that are common in each number $a_1, a_2, \cdo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Solving inequality with fraction in one side confusion In the book I'm using an example is given as follow:
$\frac{2x - 5}{x-2}< 1$
then it proceeds to say that we could multiple both sides by $x-2$ to get rid of the denominator in the left hand side (I understand that). But then it goes on to say that this method woul... |
How did the author of the book reached to the conclusion that we will need to evaluate such cases?
Because we don't know the sign of $x-2.$ All we know is that it may either be positive ($>0$) or negative ($<0$); the case $x-2=0$ not arising since then the fraction is not a real number.
why did the orientation of th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the motivation behind sigma-algebra properties? Sigma algebras are the fundamental construct that probability theory and Lebesgue integration are based on. I learned a few monographs in probability theory where the term of "sigma-algebra" shows up as a definition, utilitarian, without any discussion around it.
... | The motivation for $\sigma$- algebras is to define a family of sets to serve as the domain for a measure $\mu$. It this sense it is clear that the set itself should be measurable, and the empty set should be measurable. Also, if we know the measure of X and the measure of $A \subseteq X$, then $X\setminus A$ should hav... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Expected value of a random variable at a stopping time. Let $x_1,x_2 \dots$ be adapted to the filtration $\mathcal{F}_1, \mathcal{F}_2, \dots$.
Let $\tau$ be a stopping time that is also adapted to the filtration.
Say that
$\mathbb{E}[x_i \mid \mathcal{F}_{i-1}] = 0$.
Is it true that
$$\mathbb{E}[x_{\tau}] = 0?$$... | Suppose $\tau$ is finite with probability one. Then $$ \mathbb E x_{\tau} = \mathbb E [ \mathbb E[x_\tau | \tau]] = \sum_n \mathbb E [x_\tau | \tau = n]P(\tau = n) $$
If $\tau$ is not finite, i.e. $P(\tau <\infty) <1$, then let $N$ be a positive integer and let $\tau_N := \min(\tau,N)$. Then $\tau_N$ is finite with pro... | {
"language": "en",
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"source": "stackexchange",
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Show that the eigenvalues of $AA^T$ and $A^TA$ are non-negative. Let $A\in \mathbb{R^{m\times n}}$. Show that the eigenvalues of $AA^T$ and $A^TA$ are non-negative.
I could just apply the definition of an eigenvalue for $AA^T$ (or $A^TA$), but I don´t know how to determine the sign of the eigenvalue. Here is what I tri... | Given
$A \in \Bbb R^{n \times m}, \tag 1$
we have
$A^T \in \Bbb R^{m \times n}, \tag 2$
whence
$AA^T \in R^{n \times n}; \tag 3$
we observe that
$(AA^T) = (A^T)^TA^T = AA^T, \tag 4$
that is, $AA^T$ is a symmetric matrix operating on $\Bbb R^n$,
$AA^T: \Bbb R^n \to \Bbb R^n, \tag 5$
thus if $\mu$ is an eigenvalue of $A... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Probability Question. Which solution is right? Suppose, there is a building with $6$ floors and a ground floor. If $10$ persons get into an elevator on the ground floor, what is the probability that exactly $2$ persons will get out on the $2nd$ floor?
I can present two solutions, but which one of them is right?
Let $x_... | The first solution is incorrect, because it implies that each person’s decision to stay or get out of the elevator is not independent from other persons’ decision. Think about it like this, You have $n$ identical coins. You cannot differentiate each from the others. But still, since the result of individual coin flip i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3564889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the fewest number of testers needed to identify the poisoned wine? The King has 1000 bottles of wine, exactly one of which is poisoned. Your
job is to identify and throw out the poisoned bottle as quickly as possible by having the royal
taste-testers drink the wines. Since the poison takes a little while to tak... | As mentioned in the comments, you should use binary representation of numbers. Let's start with fewer bottles, say $6$. We can write the binary numbers for the labels as $001$, $010$, $011$, $100$, $101$, $110$. Now let's assign a tester for each bit. For example, tester 1 will check the last bit, tester 2 will check t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two independent random geometric variables
Correct Answer = 0.0495
My work:
X, Y~geom(p)
$F(2, 2) = P(1, 1) + P(1, 2) + P(2, 1) + P(2, 2) = p^2 + 2p(1-p) + p^2(1-p)^2 = 0.0441$
I think this is the right step, not sure how to solve equation involving $p^4$..
(Finan Exam P 40.24)
| Let $X$ and $Y$ be the number of attempts made by A and B. We can assume these random variables to be independent, so
$$
F(2,2)=\mathbb P(X\leq 2, Y\leq 2) = \mathbb P(X\leq 2)\cdot \mathbb P(Y\leq 2) = (p+p(1-p))^2 =(2p-p^2)^2=(1-(1-p)^2)^2 = 0.0441.
$$
The equality $(1-(1-p)^2)^2 = 0.0441$ follows also directly from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3565184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the dimension of the range of operator I have the operator $Tf(x)=\int_0^x(x-y)f(y)dy$, for $f\in\mathcal{C}([0,1])$, equipped with the supremum/infinity norm. I know that such an operator is called a "Fredholm operator", and I am aware of several theorems that can be applied to it, for instance to show that th... | We represent the operator in the form $Tf(x)=x\displaystyle\int\limits_0^xf(y)dy-\int\limits_0^xyf(y)dy$. It's easy to see that functions $x^n$, $n=2,3,4,...$ lies in the range of $T$, since the functions $n(n-1)x^{n-2}$ pass into them under the action of the $T$ (you can just solve the equation $Tf(x)=x^n$ by differen... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Area of a parallelogram using similar triangles
$ABCD$ is a parallelogram and point $M$ lies on $AB$ such that $AM:MB=2:3$. If $DM \cap AC=N$ and the area of $\triangle ADN=a$, I should find the area of the parallelogram $ABCD$.
Let $AD=BC=b$ and $NN_1\perp AD=h_1, BB_1 \perp AD = h_2$. We have $S_{\triangle ADN}=\d... | Hint: $\Delta ANM$ is similar to $\Delta DNC$, so we need to find the area of $\Delta DNC$ in terms of $a$.
Let $Area(ABCD) = A$
$$Area(AND) + Area(CND) = \dfrac{A}{2}$$
$$\frac{Area(ANM)}{Area(CND)} = \bigg(\frac{AM}{CD}\bigg)^2 = \bigg(\frac{2}{5}\bigg)^2 = \frac{4}{25}$$
Also,
$$Area(AND) + Area(ANM) = \dfrac{A}{5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3565533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Computation of Betti numbers of a given space Trying to verify my computations below. I don't have much intuition for homology or the Betti numbers computation. It's a simple case, yet somehow intuitively I'm surprised to get $\beta_1(W) = 3$.
