Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proof of Inverse Function Theorem, class k I was studying the Inverse Function Theorem, and I found this proof on the internet:
http://virtualmath1.stanford.edu/~andras/174A-2.pdf
In the proof, there is this line about $C^k$ functions:
If $F$ is $C^k$, $k > 1$, then $DF$ is $C^{k−1}$, hence $(DF)^{−1}$ is $C^{k−1}$, h... | You can express the derivative $D (F^{-1})$ of $F^{-1}$ in terms of $F$ and the derivative of $F$. If these are both differentiable then so is $D (F^{-1})$, just because it can be expressed through the composition of differentiable maps. The general case is just by induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3524900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f:[0,\infty)\to[0,\infty)$ is continuous & decreasing and $α>0$ s.t. $\int_{0}^\infty x^α f(x)dx<\infty$, $\lim_{x\to\infty}f(x)x^{α+1}=0$ Let $f: [0,\infty)\rightarrow [0,\infty)$ be a continuous and decreasing function. Suppose that exists an $\alpha>0$ such that $\int_{0}^{\infty} x^\alpha f(x) dx < \infty$. Pro... | Indeed suppose that the statement $\lim_{x\rightarrow \infty} f(x)x^{\alpha+1}=0$ is false. Then, by definition of the limit, there exists a $\varepsilon>0$ such that for every $X\in[0,\infty[$, there exists a $x\geq X$ such that $f(x)x^{\alpha+1}\geq\varepsilon$.
Hence we can construct a sequence $x_1,x_2,x_3,\dots$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3525021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Finding limit using polar coordinates Polar coordinates are often useful in finding the limits in the case where $(x,y) $ approaches (0,0). Is it possible to use polar coordinates to find
$$\lim_{(x,y)\rightarrow (1,1)} \frac {xy-y-2x+2}{x-1} $$
This limit is -1 by direct substitution, can I use polar coordinates
| For this case you would have to use $x=r\cos\theta+1,y=r\sin\theta+1$, giving
$$\lim_{r\to0}\frac{(r\sin\theta-1)r\cos\theta}{r\cos\theta}$$ where $\theta$ can vary arbitrarily. This indeed shows that the limit exists and is $-1$, but there is no benefit to switch to polar.
An alternative approach would be to use tran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3525417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $x^d \equiv 1 \pmod{p}$ has full number of roots, $d\mid p-1$? Let $p$ be a prime.
What I know is that $d\mid p-1$ then $x^d \equiv 1 \pmod{p}$ has the maximum possible number of roots, that is, $d$.
I am wondering if the converse holds.
I have searched for $d\not\mid p-1$ that still has $d$ many roots in the equat... | Yes, the converse holds.
We have $x^m \equiv 1 \bmod{p}$ iff $x^d \equiv 1 \bmod{p}$, where $d=\gcd(m,p-1)$.
Moreover, $x^d \equiv 1 \bmod{p}$ has exactly $d$ roots because $\mathbb F_p^\times$ is cyclic.
Therefore, $x^d \equiv 1 \bmod{p}$ has $d$ roots iff $d$ divides $p-1$.
| {
"language": "en",
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Positive matrix with integer eigenvalues Is there any way of creating a positive matrix which has integer eigenvalues? Each entry $a_{ij}$ of the matrix must be strictly greater than $0$. I get how to create a matrix with certain eigenvalues using diagonal matrices, but I do not know how to make sure the matrix is stri... | The $n \times n$ matrix with diagonal entries $b$ and off-diagonal entries $a$ has eigenvalues $b-a$ (with multiplicity $n-1$) and $b + (n-1) a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3525859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Let f be a real valued continuous function such that for all real x and for all t≥0, f(x)=f(x•e^t). Show that f is a constant function. My approach to solving the problem stated in the title is as follows:
f(x)=f(x•e^t)=y(say)
Differentiating both sides w.r.t x, I got
f'(x)=e^t•f'(x•e^t)
This implies
e^t=1 [since f(x)=... | For the function $e^x$ maps $[0,\infty]$ onto $[1,\infty]$. So given any $0<x<y$ we can choose $t$ such that $y=xe^t$ and then we have $f(x)=f(y)$. In other words we have $f(x)$ equals some constant $c_1$ on the positive reals (either $x=y,x<y$ or $y<x$).
Similarly, $f(x)$ is constant $c_2$ on the negative reals. But w... | {
"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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Closed unit ball of $X^*$ has extreme points. Let $X$ be a Banach space. I have to show that the closed unit ball of the dual space $X^*$ has extreme points.
I just used the Banach-Alaoglu and Krein-Milman theorems to prove this but I'm not sure if this is correct because we haven't specified a topology on $X^*$. Beca... | The concept of extreme points doesn't depend on which topology you choose. So we could restate Krein-Milman as follows:
Let $V$ be a vector space, and $K \subset V$ a nonempty convex set. Suppose there exists a locally convex topology $\tau$ on $V$ such that $K$ is compact with respect to $\tau$. Then $K$ has extre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3526163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$I_n=\int_0^1{\frac{x^n}{x^n+1}}$. Prove $I_{n+1} \le I_n$ for any $n \in \mathbb N$ $$I_n=\int_0^1{\frac{x^n}{x^n+1}}$$
Prove $\lim_{n\to\infty}{I_n} = 0$
Here is what I tried.
First, I rewrite $I_n$.
$$I_n=\int_0^1{1-\frac{1}{x^n+1}}=1 - \int_0^1{\frac{1}{x^n+1}}$$
Now the limit becomes:
$$L=1-\lim_{n\to\infty}\int_... | For any $\epsilon \gt 0$, $\int_0^1\frac{1}{x^n+1}dx\gt K_n(\epsilon)=\int_0^{1-\epsilon}\frac{1}{x^n+1}dx$.
For every $\delta \gt 0$, there exists an $N$, so that for all $n\gt N$, $x^n\lt \delta$.
In this case, $K_n(\epsilon)\gt\frac{1-\epsilon}{1+\delta}$. Since $\delta$ is arbitrarily small, $\lim_{n\to \infty}K... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3526292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Simplification of a sum of factorials I am having trouble simplifying the following series. The sum is as follows: $$\sum_{k=0}^{\infty} \frac{x^k}{k^2!}.$$
is it possible to simplify? My intuition is telling me that a simplification involving the exponential is possible. Help would be appreciated, thank you.
| I think I have a partial answer. Begin by recognizing the square factorial function can be broken up into two components:
$$
k^2 ! = k! \prod_{j=k+1}^{k^2} j
$$
The series is now:
$$
\sum_{k=0}^\infty \frac{x^k}{k!} \prod_{j=k+1}^{k^2} \frac{1}{j}
$$
The product can be expressed as
$$ \frac{\Gamma(k+1)}{\Gamma(k^2+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3526421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Inequality that holds for all $n\in \mathbb N$ and $x\in [0,1]$. I want to prove that
$$x(1-x)^n+x^n(1-x)\le \frac1{2n}$$
for $x\in[0,1]$ and any $n\in \mathbb N$ ($n\ge1$). I have plotted the function $$f(x)=x(1-x)^n+x^n(1-x)-\frac1{2n}$$ and (if I am not mistaken) I know that it holds with equality for $n=1,2$ and s... | We claim that
$$x^n(1-x)\leq \frac{x}{2n}.$$
This is equivalent to
$$x^{n-1}(1-x)\leq \frac{1}{2n}.$$
However, the maximum of the left side is attained when its derivative is $0$, as it is $0$ at both $0$ and $1$. This means
$$(n-1)x_0^{n-2}-nx_0^{n-1}=0\implies x_0=\frac{n-1}{n}.$$
So
$$x^{n-1}(1-x)\leq \left(\frac{n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3526576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Third order differential equation with variable coefficients I am wondering if there were known methods for solving this type of differential equations ?
$$x^2y'''+3xy''+2xy'+2y=0$$
Thank you in advance
| You can reduce the order of the DE:
$$x^2y'''+3xy''+2xy'+2y=0$$
$$(x^2y'')'+(xy')'-y'+2(xy)'=0$$
Integrate:
$$x^2y''+xy'+(2x-1)y=C_1$$
Now it's a second order DE. Then use reduction of order or series solution.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Order of a power series Let $f$ be the a power series over $\mathbb{C}$ of infinite radius. The order $\rho(f)$ of $f$ is defined to be the infinum of the following set :
$$ \{ A \ge 0 \mid \exists r_0\ge 0, \forall r\ge r_0, M_f(r) \le \exp(r^A)\}$$
Where $M_f(r)$ is the supremum of $f$ over all complex numbers of ... | You're on a good way, just the final estimate is missing. This is easier to do with the whole series than termwise. Let me however replace your $A$ with $1/\vartheta$ so that at the end I get exponents of $\vartheta$ instead of $1/A$. What we want to show is
$$\frac{1}{\vartheta} < \liminf_{n \to \infty} \biggl(\frac{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3527179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Surface area of an oblate spheroid using gaussian quadrature I want to compute the surface area of an oblate spheroid using gaussian quadrature, the parametrization of the oblate spheroid is given by:
$$x = a \cdot \sin\theta \cdot \cos \phi \\
y = a \cdot \sin\theta \cdot \sin \phi \\
z = b \cdot \cos\theta $$
Where $... | Even though you eventually got the right areal element you seem to have an unsteady method of deriving it. Since on the surface we have parameterized
$$\vec r=\langle x,y,z\rangle=\langle a\sin\theta\cos\phi,a\sin\theta\sin\phi,b\cos\theta\rangle$$
We can take the differential to get
$$d\vec r=\langle a\cos\theta\cos\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3527286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Sum (sigma) notation disparity this might be a stupid question but I've googled it and I can't find an answer that I "trust". I am dealing with PCA (I guess that's not relevant, but just in case it is) and I am seeing a lot of Sigma notation that I'm not used to. My whole life I've always seen the following:
$\sum_{i=1... | Basically, yes. It means that the index you are summing over is $i$, and presumably the domain for $i$ is stated elsewhere.
You might also see things like
$$ \sum_{a\in A} $$
or
$$ \int_A $$
etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3527392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given a recurrence formula, evaluate $\lim\limits_{n\to \infty} n^2 x_n^3$
Define a sequence $(x_n)_{n\geq 0}$ with a fixed initial term $x_0 > 0$ such that:
$$x_0 + x_1+\ldots+x_n=\frac{1}{\sqrt{x_{n+1}}}$$
Evaluate
$$\lim_{n\to \infty} n^2 x_{n}^3$$
My attempt: I should define a new sequence $s_n = \displaystyle\s... | Start with
$$
x_n=\left(\sum_{k=0}^{n-1}x_k\right)^{-2}\tag1
$$
Let $x_n=u_n^{-2}$, then we get
$$
u_n=\sum_{k=0}^{n-1}u_k^{-2}\tag2
$$
From $(2)$, we get
$$
u_{n+1}-u_n=u_n^{-2}\tag3
$$
Equation $(3)$ indicates that $u_n$ is increasing. If $u_n$ were bounded above, it would approach a finite limit, $\bar u$, and then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3527497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Counting (Combinations) Suppose there are 10 people apart of a club: A, B, C, D, E, F, G, H, I, and J. They decide to go to a restaurant for a club outing, but there isn't one table to seat all of them, so they decide to take one table that seats four people and two tables that seat three people. Based on this, what is... | By conditional probability, we have
$$P(J\text{ is at $G$’s table}) =
P(G\text{ at $4$-table})P(J\text{ at $G$’s table} | G\text{ at $4$-table}) + P(G\text{ at a $3$-table})P(J\text{ at $G$’s table}| G\text{ at a $3$-table})$$
$$\left(\frac4{10}\right)\left(\frac39\right)+\left(\frac6{10}\right)\left(\frac29\right) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3527698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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complete square of summation of odd numbers Suppose we have an integer(which may be odd or even), let us say for example $x=20$, also we have an odd number which is also known (let us say for example $y=39$).
Is there any formula that tell us how many consecutive odd numbers starting from $y$ that if we add them togeth... | Not an answer, but too much for a comment:
The sum of all the odd numbers below $y$ is $\left(\frac{y-1}2\right)^2$. If you add $n$ odd numbers starting with $y$ the greatest is $y+2n-2$ and the sum of all the odd numbers up to that is $\left(\frac{y+2n-1}2\right)^2$. You are then looking for
$$\left(\frac{y+2n-1}2\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3527806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that there is no negative integer with $n^2+n<0$ I'm trying to prove the statement: There is no negative integer with $n^2+n<0$. For this, I went with proving it by contradiction. Here's what I have so far:
Let us assume there is a negative integer $n$ with $n^2+n<0$. Since $n$ is negative, this means that $n<0... | In your argument it is not clear how you conclude that $n^{2}+n $ is at least $0$.
Let $m =-n$. Them $m >0$ and $n^{2}+n <0$ gives $m^{2} <m$. But then you can cancel $m$ and get $m <1$. there is no positive integer $m$ such that $m <1$ so we have a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3527937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Every compact Hausdorff space can be expressed as a disjoint union of finitely many open sets. Let $X$ be a compact Hausdorff space.
Can we express as disjoint union of open sets ?
I got a proof for that.
Since $X$ is compact there exist a finite sub collection of open sets $\{W_{i}\}_{i=1}^{n}$ that covers $X$. Con... | You can always do it with $n=1$ and $W_1=X$: if $X$ is connected, this is the one and only possibility. Otherwise, your procedure fails in general, your error lying in the fact that $\{W_i',\:\, i\in\Bbb N\}$ may not be a covering. Also, the entire idea of a non-trivial case $n\ge2$ fails catastrophically: again, see t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3528099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the difference between equality and logical identity? I'm reading the book Introduction to Logic and to the Methodology of the Deductive Sciences by Alfred Tarski and he states:
"In this book we consider the notion of equality among numbers always as a special case of the general concept of logical identity. On... | I expect that Tarski means by "logical identity" something close to "literally the same thing", which could be applied to numbers in arithmetic, sets in set theory, or anything else.
Now imagine a mathematician X (not Tarski) who might say something like "a rational number is a pair of integers called numerator and de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3528250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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If $|z-\frac 3z|=2$, Find the greatest value of $|z|$ $$|z^2-3|=2|z|$$
And $$|z^2-3|\le |z|^2+3$$
$$2|z|\le |z|^2+3$$ which isn’t a valid equation, what’s going wrong?
| Your inequality $2|z|\le|z|^2+3$ is very much valid. For example, $|z|=2$ satisfies it.
The maximum value of $|z|$ that allows this inequality is the maximum root of the corresponding equality $2|z|=|z|^2+3$, which can be solved by the usual methods for quadratic equations to give $|z|=3$.
Thus $|z|=3$ is an upper bou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3528430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating $\lim_{n \to \infty}\frac{1*2+2*3+...+n(n+1)}{1*3+3*5+...+(2n-1)(2n+1)} $ This might seem pretty straightforward, as both the numerator and the denominator seem to resemble some well-known sums, however the numerator is the only one which can actually be rewritten as $\frac{n(n+1)(n+2)}{3}$. Could you point... | Method one
We can write the summation formula for both the numerator and denominator which is not my favorite :)
Method two
We use calculus (and the definition of integral) to find the limit.
$$
\lim_{n\to\infty}\frac{1*2+2*3+...+n(n+1)}{1*3+3*5+...+(2n-1)(2n+1)}
{=
\lim_{n\to\infty}\frac{\sum_{i=1}^ni(i+1)}{\sum_{i=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3528528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Equivalence classes and subgroups of bijections of $S_3$
Consider the group $S_3 = Sym({1,2,3})$, which we recall is the group of all bijections from $\{1,2,3\}$ to $\{1,2,3\}$ and where the group operation is composition. We have seen that this group has order 6 and is not abelian. We let
S3 equal this set
For inst... | Let's write out the elements of $S_3$. They are $\{(12), (23), (13), (123), (132),e\}$. (In the picture they are $a_1=(12), a_2=(23), a_3=(13), a_4=(132)$ and $a_5=(123)$.)
Now, it's clear that there are $3$ equivalence classes. They are $[(12)]=\{e, (12)\}, [(13)]=\{(123),(13)\}$ and $[(23)]=\{(132), (23)\}$. Only... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3529013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definition of a limit between functions in metric spaces: Rudin vs Amann-Escher - excluding a limit point Amann-Escher in Analysis I define a limit of a function between metric spaces as follows:
Let $X,Y$ be metric spaces, $D \subseteq X$ and $a \in X$ a limit point of $D$. Let $f\colon D\to Y$ be a function. We writ... | In Amann -Escher definition if $x_n \to a$ (and $a \in D$) then $(x_1,a,x_2,a,x_3,a...)$ also converges to $a$. So this forces the limit of $f(x_n)$ to be $f(a)$. So this becomes definition of continuity at $a$ rather than existence of the limit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3529140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a closed form for the sequences mentioned in the body of this question? Is there any hope for closed forms for expressions like: $$\sum_{k=0}^{\infty}\frac{x^k}{(3k+i)!}\text{ and/or }\sum_{k=0}^{\infty}\frac{kx^k}{(3k+i)!}$$where $i\in\{0,1,2\}$?
I am interested because I am trying to find an answer to this q... | Considering
$$f_i=\sum_{k=0}^{\infty}\frac{x^k}{(3k+i)!}\qquad \text{and} \qquad g_i=\sum_{k=0}^{\infty}\frac{kx^k}{(3k+i)!}$$
a CAS gives
$$f_i=\frac{1}{i!}\,\,
_1F_3\left(1;\frac{i+1}{3},\frac{i+2}{3},\frac{i+3}{3};\frac{x}{27}\right)$$
$$g_i=\frac{x}{(i+3)!}\,\,
_1F_3\left(2;\frac{i+4}{3},\frac{i+5}{3},\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3529254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Better method for solving ODE $yy''+ (y')^2-(y')^3\ln (y)=0$ What's a better way to solve this equation?
$$yy''+ (y')^2-(y')^3\ln (y)=0 $$
so far, I've tried:
*
*reducing the order of the equation ($p=y'$): $yp'+ p^2-p^3ln(y)=0 $
*dividing everything by $p^3$: $y\frac{p'}{p^3} + \frac{1}{p}=ln(y)$
*solving the ho... | As I stated in the comments, use the fact that
$$y y'' + y'^2 = \frac{d}{dx} (y y')$$
Then
$$\frac{d}{dx} (y y') = y'^3 \log{y} $$
This may be rewritten as
$$\frac{d(y y')}{(y y')^2} = \frac{\log{y}}{y^2} y' dx = \frac{\log{y}}{y^2} dy$$
This equation may be integrated to produce
$$-\frac1{y y'} + C_1 = -\frac{1+\log{y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3529705",
"timestamp": "2023-03-29T00:00:00",
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} |
Prove Quotient Ring is Isomorphic to Product of Fields
Problem: (a) Show $\mathbb{Z}[i]/\langle 5\rangle$ is a product of two fields. (b) Show $\mathbb{Z}[i]/\langle 3\rangle$ is a field. (c) Show $\mathbb{Z}[i]/\langle 2\rangle$ is neither a field nor a product of two fields.