Is my computation wrong?
Here is how I construct $W$.
$W$ is construct... | That looks reasonable to me. Another way to think about it is that $W$ is homotopic to a sphere with four holes punched in it. A loop around each hole is a generator of $H_1(W)$, but the sum of all of those loops is homologous to a loop around all the holes, and that's contractible by going around the other end of th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to show that $[-2,2)$ is not compact?
How to show that $[-2,2)$ is not compact?
I can show that $(-2,2)$ is not compact since $K=\bigcup_{n\in\mathbb{N}}(-2,2-\frac{1}{n})$ has no finite subcover.
However I'm not sure how I can write a union of open sets which will include $-2$?
| Compact implies sequentially compact. Consider $x_n=2-1/n$. The limit, $2$, is not in the set.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complex conjugate of an involved expression I understand the the complex conjugate of, say, $z:=\exp({a+ib})$ is $z:=\exp({a-ib})$.
However , I have a composite expression and I'm not sure how to attack taking it's complex conjugate.
Say $z:=i\exp({ib}) / ({a + ic})$
I would be tempted to say that the denominator beco... | If $z=e^{a+ib}$ then $z=e^a(\cos b+i\sin b)$, so $\bar z=e^a(\cos b-i\sin b)$ and then $\bar z=e^{a-ib}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3566037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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convergence of $\large \int_1^{+\infty} \frac{\ln(x)}{\sqrt{1+x}}dx$ I'm trying to determine if the underneath integral converges or not.
$\large \int_1^{+\infty} \frac{\ln(x)}{\sqrt{1+x}}dx$
I know that in the interval [1, $+\infty$) the enqualities:
$ \frac{\ln(x)}{\sqrt{1+x}} \leq \frac{\ln(x)}{\sqrt{x}}$ hold... | $$I=\int_{a}^{\infty} \frac{dx}{x^{\beta}}<\infty$$ if $\beta>1$ diverges if $\beta<1$, because in your case $\beta=1/2<1$.
Hence the given integral will diverge like $$\int_{1}^{\infty} \frac{dx}{x^{0.99}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3566185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Banach space property of parabolic Sobolev space Let $X$ be a real Banach space with norm $||\cdot||$. We define for $1<p<\infty$ and $t_1<t_2$, the space $Y=L^p(t_1,t_2;X)$ to be the space of measurable functions $f:(t_1,t_2)\to X$ such that the norm
$$
||f||_{L^p(t_1,t_2;X)}:=\Big(\int_{t_1}^{t_2}||f||_{X}^p\,dt\Big)... | If you use this post, you can show the desired result by applying the result twice provided that X is separable and reflexive.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing there's no closed-form: $\sum_{n=0}^\infty(-1)^n\frac{\cos^2({3^nx})}{3^n}$ Problem_
Compute $$\sum_{n=0}^\infty(-1)^n\frac{\cos^2({3^nx})}{3^n}$$
The problem is pretty simple, but it was hard for me to segregate into the partial fractions(I wanted to make a form of telescoping).
Hmmmm... My attempts were:
$... | Lets say $$f(x)=\sum_{n=0}^\infty(-1)^n\frac{\cos^2({3^nx})}{3^n}$$
Then $$f'(x)=-\sum_{n=0}^\infty(-1)^n\sin({2*3^nx})$$
Now $$\sin(t)=t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+...$$
with $t=2*3^nx$ $$\sin(2*3^nx)=2*3^nx-\frac{(2*3^nx)^3}{3!}+\frac{(2*3^nx)^5}{5!}-\frac{(2*3^nx)^7}{7!}+...=2*3^nx-\frac{3^{3n}(2x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3566672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
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Implicit Differentiation of logarithm
Differentiate $y=\log_a(x)$ with respect to $x$
I see that $a^y=x$.
My textbook says implicit differentiation gets us \begin{align*}a'(\ln a)\frac{dy}{dx}&=1 \\\implies \frac{dy}{dx}&=\frac{1}{a'\ln a} \\ \frac{dy}{dx}&=\frac{1}{x\ln a}\end{align*}
What I don't understand is why ... | Option:
$a^y=x$;
Tale $\log_e$ of both sides:
$y \log a=\log x$;
Differentiate with respect to $x$:
$y' \log a=\dfrac{1}{x}$;
$y'=\dfrac{1}{x \log a }$;
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3566979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Radius of convergence of power series where power increases by increments of 2 I know that to determine the radius of convergence of the series
$$ \sum_{n=0}^\infty a_nx^n $$
I need to find
$$ \lim_{k\rightarrow \infty} \left| \frac{a_{k+1}}{a_k} \right| = c$$
Then the radius of convergence $R$
$$R = \frac{1}{c}$$
Howe... | Consider the series:
$\begin{equation*}
\sum_{n \ge 0} a_n x^{B n}
\end{equation*}$
From the respective theory, you know that for the series:
$\begin{equation*}
\sum_{n \ge 0} a_n y^n
\end{equation*}$
there is a radius of convergence $R$ such that it converges if $\lvert y \rvert < R$ and diverges whenever $\lvert y \r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3567076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Simplify $(1+i^\frac{1}{2})^\frac{1}{2}$ How can I simplify $\sqrt{1+\sqrt{i}}$?