My Attempt: Now from first glance, it is... | Hint:
a) In $\mathbf Z[i]$, one has $5=(2+i)(2-i)$ and these factors are irreducible since their norm is equal to $5$. Furthermore, $\mathbf Z[i]$ is a P.I.D., so each generates a maximal ideal, and you can apply the Chinese remainder theorem:
$$\mathbf Z[i]/5\mathbf Z[i]\simeq \mathbf Z[i]/(2+i)\times\mathbf Z[i]/(2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3529873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that $\frac{ 1- \sin\frac{5\pi}{18}}{\sqrt{3} \sin \frac{5\pi}{18}}= \tan\frac{\pi}{18} $ It's easy to verify on WolframAlpha, but I have difficulty deriving it.
It's easy to see
$$
\tan\left(\frac{\pi}{18}\right)=\frac{\sqrt{3}-\tan(5/18 \pi)}{1+\sqrt{3}\tan(5/18 \pi)}
$$
| We need to prove that $$(1-\sin50^{\circ})\cos10^{\circ}=2\sin50^{\circ}\sin10^{\circ}\cos30^{\circ}$$ or
$$2\cos10^{\circ}-\sin60^{\circ}-\sin40^{\circ}=2(\cos40^{\circ}-\cos60^{\circ})\cos30^{\circ}$$ or
$$2\cos10^{\circ}-\sin40^{\circ}=2\cos40^{\circ}\cos30^{\circ}$$ or
$$2\cos10^{\circ}-\sin40^{\circ}=\cos70^{\circ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3530024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluate $ \lim_{n\to\infty} \frac{n^n}{3^n} a_n$ For
$$ f(z)= \sum_{n=0}^{\infty } a_n z^n $$
with
$$ |f(z)| \leq M e^{|z|}, $$
calculate
$$ \lim_{n\to\infty} \frac{n^n}{3^n} a_n$$
Tried several methods but I don't know how to approach this correctly, thanks for your time.
| This is for $f$ being an entire function. By Cauchy's formula
$$
a_n = \frac{1}{{2\pi i}}\oint_{\left| t \right| = r} {\frac{{f(t)}}{{t^{n + 1} }}dt} ,
$$
with some $r>0$. Thus,
$$
\left| {a_n } \right| \le \frac{1}{{2\pi }}\oint_{\left| t \right| = r} {\frac{{\left| {f(t)} \right|}}{{\left| t \right|^{n + 1} }}\left|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3530149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof review/explanation: Let $\text{char}(\mathbb{K}) = 0$. It then follows that $AB-BA \ne 1 \, (A, B \in \Bbb K^{n \times n})$
Let $\text{char}(\mathbb{K}) = 0$. It then follows that $AB-BA \ne 1 \, (A, B \in \Bbb K^{n \times n})$.
I first showed that $\text{trace}(AB) = \text{trace}(BA)$ for every $A, B \in \Bbb ... | If $\text{char}(\mathbb{K})=p$ and $p \mid n$, then you get $0=0$, which doesn't give you a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is $\det(I + A^{50}) = 4$ here? I’m trying to understand the reasoning behind the following:
Let $A \in \mathbb{R}^{3\times3}$ with eigenvalues $1$, $-1$, $0$. What is $\det \left(I + A^{50} \right)$?
The given answer is 4.
A is obviously singular due to eigenvalue 0. Moreover, spectral decomposition of $A$ exist... | As you said, you can diagonalize $A=PDP^{-1}$ with $D=\operatorname{diag}(1,-1,0)$. Then $A^{50}=PD^{50}P^{-1}$. $D^{50}=\operatorname{diag}(1,1,0)$.
So,
$$\det(I+A^{50})=\det(P^{-1}(I+A^{50})P)=\det(I+D^{50})=\det(\operatorname{diag}(2,2,1))=4.$$
The first equality follows from the fact that determinants are invaria... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Finding a constant to make this function periodic So I have the function $$ f(x) = x^2 + b $$ which is $ 2\pi $ periodic for $ 0 \leq x < 2\pi $
If $ F $ is the antiderivative of $ f $ with $ F(0) = 1 $, what value of $ b $ is $ F $ periodic?
So I found that $$ F(x) = \frac{1}{3}x^{3}+bx+1 $$ so in order to prove some... | Note that $f(x)=x^2+b$ only in the interval $0\leq x<2\pi$. For all other $x$ we have to compute first the "fractional part mod $2\pi$ of $x$". In other words we have
$$f(x):=\left(x-2\pi\left\lfloor{x\over2\pi}\right\rfloor\right)^2+b\qquad(-\infty<x<\infty)\ .$$
The antiderivative $F$ is obtained by inserting this $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3530616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cubic equation with three distinct roots Let
$$g(x)=ax^3+bx^2+cx+d$$
be a polynomial of degree $3$ with $a,b,c,d,e\in\mathbb{R}$ and $a>0$. Suppose that the cubic equation $g(x)=0$ has three distinct real roots, i.e. the discriminant $\Delta>0$.
Let $f(x)=\frac{a}{4}x^4+\frac{b}{3}x^3+\frac{c}{2}x^2+dx$. Can we expres... | So, (lots of this is on the Wikipedia page linked in above comments), if we set $$x=t-\frac{b}{3a} ..[1]$$ then $g(x)$ becomes $G(t)=t^3+pt+q$ where $p=\frac{3ac-b^2}{3a^2}$ and $q=\frac{2b^3-9abc+27a^2d}{27a^3}$.
Provided $4p^3+27q^2<0$, it has three real roots given by setting $k=0, 1, 2$ in $$t_k=2\sqrt{\frac{-p}{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3530757",
"timestamp": "2023-03-29T00:00:00",
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Sum of two perfect squares is also a perfect square. Proof that one of these numbers is divisible by 3 At first, I tried to proof by contradiction: I considered two numbers that are not divisible by $3$. Than I tried to write consecutive perfect squares that are divisible by $3$ to see some pastern, but it didn't get m... | HINT. Write $c^2=a^2+b^2$ and consider the equation $\mod 3$, remembering that $x \equiv 0 \mod 3$ implies that $x$ is divisible by $3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3530934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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write down the transition probability matrix for a certain pattern Suppose we're generating a sequence of 0s and 1s. We want a specific pattern 0011, and we are given this transition matrix with arbitrary probabilities
$$
\begin{bmatrix}
p & q \\
q & p \\
\end{bmatrix}
$$
where the first column and ... | the first 2x2 matrix tells that you switch symbols (from 0 to 1 or from 1 to 0) with probability $q$.
The second 5x5 matrix provides a path towards state 0011. Let us analyze it more carefully:
*
*from state "1", you switch to "0" with probability $q$
*from state "0", you switch to state "00" with probability $p$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3531094",
"timestamp": "2023-03-29T00:00:00",
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Why is length often a unitless quantity in mathematics? This is something that's bothered me for a long time: Why is length often not given any units (e.g. inches or meters) in mathematics? The unit circle is said to have a "radius of 1," with no unit given. A vector is said to have a magnitude of, say, 27, with no uni... | Distance in a Cartesian coordinate plane is a function which inputs two points $p_1 = (a_1,b_1)$ and $p_2 = (a_2,b_2)$ and outputs their distance $d(p_1,p_2) = \sqrt{(a_1-a_2)^2 + (b_1-b_2)^2}$, which is a real number. So you can think of the unit of distance abstractly as the numeral $1$, which is the multiplicative i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3531200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove or disprove the statements about linear map Let $m,n\in \mathbb{N}$ and let $V, W$ be $\ \mathbb{R}$-vector spaces with $\dim V=n$ and $\dim W=m$.
Let $f:V\rightarrow W$ be a linear map.
I want to prove or disprove the following:
There is a linear map $f:V\rightarrow W$ with the following properties:
*
*$f... | Since these are linear maps between finite dimensional vector spaces $V_n \longrightarrow W_m$ so each of them (if it exists) can be represented by a $m \times n$ matrix $A$.
For (1): Here $A$ must be $4 \times 3$ matrix. For injective map, no free columns in $A$, so take
$$A_{4 \times 3}=\begin{bmatrix}1&0&0\\0&1&0\\0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3531296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For any ring $R$, prove that $R$-$\mathbf{Mod}$ has no subobject classifier. This is Exercise I.3 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]".
The Question:
For any ring $R$, prove that the category $R$-$\mathbf{Mod}$ of all left $R$-modules has no subobject classifier.
I assume that the morph... | The terminal and initial object is the $0$-module, $\{0\}$. Addition/multiplication with elements in $R$ given bu the only way possible. Consider $S = 0$. Then we get that $\ker (\phi) = 0$ and every $X$ embeds into $\Omega$. A morphism with zero kernel in $R$-$\mathbf{Mod}$ has to be a monomorphism and because right a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Linear isomorphism $f:U_1\to U_2$ with $U_2\subset U_1$ Let $V$ be a finite-dimensional vector space and $U_1,U_2\subset V$ be subspaces of $V$ such that $U_2\subset U_1$ and let $f:U_1\to U_2$ be a linear isomorphism. Prove (or give a counterexample) whether the following holds: $$U_1=U_2$$
I can't think of a countere... | Hint:
Since we have finite dimensional spaces,
$$\dim (U_1/U_2)=\dim U_1 -\dim U_2. $$
Now, since $U_1$ and $U_2$ are isomorphic, they have the same dimension.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3531661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Improper Integral - If there is an asymptote, does it necessarily have to diverge? Context: After learning that the harmonic series diverges, I have begun to doubt just merely looking at graphs
Here's an improper integral with two vertical asymptotes: $x=0$ and $x=0.5$.
$$\int_{0}^1\frac{1}{2x^2-x}dx$$
It diverges.... | I assume by "diverges" you mean is not absolutely (and so Lebesgue) integrable? Because in this case you can just compute the integral of its absolute value
$$ \int_0^\frac{1}{2} \frac{1}{x-2x^2} + \int_\frac{1}{2}^2 \frac{1}{2x^2-x} $$
and notice that, for instance, the second integrand is $+\infty$. Because the integ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\prod_{k=1}^\infty (1-1/2^k)$ converge to zero? I know this product converge
$$\prod_{k=1}^\infty (1-1/2^k),$$
but I don't know how to prove that this limite is different to zero. This is equivalent to prove that
$$\sum^{\infty}_{k=1} \log(1-1/2^k)$$ converge. I can't get it. Could you give me any hint? Thank!
I'm tr... | $\log(1-x)$ is a concave function on $[0,1)$, so for any $x\in[0,1/2]$ we have
$$ \log(1-x)\geq -2\log(2) x $$
immediately implying
$$ \sum_{k\geq 1}\log\left(1-\frac{1}{2^k}\right) \geq -2\log(2)\sum_{k\geq 1}\frac{1}{2^k} = -2\log(2) $$
and
$$ \prod_{k\geq 1}\left(1-\frac{1}{2^k}\right) \geq \frac{1}{4}.$$
Much bette... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove $\sin x + \arcsin x > 2x$ using Maclaurin series My teacher asked us to solve this problem using the Maclaurin series, but I could not figure out how to approach..
Prove that the inequality sin x + arcsin x > 2x holds for all values of x such
that 0 < x ≤ 1.
I know that the Maclaurin series of
sin(x) = x - $\fra... | For $0\le x\le1$ we have
$$\sin x\ge x-{1\over6}x^3\ge0\quad\text{and}\quad\arcsin x\ge x+{1\over6}x^3+{3\over40}x^5\ge0$$
which imply
$$\sin x\arcsin x\ge x^2+\left({3\over40}-{1\over36} \right)x^6-{1\over80}x^8=x^2+\left(34-9x^2\over720\right)x^6\ge x^2$$
By AGM we have
$${\sin x+\arcsin x\over2}\ge\sqrt{\sin x\arcsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3532201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's the probability that you wait $2$ hours for the train when you have already waited $1$ hour?
Let's say there are independent waiting times of trains which are
exponentially distributed with mean value $\frac{1}{2}$ hours. If you
have already waited $1$ hour for the train, what's the probability
that you w... | Let $T_n\stackrel{\mathrm{i.i.d.}}\sim\mathrm{Expo}(\lambda)$. Define $S_0=0$ and $S_n=\sum_{i=1}^n T_n$. Then $S_n$ is a renewal process, in fact a Poisson process, with associated counting process $N(t) = \sup\{n: S_n\leqslant t\}$. Define the age process by $A_t = t - S_{N(t)}$ and the residual process by $R_t = S_{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help Proving Linear Regression Model in Matrix Notation I need to show that:
$\hat \mu ^{'} \hat \mu = \hat \mu ^{'} y$
Given the matrix notation of the linear regression model: $y= X\beta + \mu$
Also given: $\hat y = X \hat \beta $ and $\hat \mu = y - X\hat \beta $
I have tried:
$(y-X \hat \beta)^{'} (y-X\hat \beta)... | We know that $$X'(y-X\hat{\beta})=0\tag{1}$$
We want to show that $$\hat{\mu}'\hat{\mu}=\hat{\mu}'y$$
which is equivalent to
$$(y-X\hat{\beta})'X\hat{\beta}=0$$
Let's take the transpose,
$$\hat{\beta}' \color{red}[X'(y-X \hat{\beta})\color{red}]=0$$
From $(1)$, we can see that the result is true.
Regarding your attemp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Logistic map confusion Logistic map is a simple example of discrtete dynamical systems defined as $$x_{i+1}=\lambda*x_{i}*(1-x_{i}).$$
It is known that for $\lambda=4$ this map shows a chaotic behavior for $x\in (0,1)$.
The question:
How could it be that the logistic map at $\lambda=4$ is chaotic, while, starting wit... | What is actually meant with "chaotic" is that almost all start values lead to a (theoretically) non-periodical sequence and the sequence shows no structure which means that it can be used as a pseudo-random-generator.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3532590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that :
$f(1)+f(-1)-2(f(0)+1)\equiv 0\pmod{2f(-1)}$ Problem :
$$f(t)=t^{3}+\alpha t^{2}+\beta t+\gamma , \alpha ,\beta \operatorname{and} \gamma \in\mathbb{Z}$$
with root $x_{1},x_{2},x_{1}x_{2}$
Question is :
Prove that :
$$f(1)+f(-1)-2(f(0)+1)\equiv 0\pmod{2f(-1)}$$
My try:
We have :
$$\begin{cases}x_{1}... | COMMENT.-I am afraid your question is not true. Choose, for instance, $(x_1,x_2)=(3,5)$ so $$f(t)=t^3-23t^2+135t+225$$ and $f(1)+f(-1)-2(f(0)+1)=-48$ and $2f(-1)=-768$. However if you interchange the modulus you have $48$ divides $768$ and maybe you could have a correct question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3532740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\int_0^1\ln^2\Gamma(x)\,\mathrm{d}x$ I want to solve the following integral but after some work I didn't find a way to go. Could anyone give me a hint?
\begin{equation}
I=\int_{0}^{1}\ln^2\Gamma(x)\,\mathrm{d}x
\end{equation}
The answer is
\begin{equation}
I=\frac{\ln^2 (2\pi)}{3}+\frac{\pi^2}{48}+\frac{\gamma... | Use Parseval's Theorem as @James Arathoon metioned and use the Fourier Series given here:
Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3532877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Wronskian of $x|x|$ and $x^2$. Wikipedia says wronskian of $x|x|$ and $x^2$ is identically zero.
But it is not LD.
I know why these two are LI and not LD.
since x|x| is not differentiable function,how to find their wronskian????
And plz suggest ways to check LI and LD when functions are not differentiable.
Thanks in a... | The function $f(x)=x|x|$ is is differentiable everywhere. For $x>0$, you have $f(x)=x^2$, differentiable. For $x<0$ you have $f(x)=-x^2$, differentiable. At $0$, you have
$$
\frac{f(h)-f(0)}h=\frac{h|h|}h=|h|\to0,
$$
so the derivative exists and is zero.
For two functions, using the Wronskian is overkill. Linear de... | {
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"timestamp": "2023-03-29T00:00:00",
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All squarefree semiprime numbers that less than a certain number How can i find all number $N$, where $N=p*q$. (Here, $p$ and $q$ are primes and $p<q$) ?
For example: I want a formula that help me to find all $N<1000$
| Apply the sieve of sundaram's logic, $$(2a+1)(2b+1)=2(2ab+a+b)+1$$ one of $a,b$ must less than 16, because if not $a=b$ gives $$4a^2+4a+1>1000$$ Then applying it again you'll need $c$ less than 3, because you can sieve out all values that would not create primes in the product. Finally you would double all primes le... | {
"language": "en",
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"source": "stackexchange",
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Volume between surfaces
Find $V(T),$ where $T$ is the region bounded by the surfaces $y = kx^2+kz^2$ and $z=kx^2+ky^2,$ where $k\in\mathbb{R}, k > 0.$
I tried solving for the area over which these curves intersect, which gave me $y-kz^2 = z-ky^2.$ Solving gives $y+ky^2 - (z+kz^2) =0\Rightarrow (y-z)(1 + k(y-z)) = 0.$... | Observe that the enclosed volume by $y = kx^2+kz^2$ and $z=kx^2+ky^2$ are symmetric with respect to the plane $y=z$. So, the total volume is twice the volume between the surfaces $y=z$ and $z=kx^2+ky^2$.