I thought about making $z^2=1+\sqrt{i}$ and then $w=z^2$
But I'm not really sure
| $$(1+i^\frac{1}{2})^\frac{1}{2}=(1+(e^{i\frac\pi2+i2\pi n})^\frac12 )^\frac12
=(1+e^{i\frac\pi4+i\pi n} )^\frac12$$
$$=\left(1+\cos(\frac\pi4 +\pi n)+ i \sin(\frac\pi4 +\pi n) \right)^\frac12
=(re^{i\theta})^\frac12\tag 1$$
where,
$$r=\sqrt{\left(1+\cos(\frac\pi4 +\pi n)\right)^2+\sin^2(\frac\pi4 +\pi n) }
=\sqrt{2+2\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3567232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why are rings with identity $\mathbb{Z}$-algebras? According to Dummit and Foote's definition, an $R$-algebra ($R$ is a commutative ring with identity) is a ring A with identity together with a ring homomorphism $f: R \rightarrow A$ mapping $I_R$ to $I_A$ such that the subring $f(R)$ of $A$ is contained in the center o... | Hint: where do you send $2 = 1 + 1$?
| {
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"timestamp": "2023-03-29T00:00:00",
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How to find out number of positive eigenvalues of a symmetric matrix? Suppose $A$ is a $3 \times 3$ symmetric matrix such that $$[x,y,1]A\left[\begin{array}{c}
x \\
y \\
1
\end{array}\right]=xy-1.$$
Let $p$ be the number of positive eigenvalues of $A$ and let $q = rank (A) - p$. Then
(1) $p=1.$
2) $p=2.$
(3) ... | We have $$A=\begin{bmatrix} 0 & \frac12 & 0 \\ \frac12 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$
Clearly $-1$ is an eigenvalue. The other eigenvalues satisfiesy $\lambda_1+\lambda_2=0$ and $\lambda_1\lambda_2=-\frac14$. Hence the remaining eigenvalues are $\frac12$ and $-\frac12$.
$p=1$ and $q=3-1=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3567443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Division of $ f= X^4+X^3+X^2+X+2$ by $g(X)=X-\cos(\alpha)+i \sin(\alpha)$ We have the polynomial $ f= X^4+X^3+X^2+X+2$ with $f\in \Bbb C[X] $, it asks to determine the quotient of the division of the polynomial $f$ by the polynomial $g$, $g(X)=X-\cos(\alpha)+i \sin(\alpha) \in \Bbb C[x] $, $α \in(0,π/2)$, knowing that ... | By remainder theorem,
$$1 + i(1 + \sqrt{2}) = f(\cos \alpha - i \sin \alpha) = f(e^{-i\alpha}).$$
Therefore
\begin{align*}
&(e^{-i\alpha})^4 + (e^{-i\alpha})^3 + (e^{-i\alpha})^2 + e^{-i\alpha} + 2 = 1 + i(1 + \sqrt{2}) \\
\iff \, &(e^{-i\alpha})^4 + (e^{-i\alpha})^3 + (e^{-i\alpha})^2 + e^{-i\alpha} + 1 = i(1 + \sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3567563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Variance of mixed random variable $X \sim F(x)$
$$F(x)=\begin{cases}0,&x<0\\x^2,&0\leq x<1/2\\x,&1/2\leq x<1\\1,&x>1\end{cases}$$
(not right-continuous)
I want to compute $\operatorname{Var}(X)$.
Is this correct:
$$\mathbb E[X]=\int_0^{1/2}2x^2\,dx+\int_{1/2}^1x\,dx+1/2\cdot \mathbb P(1/2)$$
How can I evaluate $\mathbb... | Your expression for $\ \mathbb{E}\left[X\right]\ $ is correct if $\ \mathbb{P}\left(\frac{1}{2}\right)\ $ is taken to mean the same thing as $\ \mathbb{P}\left(X=\frac{1}{2}\right)\ $. And yes, the value of $\ \mathbb{P}\left(X=\frac{1}{2}\right)\ $ is the size of the jump in $\ F\ $ at $\ x=\frac{1}{2}\ $:
$$
\mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3567897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum of three perfect cubes is equal to a perfect fourth How many answers does the following equation have?
$a^3+b^3+c^3=d^4$ where $a, b, c, d\in \mathbb{Z}^+$.
This was asked in a test for gifted math students in 7th grade in Finland. I have been thinking about this and couldn't solve this for my number theory isn't t... | Note that if $a^3 + b^3 + c^3 = n$, then $(na)^3 + (nb)^3 + (nc)^3 = n^4$.
So you have infinitely many solutions even if you require $a,b,c$ to be distinct.
There are also solutions where $a,b,c$ are pairwise coprime, e.g.
$$ \eqalign{19^3 + 89^3 + 117^3 &= 39^4\cr
107^3 + 163^3 + 171^3 &= 57^4\cr
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3568046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Inverse Laplace transform of $F(s)=\frac{3s+7}{s^2-2s-3}$ I have to calculate the inverse Laplace Transform of this image:
$$F(s)=\frac{3s+7}{s^2-2s-3}$$
I try decomposing it in this way:
$$F(s)=\frac{3s-3+10}{(s-1)^2-4}=3\frac{s-1}{(s-1)^2-4}+5\frac{2}{(s-1)^2-4}$$
where I can identify that the original function is $$... | You are correct! Indeed, we have that
$$f(t)=3e^t\cosh(2t)+5e^t\sinh(2t)=3e^t\frac{e^{2t}+e^{-2t}}{2}+5e^t\frac{e^{2t}-e^{-2t}}{2}=4e^{3t}-e^{-t}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3568201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proving a union of sets How to prove:
$\cup_n[\frac{1}{n},1] = (0,1]$, where $n \in \mathbb N $
The only thing I'm aware of is that we have to prove both left to right and right to left as I'm dealing with sets, and couldn't find a starting point.
Can anyone help with this?
| If $x\in\cup_n[\frac{1}{n},1],$ then $x\in[\frac1n,1]$ for some $n$, so, since $\frac1n>0$, it follows that $x\in(0,1]$.
On the other hand, if $x\in(0,1]$, then $x\in[\frac1n,1]$ for all $n>\lfloor \frac1x\rfloor$, so $x\in \cup_n[\frac{1}{n},1] $.
Here $\lfloor y\rfloor$ is the floor function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3568402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find ways from $(0,0)$ to $(8,8)$ You are allowed only to go east or north. Because of road construction, you cannot touch the points $a, b, c$ and $d$. Under these restrictions, the number of ways that you can go from $(0, 0)$ and finish at $(8, 8)$ in the following figure is:
First, I used ${16\choose 8}$ to get the... | Through $a$. $\binom{6}{3}\times\binom{10}{5}$.
Through $b$ without going through $a$. $\binom{6}{4}\times\binom{9}{4}$.
Through $d$ without going through $a$. $\binom{6}{2}\times\binom{9}{5}$.
$\binom{16}{8}-\binom{6}{3}\times\binom{10}{5}-\binom{6}{4}\times\binom{9}{4}-\binom{6}{2}\times\binom{9}{5}=4050$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3568613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
The system of three DE I want to solve the following system of DE:
$$
\begin{cases} \dot{x} = 2x+6y -15z, \\ \dot{y} =x+y-5z,\\ \dot{z} = x+2y-6z, \end{cases}
$$
First, I rewtite the coefficents in the matrix form:
$$A = \begin{bmatrix}
2 & 6 & -15\\
1 & 1 & -5\\
1 & 2 & -6
\end{bmatrix}$$Then, I find $$det(A-\lambda ... | When the coefficient matrix $A$ has only one (repeated) eigenvalue $\lambda$, you’re in luck: the exponential $e^{tA}$ is easily computed without having to find any eigenvectors, generalized or otherwise. If the eigenvalue’s algebraic and geometric multiplicities are equal, then it must be a multiple of the identity ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3568769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding if $\int_{1}^{\infty} \frac{\sin(x+2)}{x^2} \, dx $ converges, with two conficting solutions? Consider the problem where the following integral converges or not:
$$\int_{1}^{\infty} \frac{\sin(x+2)}{x^2} \,dx $$
I tried to solve it in two different ways but the results conflict. I am not sure why.
First Soluti... | You are writing
$$\int f(x)dx\le \int g(x)dx$$
and conclude that if $$ \int g(x)dx$$ diverges, so does $$\int f(x)dx.$$
This is wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3568936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$\mathbb{Z}[T]$-module and extension I consider the structure of $\mathbb{Z}[T]$-module on $\mathbb{Z}$ given by the multiplication $P \times a := P(0) \times a$.
Now i consider a morphism of ring $\phi : \mathbb{Z}[T] \to R$, that give a structure of $\mathbb{Z}[T]$-module on $R$, i denote by $t := \phi(T) \in R$. I ... | Yes that is true, because $\mathbb Z \cong \mathbb Z[T]/(T)$ as $\mathbb Z[T]$-modules with the structure that you described. Then, we may apply the general identity $A/I\otimes_A M \cong M/IM$, which holds for $A$ a ring, $I$ an ideal and $M$ an $A$-module.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3569093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Function Plus a Constant as a Parameter of a Function We spent about 10 minutes arguing this in Calculus class, but we ended up dismissing the problem. Here it is:
We were trying to prove the chain rule from 1st principles, but we weren't sure which equation was right:
$h'(x)=\lim\limits_{x\to 0}\frac{f(g(x) + h) - f(g... | It appears your $h$ function is $h(x) = f(g(x))$. If so, then note that whatever you replace $x$ with on one side must be the same on the other side, e.g., $h(y) = f(g(y))$, $h(x + j) = f(g(x + j))$, etc. Thus, the correct way to express its derivative is
$$\begin{equation}\begin{aligned}
h'(x) & = \lim_{j \to 0}\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3569388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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May I divide by number n in order to solve $2n = n^2$ ( even in a case where $n$ is not equal to $0$)? Suppose I have the equation : $2n = n^2$.
Dividing by $n$ ( provided $n$ is not $0$) , I get (apparently): $n = 2$. However, from another point of view, I have:
$2n = n^2 \rightarrow n^2 = 2n $
$\rightarrow \sqrt{n... | Assuming we work in $\Bbb Z$, I would proceed as follows:
given that
$n^2 = 2n, \tag 1$
we may write
$n(n - 2) = n^2 - 2n = 0; \tag 2$
now since $\Bbb Z$ is an integral domain, we have
$n \ne 0 \Longrightarrow n - 2 = 0 \Longrightarrow n = 2; \tag 3$
this shows that
$n = 0, 2 \tag 4$
are the only solutions to (1).
As ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3569565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$\ln(\ln n) / \ln n$ inequality I am reading a book, Randomized Algorithms by Motwani.
In the Section 3.1 Occupancy problems, there is one step in the analysis that really puzzles me:
Let $k=\lceil (e \ln n) / (\ln\ln n) \rceil$,
$$(e/k)^k \; 1/(1-e/k) \le n^{-2}.$$
The book does not mention a single word about the abo... | $$\begin{align}
\ln \frac{(e/k)^k}{1-e/k} &= k - k \ln k - \ln(1-e/k)\\
&\le k - k\, \ln \frac{e \ln n}{\ln\ln n} - \ln(1-e/k)\\
&= - k \,\ln \frac{\ln n}{\ln\ln n} - \ln(1-e/k)\\
&\le - \frac{e \ln n}{\ln\ln n} \,\ln \frac{\ln n}{\ln\ln n} - \ln \frac{\ln n}{\ln\ln n} - \ln(1-e/k)\\
&\le - \frac{e \ln n}{\ln\ln n} \,\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3569674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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For any natural numbers $a,b,c$, prove the associativity property $(a + b) + c = a + (b + c)$. For any natural numbers $a,b,c$, we have $(a + b) + c = a + (b + c)$.
MY ATTEMPT
We shall prove it by induction on $c$. For $c = 0$, we have that $(a + b) + 0 = a + b$ and $a + (b + 0) = a + b$. Let us assume that $(a + b) +... | Proof by induction :
$\displaystyle (a + b) + c^+$ $\hspace{0.25in}$
$=\displaystyle((a + b) + c)^+$$\hspace{0.195in}$Definition of Addition in Minimal Infinite Successor Set
$=\displaystyle (a + (b + c))^+$$\hspace{0.195in}$Induction Hypothesis
$=\displaystyle a + ((b + c)^+)$$\hspace{0.25in}$Definition of Addition in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3569794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I prove that if $x^2+bx+c$ is factorable, then $x^2-bx+c$ is also factorable? I want to prove that if $x^2+bx+c$ is factorable, then $x^2-bx+c$ is also factorable(by factorable I mean that it can be expressed with the product of $2$ binomials $(x+y)(x+z)$, where $y,z\in\mathbb Z$). Also, $b,c\in\mathbb Z$. It s... | If $x^2 + bx + c$ factors then by quadratic equation it must factor to
$(x - \frac {-b+ \sqrt{b^2 - 4c}}2) (x - \frac {-b- \sqrt{b^2 - 4c}}2)$ and this factors if and only if $b^2 - 4c$ is a perfect square (note: if $b$ is even/odd then $b^2 -4c$ is even/odd so $-b\pm \sqrt{b^2-4c}$ will be even so either $\frac {-b \p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3569928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Evaluate : $\lim\limits_{n\to +\infty}\int\limits_n^{2n}\frac{\ln^{3} (2+\frac{1}{x^{2}})}{1+x}dx$ Problem :
Evaluate :
$$\lim\limits_{n\to +\infty}\int\limits_n^{2n} \frac{\ln^{3} (2+\frac{1}{x^{2}})}{1+x}dx$$
My attempt :
$$y=\frac{x}{n}$$
Then :
$$I(n)=\int\limits_1^2 n\frac{\ln^{3}(2+\frac{1}{(ny)^{2}})}{1+nx}d... | By the Mean Value Theorem for integrals, one has
$$ \int\limits_n^{2n} \frac{\ln^{3} (2+\frac{1}{x^{2}})}{1+x}dx=\ln^{3} (2+\frac{1}{\xi^{2}(n)})\int\limits_n^{2n} \frac{1}{1+x}dx=\ln^{3} (2+\frac{1}{\xi^{2}(n)})\ln(\frac{1+2n}{1+n})$$
for some $\xi(n)\in(n,2n)$. Noting that, as $n\to\infty$, $\xi(n)\to\infty$, one has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3570067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to plot the polar equation theta = pi/6 on wolframalpha I need to plot the polar equation
theta = pi/6
My question has two parts.