Recognize that the integration region in $xy$-coordinates is the circle given by
$$x^2+ \left(y -\frac{1}{2k}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3533481",
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Given a positive, finite, regular measure $\lambda$ and $g\in L^1(\lambda)$, the measure $\mu$ given by $\mu(E)=\int_E g~d\lambda$ is regular Suppose $\lambda$ is a positive, finite, regular measure, and suppose $g \in L^1(\lambda)$. Define a measure $\mu$ by letting $\mu(E)=\int_E g~d\lambda$. Then is it true that $\m... | Let $\epsilon >0$. Since $\mu << \lambda$ there exists $\delta >0$ such that $\lambda (E) <\delta$ implies $|\int_E gd\lambda| <\epsilon$. Let $E$ be any Borel set and $K$ be a compact subset of $E$ with $\lambda (E\setminus K) <\delta$. Then $|\mu (E\setminus K)| \leq \int_{E\setminus K} |g| d\lambda <\epsilon$.
| {
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How many words can be formed out of the letters of the word GRANDMOTHER, such that each word starts with G and ends with R? This is a problem from a specimen question paper:
"How many words can be formed out of the letters of the word GRANDMOTHER, such that each word starts with G and ends with R?"
My problem is that t... | As you say, the problem is not particularly clear. What you have is correct if you are supposed to use all the letters: once you fix the G and R at the start and end, all the $9$ internal letters are different, so it is just the number of ways to permute them.
However, another possible interpretation is that you have ... | {
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Approach the covariance matrix I have this problem:
Let the random vector $\binom{X}{Y}\sim
N_{2}(\binom{0}{0},\bigl(\begin{smallmatrix} 1 & \rho\\ \rho & 1 \end{smallmatrix}\bigr))$
1) Find the distribution of $Z=X+Y$.
2) Find the distribution of $W=X^2$, the mean and the variance.
3) Calculate $Cov(X,W)$ and $Cov(Z... | It is given that $X$ and $Y$ are jointly normal, $EX=EY=0$, $EX^{2}=EY^{2}=1$ and $cov(X,Y)=\rho$ which gives $EXY=\rho$.
$X+Y$ is normal with mean $0$ and its variance is $E(X+Y)^{2}=EX^{2}+EY^{2}+2EXY=1+1+2\rho$.
$P(W \leq w)=P-\sqrt w \leq X \leq \sqrt w)-\int_{-\sqrt w} ^{\sqrt w} f(x)dx$ where $f$ is the standard... | {
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An expression for $\lim_{n\to\infty}\frac1{2^n}\left(1 + x^{1/n}\right)^n$
I am looking for a closed form answer to the limit ($x<1$): $$\lim_{n\to\infty}\frac1{2^n}\left(1 + x^{1/n}\right)^n$$
For context, I was studying weighted averages and considered $$\left(\frac12(x^{1/n} + y^{1/n})\right)^{n}$$ to be a good w... | This is a bit of a hand-wavy argument but I thought it was fun so I'll share it. Let $f_n(x)=\dfrac{\left(1+x^{\frac{1}{n}}\right)^{n}}{2^{n}}$, and assume $\lim\limits_{n\to\infty}f_n(x)=f(x)$ exists, and that the limit of derivatives, $\lim\limits_{n\to\infty}f_n'(x)$, exists and is equal to $f'(x)$. Then,
$$\begin{a... | {
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Is this alternative representation of $f(x)=xe^x$ as Maclaurin series correct? Let $f(x)=xe^x$. I know
$$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$$
and so
$$f(x)=x\sum_{n=0}^\infty \frac{x^n}{n!}=x(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...)=x+x^2+\frac{x^3}{2!}+\frac{x^4}{3!}+...=\sum_{n=0}^\infty \frac{x^{n+1}}{n!}$$
But sh... | Indeed $f^{(n)}(0)=n$. To make the connection, note that in the first case the first term is zero, so you have $\sum_{n=1}^\infty \frac{1}{(n-1)!} x^n$, and now you can shift the index to start at $0$ again by replacing $n$ with $n+1$ everywhere. This gives $\sum_{n=0}^\infty \frac{1}{n!} x^{n+1}$ as you expected.
| {
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Let $V$ be a vector space. If $U \leq V$, $Z \leq V$, $U \cap Z = 0$ and $Z \simeq V/U$, then $V = U \oplus Z$. Let $V$ be a vector space (possibly of infinite dimension).
I know that given a subspace $U$ of $V$, we can always write $V = U \oplus Z$, where $Z$ is some subspace of $V$ such that $Z \simeq V/U$.
Let $U$ a... | The issue here is which map is giving you the isomorphism $Z\simeq V/U$. Let $\pi:V\to V/U$ denote the projection map.
Since every short exact sequence of vector spaces splits, there is a map $i:V/U\to V$ such that its composition with the projection above $i\circ\pi=id_{V/U}$ is the identity map. This gives you a way ... | {
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How many 7-digit phone numbers are possible, assuming that the first digit can’t be a 0 or a 1 I use the multiplication rule.
For the first digit I have 8 choices. For the last 6 digits I have 10 choices for each. So answer is $8 \cdot 10 ^6$.
Is there any other way to solve this problems. I usually gain a lot of insi... | number of 7 digit numbers ( including leading 0): 10,000,000
number of 7 digit numbers including lead 0 or 1 : - 2,000,000
number of 7 digit numbers not lead by 0 or 1 : 8,000,000
This more just taking a complement of a set. ( so an interior form of inclusion-exclusion)
You could realize all 7 have at least ... | {
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Summation of $\log\left(\frac i2\right)$ Could someone just briefly explain why this summation is true?
$$\sum_{i = 1}^n\log\left(\frac i2\right) = \frac n2(\log n - 1)$$
I'm having a hard time wrapping my head around this. Any help is appreciated, thanks.
| $$\sum_{i=1}^n \log(i/2) = \log \prod_{i=1}^n (i/2) = \log \frac{n!}{2^n} = \log(n!) - n \log 2.$$
Stirling's approximation (applied somewhat crudely) gives $\log(n!) \approx n \log n - n$, so
$$\sum_{i=1}^n \log(i/2) \approx n \log n - (1 + \log 2) n = n(\log n - (1+\log 2)).$$
If the original quantity is instead sli... | {
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Is there a standard name for the function $f_m(x,y) = mx + y$? When formalizing the base-$m$ positional numeral system, the function $$f_m : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$$ $$x,y \mapsto mx + y$$ is extremely useful. For example, observe that $$365 = f_{10}(f_{10}(3,6),5).$$
Question. Is there an... | The inverse of the function you gave $$f_m: \mathbb{N} \to \mathbb{N} \times \mathbb{N}$$ $$a \mapsto (b,r) \backepsilon a=bm + r $$
is called Euclidean division so logically your function should be called Euclidean multiplication. Though i've never heard that term used.
| {
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If $H,K$ are subgroups of $G$ s.t. $o(H), o(K)$ are relativily prime $\implies H \cap K = \{ e \}$.
If $H,K$ are subgroups of $G$ s.t. $o(H), o(K)$ are relativily prime
$\implies H \cap K = \{ e \}$.
Here $o(H)$ means the order of $H$.
Below is my attempt but I am afraid I might be jumping some hoops here:
Suppose ... | Note, $H\cap K$ is group and moreover subgroup(since, intersection of two subgroup of a same group is again a group ) of both $H, K$. So, $o(H\cap K)|o(H), o(H\cap K)|o(K)$, but since $o(K), o(H)$ are co-primes, so $1$ is the only common divisor to them.
| {
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Why we need branch cut?[Continuity and analytic on Branch cuts] Why we need branch cut in Complex analysis? It just my guess though, complex function itself is multi-valued function, so To define the one-valued function we take the branch cut, right?
Also what about the continuity and analytic on the branch cut for th... | You are right: "To define the one-valued function". Branch cut has to start at branch point and then goes arbitrary path to infinity. In this way Riemann surface is cut into Riemann sheets. On a given sheet function is on the cut discontinuous and non-analytic. Of course, function remains analytic if you go from one sh... | {
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How many numbers are required to define a sequence without stating a rule/function for generating the next term in the sequence? I'm wondering if there is some minimum number of numbers required to define a sequence, without explicitly stating the rule that generates the next term in the sequence. For instance if I wri... | Consider the sequence $1,1,2,3.$ At first look, it seems to be the first terms of Fibonacci numbers, but it's not true that the only sequence which starts with $1,1,2,3,5$ are Fibonacci numbers.
Here we can say that these are the first terms of triangle read by rows in which row n lists A000041(n-1) 1's followed by ... | {
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A question about the proof of Godel's Incompleteness Theorem The article A Computability Proof of Gödel’s First Incompleteness Theorem, by Jørgen Veisdal on Cantor's Paradise contains the following passage:
The second property regards the complement of a set $E$, that is, all the strings which are not in set $E$. Firs... | Decidable means there's a decision procedure for determining whether an element is in the set or not. By symmetry, this holds for the complement too: we can determine whether the element is not in the set or not.
However, it is false that the complement of a computably enumerable set is computably enumerable. In fact, ... | {
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How strongly does $\int_1^X \frac{\exp(B/\sqrt x)}{\sqrt x} dx$ depend on $B$? As part of analysing an algorithm I stumbled upon the integral $\displaystyle \int_1^X \frac{\exp(B/\sqrt x)}{\sqrt x} dx$ as an approximation for the corresponding sum. Wolfram alpha gives the antiderivative in terms of non-elementary funct... | I do not know if this will answer your question.
Effectively, after one integration by parts, we end with
$$I=\int_1^X\frac{e^{\frac{B}{\sqrt{x}}}}{\sqrt{x}}\, dx=-2 \left(B \,\text{Ei}\left(\frac{B}{\sqrt{X}}\right)-B\, \text{Ei}(B)-\sqrt{X}
\, e^{\frac{B}{\sqrt{X}}}+e^B\right)$$
What is doable is to expand as a ser... | {
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Parameter $d$ that makes the probability of the graph $G(n,\frac{d}{n})$ being $k$-colorable tends to $0$, as $n \to \infty$ , for $k \leq 2$. This is an exercise that I'm doing.
Let $\epsilon > 0$ and $d > 0$ be fixed. Prove that for $k \geq 2$, if $d \geq (1 + \epsilon)2k(\text{log} k + 1)$, then, $\lim\limits_{n \... | Whether the graph becomes "more sparse" or not is a question of how you measure sparseness; certainly the fraction $\frac{\text{total edges}}{\text{possible edges}}$ goes to $0$, but on the other hand, the average degree remains fixed at $d$. (Well, almost; if we were pickier, we'd set the edge probability to $\frac{d}... | {
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How does population growth relate to $e$? Say that there is a population of $10,000$. From year $0$ to year $1$, we know that the population has grown to $20,000$. How might I model this growth?
It clearly isn't sensible to say that the population suddenly 'jumped' from $10,000$ to $20,000$ as the year drew to a close.... | Your expectation is not unfounded.
We would model it as such: let $f(t)$ be the population at time $t$.
Then $f(0) = 10000$ and $f(1) = 20000$.
We implicitly assume that the growth of the population is proportional to to the population, and moreover that this proportion is constant in time.
This would translate symboli... | {
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What's the best way to solve $1 = A(x^2+1) + (Bx+C)(x+1)$ What's the best way to solve $$1 = A(x^2+1) + (Bx+C)(x+1)$$
I let $x=-1$ and got $A=\frac{1}{2}$
But what sub is ideal to find B&C
This gets messy quick, I think.
Instead, I started over, and I grouped like terms
$$1 = Ax^2+A+Bx^2+Bx+Cx+C$$
This led to $$A+B=0$... | At each step take advantage of your previous findings.$$1 = A(x^2+1) + (Bx+C)(x+1)$$
$$x=-1 \implies A=\frac {1}{2}$$
$$ x=0 \implies A+C=1 $$
Thus $$C =1/2$$
$$x=1 \implies 2A+2(B+C)=1$$
Thus $$B+C=0 \implies B=-\frac {1}{2}$$
| {
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Find the number of ways you can invite $3$ of your friends on $5$ consecutive days. Find the number of ways you can invite $3$ of your friends on $5$ consecutive days, exactly one friend a day, such that no friend is invited on more than two days.
My approach: Let $d_A,d_B$ and $d_C$ denote the total number of days $A... | The fact that the only way to achieve this is by inviting one friend over once and the other two twice will make this problem simpler.
How many ways are there to pick the one friend (from $3$) that will be only visiting one day instead of two?
Now if we assume the days are Monday through Friday, how many ways are there... | {
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Prove that $G=$SL$(2, \mathbb{F}_5)$ is an extension of $\mathbb{Z}_2$ by $A_5$ which is not a semidirect product. Question:
Prove that $G=$SL$(2, \mathbb{F}_5)$ is an extension of $\mathbb{Z}_2$ by $A_5$ which is not a semidirect product.
(This is a question from Rotman's Advanced Modern Algebra which I am trying t... | Your work is totally fine.
Now, for your last question: if the sequence were split, then $A_5$ would be isomorphic to a subgroup of index $2$ of $G$, and you would get a surjective morphism $G\to \mathbb{Z}/2\mathbb{Z}$ with kernel isomorphic to $A_5$.
Now since $\mathbb{Z}/2\mathbb{Z}$ is abelian, this morphism sends ... | {
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When does $2n-1$ divide $16(n^2-n-1)^2$? Find all integers $n$ such that
$\dfrac{16(n^2-n-1)^2}{2n-1}$
is an integer.
| Starting off with the initial expression:
$$\frac{16(n^2-n-1)^2}{2n-1}$$
Completing the square:
$$=\frac{4^2(n^2-n-1)^2}{2n-1}$$
$$=\frac{(4n^2-4n-4)^2}{2n-1}$$
$$=\frac{((2n-1)^2-5)^2}{2n-1}$$
Some manipulation:
$$=(2n-1)^3(1-\frac{5}{(2n-1)^2})^2$$
Expanding the expression:
$$=(2n-1)^3-10(2n-1)+\frac{25}{2n-1}$$
Sinc... | {
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combinatorics, you can win prizes from different categories but not from the same one. The question goes like this:
There is a conference for psychology in which $12$ researchers are participating. In the conference, two different companies are giving out prizes in two different categories. Three researchers will rece... | The first three prizes are distinct (TV, DVD, radio), so we use permutation instead of combinaison. There are $P(12,3)=1320$ ways to give the first three prizes.
The second prizes were the same, si combinaison was right here. There are $C(12,2)=66$ ways the give the second prizes.
Finally, there are $P(12,3)\times C(12... | {
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Quadratic function with roots in $[0,1]$. Prove that $f(0) \geq \frac49$ or $f(1) \geq \frac49$ Let $a,b$ in $[0,1]$ be such that the polynomial $f(x) = (x-a)(x-b)$ satisfies $f(\tfrac12) \geq \frac1{36}$.
I have found a quite complicated proof of the following inequality using calculus:
$$f(0) \geq \frac49 \quad\text... | I'm gonna prove this algebraically. First some observations:
$$f\left(\frac{1}{2}\right) \geq \frac{1}{36}\Leftrightarrow 2ab-(a+b)+\frac{4}{9} \geq 0 \ \ \ \ \ \ \ (1)$$
$$f(0) \geq \frac{4}{9} \Leftrightarrow ab \geq \frac{4}{9}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$
$$... | {
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Probability of Broken Computers A school orders 7 new computers for a classroom but they are told 3 will not work properly when they recieve them. The school begins to turn each computer on, one by one, to figure out which computers do not work.
There are quite a few questions but most of them I am just having troubl... | For the first question, it helps to consider the opposite event. What is the probability that more than five computers need to be turned on to find the three that do not work? In other words, what is the probability that after knowing the state of five computers, you still don't know the state of the two remaining comp... | {
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If $a, b > 0$ and $ b \neq 1$ prove that $\displaystyle {\int_1^b a^{\log_b x} dx > \ln b}$ Now, $a^{\log_b x} = x^{\log_b a},$
therefore $$ {\int_1^b a^{\log_b x}dx} = {\int_1^b x^{\log_b a}dx} = {\frac {ab - 1}{\log_b ab}} = {\frac {ab - 1}{\ln ab - \ln 1} \ln b} = {\frac {\ln b}{c}}$$ where $c \in (1 , ab)$ whe... | Consider the inequality
$$\tag1
\frac{ab-1}{\ln ab}>1,
$$
for all $a,b>0$.
This is equivalent to
$$\tag2
ab>1+\ln ab.
$$
The function $g(t)=t-1-\ln t$ has derivative $g'(t)=1-\tfrac1t$, so $g$ is decreasing on $(0,1)$ and increasing on $(1,\infty)$, with the only critical point, a minimum, at $t=1$. Thus $g(t)\geq g(... | {
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Dimension of connected components of $O(n)$ The special orthogonal group $SO(n)=O_+(n)$ is the subgroup of the orthogonal group $O(n)$ containing matrices with determinant $+1.$ The other connected component of $O(n)$, call it $O_-(n)$, has matrices with determinant -1.
I always thought $O_+(n)$ and $O_-(n)$ had the sa... | Actually, if $M\in O_-(n)$, then$$\begin{array}{ccc}O_+(n)&\longrightarrow&O_-(n)\\N&\mapsto&MN\end{array}$$is a diffeomorphism. So, yes $\dim O_+(n)=\dim O_-(n)$.
| {
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Integral with binomial to a power $\int\frac{1}{(x^4+1)^2}dx$ I have to solve the following integral:
$$\int\frac{1}{(x^4+1)^2}dx$$
I tried expanding it and then by partial fractions but I ended with a ton of terms and messed up. I also tried getting the roots of the binomial for the partial fractions but I got complex... | Hints:
$$\frac1{(x^4+1)^2}=\frac{x^4+1-x^4}{(x^4+1)^2}=\frac1{x^4+1}-\frac{x^4}{(x^4+1)^2}$$ and by parts
$$4\int\frac{x^3x}{(x^4+1)^2}dx=-\frac x{x^4+1}+\int\frac{dx}{x^4+1}.$$
This way we can get rid of the square at the denominator, and we are left with
$$\frac1{x^4+1}.$$
Now using the factorization of the quartic b... | {
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Find the number of group isomorphisms from the group $(\mathbb{Z}_3, +)$ to itself. I have to find the number of isomorphisms from the group $( \mathbb{Z}_3, + )$ to itself.