1) Is it a line? I'm pretty sure it is, since the angle theta in the polar equation is a constant, but since I was not able to plot this on wolframalpha, I'm not 100% sure and I would l... | Since $r$ does not depend on $\theta$ trying to graph it in polar coordinate is not possible. Since the slope of your graph is $\tan (\theta)$,you may try to graph it in Cartesian coordinates $y=\dfrac x{\sqrt 3}$ and let $x\ge 0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3570418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Determine if integer contains another integer Is there a numeric method one can use to determine if a non-negative integer contains another non-negative integer?
For example, the integer 1472 contains 47. (Any number A that is a substring of another number B would be contained by B.)
My specific application is for subs... | Can't think of a clever way,
so I'll try brute force.
For any positive integer $n$,
let $nd(n)$ be the number of
digits in $n$
so
$nd(n)
=\lfloor \log_{10}(n) \rfloor + 1$.
To see if $n$
is a part of $m$,
check if
$10^{nd(n)}\lfloor \dfrac{m}{10^{k+nd(n)}} \rfloor
=\lfloor \dfrac{m}{10^{k}} \rfloor-n
$
for
$k = 0
$
to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3570820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let X be a Hausdorff space and Y be a subset of X. Then, Y with the subspace topology is a Hausdorff space.
Question: Let X be a Hausdorff space and Y be a subset of X. Then, Y with the subspace topology is a Hausdorff space.
This is what I did, can someone verify this and let me know if I am correct or wrong? Also, ... | Everything that needs to be there is there, so it's a valid proof. My only comments are about the style.
*
*The line where you recall the definition of the subspace topology on $Y$ is out of place. You've already used this definition once; either you should state it at the top before you use it the first time, or ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3571007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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How to prove following equality if $I = \left\{\alpha \right\}$ and $J = \left\{\beta\right\}$ sets of indecies Let $I = \left\{\alpha \right\}$ and $J = \left\{\beta\right\}$ arbitrary sets of indices
$$(\bigcup_{\alpha \in I}{A_{\alpha}}) \bigcap{(\bigcup_{\beta\in J}{B_{\beta}})} = \bigcup_{\alpha, \beta \in I\times... | The most usual way of showing equality between sets is two show two inclusions:
Let $x \in \left( \bigcup_{\alpha \in I} A_\alpha \right) \cap \left( \bigcup_{\beta \in J} B_\beta \right)$
Then $x \in \bigcup_{\alpha \in I} A_\alpha$ so there is some $\alpha_x$ such that $x \in A_{\alpha_x}$, and similarly there is a $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3571188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $\int_{0}^{\pi} \frac{1}{3+2\cos(t)}\mathrm{d}t = \frac{\pi}{\sqrt{5}}$ I need to proof that
\begin{align}
\int_{0}^{\pi} \frac{1}{3+2\cos(t)}\mathrm{d}t = \frac{\pi}{\sqrt{5}}
\end{align}
is correct. The upper limit $\pi$ seems to cause me some problems. I thought about solving this integral by using the re... | Using function transformations, compress the integral by a factor of $2$ in the $x$-axis, then multiply by $2$ to get:
$$2 \int_{0}^{\pi/2} \frac{\mathrm{d}t}{3+2\cos(2t)} = 2 \int_{0}^{\pi/2} \frac{\mathrm{d}t}{4 \cos^2 t+1} = 2 \int_{0}^{\pi/2} \frac{\mathrm{\sec^2 t\ d}t}{4 + \tan^2 t+1}$$
and substituting $u = \tan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3571410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solution to "Heat Equation" with Fractional Laplacian in 2 Dimensions Statement of the Problem
We consider the equation:
$ \partial_t u + (- \Delta)^{1/2}u = 0 $
for $ u : \mathbb{R}^2 \rightarrow \mathbb{R} $.
I would like to find a non-trivial solution to this equation, using the Fourier Transform.
I believe I have ... | The problem is that a radially symmetric function doesn't allow you to ignore the dependence on the angle between $x$ and $\xi$. The integral can be solved without this assumption. Finding the radial symmetry is very cumbersome, but it is possible (full solution below):
We know that
$$ \mathcal{F}^{-1}[e^{-t|\xi|}] = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3571607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Prove that $p(x)=x^4-x+\frac{1}{2}$ has no real roots.
What is the simplest way to prove that the polynomial
$p(x)=x^4-x+\frac{1}{2}$ has no real roots?
I did with Sturm's theorem:
$p_0(x)=x^4-x+\frac{1}{2}$
$p_1(x)=4x^3-1$
$p_2(x)=\frac34x-1$
$p_3(x)=-\frac{229}{27}$
The signs for $-\infty$ are $+,-,-,-$ and for $... | $f(x)=x^4$ is a convex function, hence its graph lies above the graph of the tangent line at $x=\frac{1}{2^{2/3}}$, whose equation is $g(x)=x-\frac{1}{2^{2/3}}+\frac{1}{2^{8/3}}$. $f(x)\geq g(x)$ implies
$$ x^4-x+\frac{1}{2}\geq -\frac{1}{2^{2/3}}+\frac{1}{2^{8/3}}+\frac{1}{2}=\frac{4-3\sqrt[3]{2}}{8} $$
but $64>27\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3571703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Spivak Calculus on Manifolds, Definition of Boundary In Chapter 4's section on Geometric Preliminaries, I am confused by the definition of "boundary."
The standard $n$-cube $I^n$ is defined to be $I^n(x^1,...,x^n) = (x^1,...,x^n)$, and two associated $n-1$-cubes are defined as
$$I^n_{(i,0)}(x^1,...,x^{n-1}) = I^n(x^1,... | It's indeed not the best choice of notation.
Geometrically it's clear what should happen (modulo the alternating sum): The boundary of an $n$ dimensional box is the 'collection' of its faces, each of which is an ($n$-$1$)-cube.
Now, instead of taking their collection we take their (alternating) formal sum (which can b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3571886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f \in L^p(R)$, then $\lim_{y \to \infty}\|f(x+y)+f(x)\|_p=2^{1/p}\|f\|_p$ If $f \in L^p(R)$, then $\lim_{y \to \infty} \|f(x+y)+f(x)\|_p = 2^{1/p}\|f\|_p$
I am not sure how to proceed. To me, it seems like a density argument problem, and I can show this is true for continuous functions with compact support. Howeve... | You can extend to general functions as follows. Given $\epsilon > 0$, let $g$ be continuous with compact support such that $f = g + h$ with $\|h\|_p < \epsilon$. Then
$$\|f(x+y) + f(x)\|_p = \|g(x+y) + g(x) + h(x+y) + h(x)\|_p$$
By the triangle inequality you have
$$\|g(x+y) + g(x) + h(x+y) + h(x)\|_p \leq \|g(x+y) + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3572083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The equation $a^3+b^3+c^3=kabc$ I am interested in the equation $a^3+b^3+c^3=kabc$ for $a,b,c \in \mathbb{N}$. We have by AM-GM:
$$a^3+b^3+c^3 \geqslant 3abc \implies k \geqslant 3$$
Since $k=3$ is the equality case, the solutions for $a^3+b^3+c^3=3abc$ is $(a,b,c)=(x,x,x)$ for some $x \in \mathbb{N}$. However, it is n... | We can get solutions one by one through seeking rational roots of a cubic equation.
Assume wlog $a\le b\le c$. Pick values of $a$ and $b$ that meet the above ordering requirement. Then render a cubic equation for $c$:
$c^3-(kab)+(a^3+b^3)=0$
And solve the original equation for $k$:
$k=(a^3+b^3+c^3)/(abc)$
We then hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3572248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to prove the eigenvalues of $(A+B)^{-1}A$ are within $[0, 1)$, when $A$ is positive semidefinite and $B$ is positive definite? The eigenvalue $\lambda_i$ of $(A + B)^{-1}A$ is within $[0,1)$, where $A$ is positive semidefinite and $B$ is positive definite. How to prove this?
Thank you in advance.
| I figured it out.
$(A+B)^{-1}Ax = \lambda x$ and multiply $x^T(A+B)$ on the left to get:
$x^TAx = \lambda x^T(A + B)x$ and now it's easy to see the range of $\lambda$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3572430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to prove " multiplicative inverse of inverse of $a$ is $a$ itself" ( with $a$, say, a real number) in basic arithmetics? ( $\frac{1}{1/a}$ = $a$.) Suppose I want to treat basic arithmetics on real numbers as a little deductive system ( without using abstract algebra).
In order to prove the " divide by a fraction "... | Multiply both sides by the inverse of $a$. You know from the definition of inverse that $$a\cdot\frac 1a=\frac1a \cdot a=1$$
Then the right hand side is $1$
On the left hand side use that the inverse of $a$ is $b$. then you have $$\frac 1{1/a}\cdot \frac 1a=\frac 1b\cdot b=1$$
Subtract first term and last term from the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3572555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the probability that no more than three attempts will be required to open the lock
Of the five keys, one is suitable for the lock. The key that did not
fit when trying to open the lock is put aside.
We need to find the probability that no more than three attempts will
be required to open the lock.
I tried t... | Let me provide a different solution that is often applied to such problems.
The trick is to use the opposite of what we want to find first. And then subtract its probability from $1$.
Let:
P(A)=P(1st attempt successful)=1/5
P(B)=P(2nd attempt successful)=1/4
P(C)=P(3rd attempt successful)=1/3
You found - correctly - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3572648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
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} |
Any positive real polynomial $p$ can be written as $p(x)=x|u(x)|^2+|v(x)|^2$ where $u$ and $v$ are two complex polynomials
Suppose $p:\mathbb{R}\to\mathbb{R}$ is a polynomial such that $p(x)\geq 0$ for all $x\geq 0$. There exists complex polynomials $u$ and $v$ such that
$$
p(x)=x|u(x)|^2+|v(x)|^2.
$$
A naive attem... | That seems to be a result from Pólya–Szegő, I found the following proof in VICTORIA POWERS AND BRUCE REZNICK, POLYNOMIALS THAT ARE POSITIVE ON AN INTERVAL, TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY, Volume 352, Number 10, Pages 4677–4692, Proposition 2.
Let $\Sigma \subset \Bbb R[x]$ denote the set of all polyn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3572764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that f disappears at n points Let $a, b \in \mathbb{R}$ with $a<b$ and $f:[a,b] \rightarrow \mathbb{R}$, and that there's some point $x \in [a,b]$ with $f(x)$ nonzero. If there exists $n \in \mathbb{N}$ such that for all $k\leq n, \int_{a}^{b} t^kf(t)dt = 0$. I need to show that there are $n+1$ distinct points ... | From the problem statement, we believe that $f$ should be continuous for the question to make sense. Furthermore, $f$ has to be non-zero (otherwise $f$ vanishes everywhere but changes signs nowhere).
Let $t\in (a,b)$ and suppose that $g:[a,b]\to\Bbb R$ is a non-zero continuous function on $[a,b]$ s.t. $g(t)=0$. Set... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3572963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A linear transformation such that $T(AB)=T(BA)$ The question goes as follows:
Let $V$ be a vector space and let $T: M_{2 \times 2} (R) —> V$ such that $T(AB)=T(BA)$ for all $A, B \in M_{2 \times 2}$. Show that $T(A) = 1/2(trA)T(I2)$ for all $A \in M_{2 \times 2}$.
I have no clue how to approach this. I’ve tried every... | First,we know that $T$ is a linear transfromation.
Then,we just need to considering a basis of $M_{2\times 2}$.There we choose a basis as following.
$$\left(\begin{array}{c}1 & 0\\0& 0\\\end{array}\right),\left(\begin{array}{c}0 & 1\\0& 0\\\end{array}\right),\left(\begin{array}{c}0 & 0\\1& 0\\\end{array}\right),\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3573125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to evaluate $C$?