I don't know of any procedure to do this so I basically just tried to guess functions until I cannot find anything else. I found that the function... | Hint: $${\rm Aut}(\Bbb Z_n)\cong U(n),$$ where $U(n)$ is the group of units modulo $n$.
| {
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Can a regular tetrahedron cast a square shadow given parallel light rays? How do you prove this? And, more generally, can a $n$-simplex cast a hypercube shadow in one lower dimension?
| Yes.
Viewing direction is along center of two skewed sides onto a plane perpendicular to it in orthographic projection. It also passes through tetrahedron center. The viewing point is at infinity.
Spherical symmetry restraint of four connected lines of equal projected length makes for a square for such a viewing direct... | {
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Existence of a number field containing $\alpha$ where $p$ is unramified I want to better understand the structure of algebraic number fields and for this purpose I am thinking of various problems. One of them I cannot solve, is the following:
Let $p$ be some prime number and $\alpha \in \mathbb{C}$ integral over $\math... | This is not always possible. We have:
Let $L/K/F$ be a tower of number fields and $p \subset F$ a prime. TFAE:
*
*$p$ is unramified in $L$;
*$p$ is unramified in $K$ and all primes of $K$ above $p$ are unramified in $L$.
This is a consequence of unique factorization of ideals.
In particular, in the sit... | {
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Bank vaults and probability
A burglar breaks into a bank with the intention to open the vaults and steal some gold coins.
He knows that each vault contains a number of 1 to 100 such coins with equal probability for each number. Since he is what we call an “ethical burglar”, he will only get 100 coins to help some pe... | Very nice problem! You can safely assume that there are exactly $100$ vaults since this is the maximal number of vaults the burglar may have to open. (See below for an argument.)
My take on the interpretation is as follows: Number the vaults $V_1, V_2, \ldots, V_{100}$ and say that the burglar opens $V_1$ first, then $... | {
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"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Increasing and decreasing function doubt This is a question of increasing and decreasing functions.
$f(x)=\sin x + \cos x$, $x$ belongs to $[0, 2\pi]$
Derivative of this function $f'(x) = \cos x - \sin x$
For increasing function we put $f'(x) > 0$.
I tried to solve it this way:
$\cos x - \sin x > 0$
$\cos x > \sin x$
$... | What you did up to$$\cos x>\sin x\tag1$$is fine. After this point, you need to consider two possibilities:
*
*$x\in\left(0,\frac\pi2\right)\cup\left(\frac{3\pi}2,2\pi\right)$: then $\cos x>0$ and what you did is fine;
*$x\in\left(\frac\pi2,\frac{3\pi}2\right)$: then $\cos x<0$ and what you deduce from $(1)$ is that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3538014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find $P(A \cap B)$ with $P(A)=0.4$, $P(B) = 0.3$ and $P(A \cup B) = 0.6$
Find $P(A \cap B)$ with $P(A)=0.4$, $P(B) = 0.3$ and $P(A \cup B) =
0.6$.
My professor doesn't specify whether these events are mutually exclusive. If I solve this as if they are, I get
$$P(A\cap B) = P(A)*P(B) = 0.12$$
If I solve this as if t... | When $A$ and $ B$ are mutually exclusive, $P(A \cup B) = P(A) + P(B)$.
In this case, that would mean $0.6=0.4+0.3$, which is clearly not so.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3538176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Behaviour of $L^1$ function at $\infty$ I have the following doubt..
If $f$ is $L^1(\mathbb{R}),$ then can we show $\lim\limits_{k \rightarrow \infty} \int\limits_k^\infty f(s)\,ds=0$
If so please outline the proof..
| Let $f \in L^1(\mathbb{R})$. Then $\int_{\mathbb{R}} f(s) ds = I < \infty$ and furthermore $\forall k \in \mathbb{N}$:
$$\int_{\mathbb{R}} f(s) ds = \int f(s) 1_{(- \infty, k]} ~ ds + \int f(s) 1_{[k, \infty)} ~ ds$$
Now we have
\begin{align*}
\lim_{k \rightarrow \infty} \vert \int_{k}^{\infty} f(s) ds \vert &\leq \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3538390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integration of $\exp(-z^2)/(z-z_0)$ I am interested into the following integral
$I = \int_{-\infty}^{+\infty} \frac{f(z)}{z-z_0} dz$
where $f$ is typically a Gaussian function $e^{-z^2}$.
I am tempted to simply use Cauchy formula to get
$I = 2\pi i f(z_0) = 2\pi i \ e^{-z_0^2}$
But I am not sure that the contribution o... |
One can deform the real line contour in the complex plane, but it is rather dubious that such a defomation facilitates evaluation of the integral of interest via Cauchy's Integral Theorem. However, we can use Feynman's trick to evaluate the integral in terms of the Imaginary Error Function. To that end we now procee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3538522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A real-analytic function at $p$ is $C^\infty$ at $p$ (in "An Introduction to Manifolds" by Loring W. Tu.) I am reading "An Introduction to Manifolds" by Loring W. Tu.
There is the following sentence in this book.
I cannot understand what the sentence is saying.
A real-analytic function is necessarily $C^\infty$, bec... | You are confusing
forall $K$, $f(x)= \sum_{|a|\le K} c_a x^a+O(|x|^{K+1})$
with
forall $K$ and for all $y$ close to $0$, $f(x)= \sum_{|a|\le K} c_a(y) (x-y)^a+O(|x-y|^{K+1})$ and the $c_a(y)$ are continuous
$C^\infty$ at (near) $0$ is the latter.
That a (convergent) power series is $C^\infty$ follows from that (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3538668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that a function which is defined by integral is entire by using Morera’s theorem.
Let $f : [-1,1] \rightarrow \mathbb{C}$ be a continuous function and we define $F$ on $\mathbb{C}$ by
$$\displaystyle F(z) = \int_{-1}^{1} f(t) e^{itz} dt $$
1) Show that $F$ is well-defined and continuous on $\mathbb{C}$.
2) ... | Your intuition on separating integrals is correct, but you have pulled out the wrong thing :)
Let's do it in this manner:
$$\int_{C}^{\ } \Bigl ( \int_{-1}^{1}\,f(t)e^{itz}dt \Bigr ) dz {=\int_{C} \int_{-1}^{1}f(t)e^{itz}dtdz\\=\int_{-1}^{1}\int_{C}f(t)e^{itz}dzdt\\=\int_{-1}^{1}f(t)\int_{C}e^{itz}dzdt}$$and $\int_{C}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3538815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Two Lie Group homomorphisms are equal if their induced Lie algebra homomorphisms are equal and $G$ is connected Let $f, g$ be two Lie Group homomorphisms from G to H (smooth group homomorphisms).
Let the induced Lie Algebra homomorphisms $Df(e)$ and $Dg(e)$ be equal. Then if $G$ is connected show that $f=g$.
I know tha... | Let $\varphi:G\to H$ a Lie group homomorphism and $d\varphi:\mathfrak{g}\to\mathfrak{h}$ the induced Lie algebra homomorphism, given by $(d\varphi X)_{e_H}=(d\varphi)_{e_G} X_{e_G}$.
Given $\varphi$, let $\Gamma_{\varphi}$ the graph of $\varphi$, namely $\Gamma_{\varphi}=\{(g,\varphi(g))\mid g\in G\}$. You can verify t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3538919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
determination coefficients of quadratic equation using sum and product of its solutions? let $x_1$ and $x_2$ are real solutions of the quadratic equation $ f(x)=ax^2+bx+c=0$ with $a, b, c$ are real such that its solutions satisfy the following system :
$$\begin{cases}x_1+x_2=16, \\ x_1 x_2=55.\end{cases}$$
Now my ques... | Let's assume the solutions are $x_1$ and $x_2$. Now, we can write:
\begin{align*}
f(x) &= ax^2 + bx + c \qquad (1)\\
f(x) &= (x - x_1)(x - x_2) = x^2 - x(x_1 + x_2) + x_1 x_2 \qquad (2)
\end{align*}
Since the first term of $(1)$ has a highest power of $1$, we scale equation $(2)$ into:
$$
f(x) = a(x - x_1)(x - x_2) = a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3539066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Preference relation $\succsim$ continuous if and only if the upper and lower contour sets are both closed I'm trying to show that a preference relation $\succsim$ is continuous if and only if the upper and lower contour sets are closed.
The direction $\Rightarrow$, i.e. continuity implies the upper and lower contour se... | As you have argued, if the claim is false, then $y^{n} \succ x$ for all but finitely many $n$.
Analogously, because $U(y)$ is closed, $y \succ x^{n}$ for all but finitely many $n$.
Therefore, for all $n$ larger than some $n_{0}$, we have $y \succ x^{n} \succeq y^{n} \succ x$.