Let B be the unit ball in the plane and let $u$ be a solution of the boundary
value problem:
$∆u = C$ in $B$
$\frac{∂u}{∂n} = 1 $ on $ ∂B$
where $∆ $denotes the Laplace operator, ∂B denotes the boundary of $B$ and
$\frac{∂u}{∂n}$ denotes the outer normal derivative on the boundary. Evaluate $C... | Very close, to wit:
The divergence theorem--AKA Green's identity--states
$\displaystyle \int_B \nabla \cdot \nabla u \; dA = \int_{\partial B} \dfrac{\partial u}{\partial n} \; dS, \tag 1$
$dA$ and $dS$ being the area and length elements on $B$ and $\partial B$, respectively. Given that
$ \dfrac{\partial u}{\partial n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3573218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $x=y=z$ where $\cos^{-1} x+ \cos^{-1} y + \cos^{-1} z = \pi$ Given that $$\cos^{-1} x+ \cos^{-1} y + \cos^{-1} z = \pi.$$ Also given that $$x+y+z=\frac{3}{2}.$$ Then prove that $x=y=z.$
My attempt: Let us assume $$\cos^{-1} x=a,\> \cos^{-1} y =b, \> \cos^{-1} z=c.$$ Then we have $$a+b+c=\pi \implies a+b = \pi... | Another way to solve the same could be $\cos^{-1}x=A$, $\cos^{-1}y=B$ and $\cos^{-1}z=C$
and thus $A+B+C=\pi$ and the condition becomes $\cos A+\cos B+\cos C=\frac{3}{2}$,
which can be simplified to $2\cos\frac{A+B}{2}\frac{A-B}{2}+1-2\sin^2{\frac{c}{2}}=\frac{3}{2}$, and we know $\cos\frac{A+B}{2}=\cos\frac{\pi-C}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3573374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Understanding 'trivial' step in calculating the graded cohomology ring $H^*(\mathbb{R}P^n; \mathbb{Z})$ I have a text that says it is obvious that $H^*(\mathbb{R}P^1; \mathbb{Z}/2)$ is isomorphic to $\mathbb{Z}/2[x]/x^2$ where $x$ is of degree $1$. I do not understand why this is true. The cohomology modules are $H^0(\... | I think your description of $\mathbb{Z}/2[x]/(x^2)$ is incorrect. It is a polynomial ring over $\mathbb{Z}/2$ with one variable, whose square vanishes. Explicitly, this ring consists of only the polynomials $p(x) = a + b x$ where $a, b\in \mathbb{Z}/2$, because $x^2 = 0$. As a graded ring this is simply $\mathbb{Z}/2 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3573577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Using the Monotone Convergence Theorem Let $(X, \mathbb{F}, \mu)$ be a finite measure space. If $f$ is measurable, let $E_n = \{x \in X: (n-1) \leq |f(x)| < n\}$. Show that $f$ is integrable if and only if $\sum_{n=1}^{\infty} n\mu(E_n) < \infty$.
I proved the above using the monotonicity of the integral. However, is t... | Observe that: $$\sum_{n=1}^{\infty}(n-1)1_{E_n}\leq |f|\leq \sum_{n=1}^{\infty}n1_{E_n}$$Taking the integral on both sides leads to:$$\sum_{n=1}^{\infty}n\mu(E_n)-\sum_{n=1}^{\infty}\mu(E_n)=\sum_{n=1}^{\infty}(n-1)\mu(E_n)\leq\int |f|\;d\mu\leq \sum_{n=1}^{\infty}n\mu(E_n)$$where $\sum_{n=1}^{\infty}\mu(E_n)=\mu(X)<\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3573679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How is this ODE solution correct? I just want to solve $\frac{dy}{dt} = 2y^2.$ Should be easy right? The answer should be $y=\frac{-1}{2t_0+2t}.$ Somehow this doesn't work when I'm trying to solve my problem, somehow what works is $y=\frac{y_0}{1-2y_0t}$ and I have no idea why. The only possible explanation can be trac... | $$\int \frac{dy}{y^2} = \int 2 \, dt$$
$$-\frac{1}{y} = 2t + \textrm{const.}$$
$$y=\frac{1}{\textrm{const.}-2t}$$
If $y(0)=y_0$, then $\textrm{const.}=\frac{1}{y_0}$ and
$$y= \frac{1}{\frac{1}{y_0}-2t}=\frac{y_0}{1-2 y_0 t}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3574135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
An Uncountable Subset of a Topological Space with a Countable Base $\Rightarrow$ M contains a limit point Please provide hints and not a direct answer as I would like to figure this out myself.
Prove that if $M$ is an uncountable subset of a topological space with a countable base, then some point of $M$ is a limit poi... | Hint: if $x \in M$ is not a limit point of $M$, then there is an open set $A_x$ in the given base such that $A_x \cap M = \{x\}$. If no point of $M$ is a limit point of $M$, what can you say about the sets $A_x$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3574289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Any shortcuts for integrating $\frac{x^6}{(x-2)^2(1-x)^5}$ by partial fractions? Is there a faster way to get the partial fraction decomposition of this $\frac{x^6}{(x-2)^2(1-x)^5}$?
$\frac{x^6}{(x-2)^2(1-x)^5} = \frac{A_1}{x-2} + \frac{A_2}{(x-2)^2} + \frac{B_1}{1-x} + \frac{B_2}{(1-x)^2} + \frac{B_3}{(1-x)^3} + \frac... | The "cover-up" rule gives correct answers but it suffers greatly from lack of rigour, in fact no rigour at all. Here's a more mathematically proper way to find the coefficients.First multiply through by $$(x-2)^2(1-x)^5$$ Your new equation is
$$x^6=A_1(x-2)(1-x)^5+A_2(1-x)^5+B_1(1-x)^4(x-2)^2+B_2(1-x)^3(x-2)^2+B_3(1-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3574400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluate using differentiation under the sign of integration: $\int_{0}^{\pi} \frac {\ln (1+a\cos (x))}{\cos (x)} dx$ Evaluate by using the rule of differentiation under the sign of integration $\int_{0}^{\pi} \dfrac {\ln (1+a\cos (x))}{\cos (x)} \textrm {dx}$.
My Attempt:
Given integral is $\int_{0}^{\pi} \dfrac {\ln(... | Integrating further is not actually cumbersome, let $tan(\frac{x}{2})=t\\\implies dx=\frac{2dt}{1+t^2}$
The above follows from basic trig identities..
Thus, on changing the limits, we have $$\frac{dF(a)}{da}=\int_0^{\infty}\frac{2dt}{(1-a)t^2+(a+1)}$$
The antiderivative is given by (this is a pretty standard integral..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3574602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the minimum value of $x^2+y^2$, where $x,y$ are nonnegative integers and $x+y=k$. Question: Let $k$ be a fixed odd positive integer. Find the minimum value of $x^2+y^2$, where $x,y$ are nonnegative integers and $x+y=k$.
My approach: After trying some examples I can conjecture that, the minimum value of $x^2+y^2$ i... | Try this:
$$
\begin{aligned}
x^{2}+y^{2}&=\frac{(x+y)^{2}+(x-y)^{2}}{2}
\end{aligned}
$$
Since $(x+y)$ is fixed, we minimize $x^{2}+y^{2}$ by minimizing $|x-y|$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3574774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Solve indefinite integral $ \int \frac{x^4}{\sqrt{x^2+4x+5}}dx $ I need to solve the next problem:
$$
\int \frac{x^4}{\sqrt{x^2+4x+5}}dx
$$
I know the correct answer is
$$
(\frac{x^3}{4}-\frac{7x^2}{6}+\frac{95x}{24}-\frac{145}{12})*\sqrt{x^2+4x+5}\space+\space\frac{35}{8}\ln{(x+2+\sqrt{x^2+4x+5})}+C
$$
still, I canno... | Also what might help you out is to get the CRC Handbook for mathematics. It contains over a hundred different derivatives, and integrals that will help you out with solving.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3575167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Tessellated space defines a recursive set? Is a space which has a regular geometric pattern necessarily a recursive set?
It's obvious, for example, that $\mathbb{Z}^3$ is a recursive set and it has a "regular geometric pattern", so this motivated me to ask is every tessellated set recursive. I don't have a precise ... | It actually takes some work to make the ideas in the post precise - keep in mind that equality checking for reals is not recursive (according to the standard model of computation, anyways). So a bit of circumlocution is needed.
That said, there is a positive result here. The key is the following lemma:
Suppose $T$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3575335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to compute $\lim\limits_{n\to\infty}\frac{1}{\sqrt{4n^2-1^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$. I want to compute $$\lim_{n\to\infty}\sum_{k=1}^n \frac1{\sqrt{4n^2-k^2}}.$$
I found it on this question and the exercise appears to be from previous years of a Latvian competition.
I tried writing $\frac1{\sqrt{4n^2-k^2... | Since$$\sum_{k=1}^n\frac1{4n^2-k^2}=\sum_{k=1}^n\frac1n\times\frac1{\sqrt{4-\left(\frac kn\right)^2}},$$this is a Riemann sum. The limit is $\displaystyle\int_0^1\frac{\mathrm dx}{\sqrt{4-x^2}}=\frac\pi6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3575483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
integral of spectral measure I have the following question: let $(X,\mathcal B,\mu)$ be a finite measure space and consider the operator $T_{\varphi} \colon L^2(X,\mu)\to L^2(X,\mu)$ given by $Tf(x)=\varphi(x)f(x)$, where $\varphi \in L^{\infty}(X,\mathcal{S},\mu)$. Consider the canonical spectral measure E induced by ... | Note that $\sigma(T_\varphi)=\operatorname{ess ran}\varphi$. Given $x\in X$, there exists $j$ with with $\varphi(x)\in M_j$. Then $|\varphi(x)-z_j|<\varepsilon$. Thus $x\in\varphi^{-1}(M_j)$ and then
$$
|\varphi(x)-\sum_kz_k\, 1_{\varphi^{-1}(M_k)}(x)|=|\varphi(x)-z_j|<\varepsilon.
$$
So
\begin{align}
\|T_\varphi f-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3575599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Showing moderate decrease property in the real line for $f(z)=\frac{a}{a^2+z^2}$. Let $f(z) = \frac{a}{a^2 + z^2}$ for $a>0$. Then $f$ is holomorphic in the horizontal strip
$|\Im(z)| < a$. I would like to show that in if $|\Im (z)| < a/2$, we have some constant $A>0$ such that
$$|f(x+iy)| \le \frac{A}{1+x^2}$$ for al... | All you need is the following: If $c>0,$ then
$$\tag 1 \frac{1+x^2}{c+x^2}\, \text{is bounded on }\mathbb R.$$
Can you prove this?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3575703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to show that $f$ is surjective? Suppose that $f:M\to N$ is an immersion between smooth manifolds $M$ and $N$ of the same dimension where $M$ is compact and $N$ is connected. How to show that $f$ is surjective?
| You can show that $f(M)$ is both open and closed in $N$. Since $N$ is connected, this implies that $f(M)=N$.
Let's first check that $f(M)\subset N$ is closed. Since $M$ is compact and $f$ is continuous, also $f(M)\subset N$ is compact. Since $N$ is Hausdorff, this implies that $f(M)\subset N$ is closed.
Next, we check ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3575990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Are Riemann integrable functions on a closed and bounded interval continuous? I know continuous function and monotone functions are Riemann integrable, but I’m not sure if Riemann integrable functions are continuous?
| No, not at all. Here is the precise relation of continuity and Riemann-integrability:
Let $f: [a,b] \to \mathbb{R}$ be a bounded function.
Then $f$ is Riemann-integrable if and only if the set of points at
which $f$ is discontinuous has Lebesgue-measure $0$.
In particular, since countable sets have measure zero, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3576115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
generators and monoid homomorphisms Is it true that any homomorphism $f: M \to N$ between two monoids $M$ and $N$ maps generators of $M$ to generators of $N$? I am having trouble proving it to myself.
| It has no reason to be true :
If you consider the monoid on one generator, $M = <a>$. Note that $M$ is isomorphic to $\mathbb{N}$, associating to each word $w = aa\ldots a$ the number $n$ of $a$ in that word. We consider morphisms from $M$ to itself, then we can send $a$ to however many $a$'s we want
For instance, if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3576250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Do Homotopy Equivalent, Orientable n-Manifolds Have The Same Cohomology With Compact Support? Using Poincare duality, I believe that if $X$ and $Y$ are two orientable $n$-manifolds with $X\simeq Y$, then we should have $H^i_c(X)\cong H_{n-i}(X)\cong H_{n-i}(Y)\cong H^i_c(Y)$ for all $i$. However, in a comment to the qu... | His calculation of the compactly supported cohomology of the sphere minus 3 points is incorrect, it’s $H^1_c=\mathbb {Z}^2$ since it’s compactification is the quotient of three points on a sphere. The way to answer the original question is to just see that the one point compactifications of the spaces are not homeomorp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3576374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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