In particular, $y \succ y^{n_{0}} \succ x$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3539267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that if $ (x+y)$ is even, then $(x−y)$ is even, for integers. Did I prove correctly? is this a direct proof? For $x+y$ to be even, either $x$ and $y$ are both even, or $x$ and $y$ are both odd.
If $x$ and $y$ are both even we obtain: $x=2k$ and $y=2j$.
substituting into $x-y$ we get $2k-2j$.
$2(k-j)$ is even, so ... | Actually, you can have a direct proof that $x+y$ and $x-y$ have the same parity, since $(x+y)-(x-y)=2y$ is even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3539448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove $4\sin^{2}\frac{\pi}{9}-2\sqrt{3}\sin\frac{\pi}{9}+1=\frac{1}{4}\sec^{2}\frac{\pi}{9}$. While attempting to algebraically solve a trigonometry problem in (Question 3535106), I came across the interesting equation
$$
4\sin^{2}\frac{\pi}{9}-2\sqrt{3}\sin\frac{\pi}{9}+1=\frac{1}{4}\sec^{2}\frac{\pi}{9}
$$
which ar... | We need to prove that:
$$4\sin^220^{\circ}-4\sin60^{\circ}\sin20^{\circ}+1=\frac{1}{4\cos^220^{\circ}}$$ or
$$4\sin^240^{\circ}-8\sin60^{\circ}\sin40^{\circ}\cos20^{\circ}+4\cos^220^{\circ}=1$$ or
$$2-2\cos80^{\circ}-4\sin60^{\circ}(\sin60^{\circ}+\sin20^{\circ})+2+2\cos40^{\circ}=1$$ or
$$\cos40^{\circ}-\cos80^{\circ}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3539618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Maximum likelihood estimation in $d$-dimensional Euclidean space of a ball I have a set of points in $d$-dimension Euclidean space drawn from a ball centered at point $c$ and with radius $r$ which are unknown. I want help in formulating the maximum likelihood estimator of $r$ and $c$.
| If they're uniformly distributed in the ball, then the value of the probability density at every point in the ball is equal to the reciprocal of the volume of the ball. The values of $c$ and $r$ that maximize that density, subject to the constraint that all of the observed points lie within the ball, are the values of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3539735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can there exist $\{X_n\}_{n\ge 1}$ such that $X_n \to -\infty$ a.s. and $EX_n\to 0$? Can there exist variables $\{X_n\}_{n\ge 1}$ such that $X_n \to -\infty$ a.s. and $EX_n\to 0$?
How can we prove or disprove this analytically?
| Let $X_n$ be $-n$ with probability $p_n$ and $e^{n}$ with probability $1-p_n$ where $p_n=e^{n}/(n+e^{n})$ . To show that this example works note that $\sum (1-p_n)<\infty$ and use Borel Cantelli Lemma.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3539892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Find $A\subseteq [0,1]$ such that $\lim\limits_{\varepsilon\to0}\frac{m(A\cap[0,\varepsilon])}{\varepsilon}=\frac{1}{2}$ Find a measurable set $A\subseteq [0,1]$ such that
$$\lim\limits_{\varepsilon\to0}\frac{m(A\cap[0,\varepsilon])}{\varepsilon}=\frac{1}{2}$$
I'm also interested in a set $B\subseteq [0,1]$ such that
... | A hint for the first part: let $\{x_k\}\downarrow 0$ such that $x_0=1$ and $x_k>x_{k+1}$ for all $k\in \Bbb N_{\geqslant 0} $, also choose some $r\in [0,1]$. Now set $m_k:=r\ell _k$ for $\ell _k:=x_k-x_{k+1}$, also we set $E_k:=[x_{k+1},x_{k+1}+m_k)$. Then $|E_k|=m_k$, and setting $E:=\bigcup_{k\geqslant 0}E_k$ we fou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3540055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Limit as $n\to+\infty$ of $\prod_{k=1}^{n} \frac{2k}{2k+1}$ I'm trying to evaluate
$$\lim_{n\to+\infty} \prod_{k=1}^{n} \frac{2k}{2k+1}$$
First I notice that since $k\geq1$ it is $\frac{2k}{2k+1}>0$ for all $k\in\{1,...,n\}$; so
$$0\leq\lim_{n\to+\infty} \prod_{k=1}^{n} \frac{2k}{2k+1}$$
Then I notice that
$$\prod_{k=1... | Another way:
Using arithmetic geometric Inequality
$$\frac{k+k-1}{2}> \sqrt{k\cdot(k-1)}\Rightarrow \frac{2k-1}{2k}>\sqrt{\frac{k-1}{k}}$$
$$\frac{2k}{2k-1}<\sqrt{\frac{k-1}{k}}\Rightarrow \prod^{n+1}_{k=2}\frac{2k}{2k-1}<\prod^{n+1}_{k=2}\sqrt{\frac{k-1}{k}}=.\frac{1}{\sqrt{n+1}}$$
$$\Longrightarrow 0<\prod^{n+1}_{k=2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3540188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 3
} |
Prove that $\int_a^b f(t)f'(t)dt=\frac{1}{2}[(f(b))^2 - (f(a))^2]$ Prove that If $f(t)=u(t)+iv(t)$ is function that derivative continuous on $[a,b]$ then $\displaystyle\int_a^b f(t)f'(t)dt=\frac{1}{2}[(f(b))^2 - (f(a))^2]$
I know If $\displaystyle F'(t)dt=U'(t)dt+iV'(t)dt , \text{then}$
$$\int_a^bF'(t)dt=\int_a^bU'(t)d... | $$\int_a^bf(t)f'(t)dt=\int_{f(a)}^{f(b)}udu=\left[\frac u2\right]_{f(a)}^{f(b)}$$
using $u=f(t)\Rightarrow dt=\frac{du}{f'(t)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3540413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Prove that if $0\leq a_k \leq b_k$ then $\prod_{k=1}^n a_k \leq \prod_{k=1}^{n} b_k$ Let $0\leq a_k\leq b_k$ for all $k\in\mathbb{N}$, then show
$$\prod_{k=1}^{n} a_k \leq \prod_{k=1}^{n} b_k$$
My attempt: since $0\leq a_k\leq b_k$ for all $k\in\mathbb{N}$ we have that
$$0 \leq a_0\leq b_0, 0 \leq a_1 \leq b_1,..., 0\... | 1: Yes, what you've done is correct.
2: This follows immediately from the first part you have proven: Just define $\hat{a}_k = \left|a_k\right|$ and $\hat{b}_k = \left|b_k\right|$, then $\hat{a}_k$ and $\hat{b}_k$ satisfy the assumptions from your first point and you get
$$\prod_{k=1}^{n} \hat{a}_k \leq \prod_{k=1}^{n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3540672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does this statement hold true about square matrices $A$ and $B$: Does this statement hold true about square matrices $A$ and $B$:
If $\det (A) \neq 0$ and $\det(B) \neq 0$, then $\det(A+B) \neq 0$ or $\det(A-B) \neq 0$.
I tried researching about this but it seems not to be asked online.
| It is not the case. Counterexample:
If
$A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \tag 1$
and
$B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \tag 2$
then
$\det(A) = -1 = -\det(B), \tag 3$
but
$A + B = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}, \tag 4$
and
$A - B = \begin{bmatrix} 0 & 0 \\ 0 & -2 \end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3540880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Assume that $\int_{a}^{ab} f(x) dx$ is independent of $a$. Prove $f(x)=\frac{c}{x}$ Assume $f$ is integrable on $[0,\infty)$ and assume that for $a,b>0$, the value of $\int_{a}^{ab} f(x) dx$ is independent of $a$.
Prove that $f(x)=\frac{c}{x}$, where $c$ is a constant.
I have tried several things, like showing $g(x)=... | I suppose the assumption is for all $b$. If it holds for just $b=1$ we cannot prove that $f$ has the desired form.
By Lebesgue's theorem on differentiation of indefinite integrals we can differentiate w.r.t. $a$ and get $bf(ab)-f(a)=0$ a.e.. If $g(x)=xf(x)$ we get $g(ab)=g(a)$ a.e... Can you finish?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3541092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
$\sum_{k=1}^{n}\frac{\ln k}{(2k-1)(2k+1)}<\frac{1}{4}$
Show that $$\sum_{k=1}^{n}\frac{\ln k}{(2k-1)(2k+1)}<\frac{1}{4}$$
holds for all $n\in\mathbb{N^+}$.
Since this is a positive series, it suffices to show that $$\sum_{k=1}^{\infty}\frac{\ln k}{(2k-1)(2k+1)}<\frac{1}{4},$$which is true by machine computing. WA g... | By summing by parts,
$$\begin{align}
S_n:&=\sum_{k=1}^{n}\frac{\ln(k)}{(2k-1)(2k+1)}\\
&=\frac{1}{2}\sum_{k=2}^{n}\left(\frac{\ln(k)}{(2k-1)}-\frac{\ln(k)}{(2k+1)}\right)\\
&=\frac{1}{2}\sum_{k=2}^{n}\frac{\ln(k-1)}{(2k-1)}-\frac{1}{2}\sum_{k=2}^{n}\frac{\ln(k)}{(2k+1)}
+\frac{1}{2}\sum_{k=2}^{\infty}\frac{-\ln(1-1/k)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3541416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
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