Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Show how $\sum _{n=0}^{\infty }\:\frac{(1)^n}{ln\left(n+1\right)}$ diverges $$\sum _{n=0}^{\infty }\:\frac{(1)^n}{\ln\left(n+1\right)}$$
I've tried using Ratio Test as it seems to be the most obvious but it doesn't work.
edit: actually it should be $1^n $ which is just 1. It was just a part of a question where I had to... | Not sure if the question is about $1^n$ or $(-1)^n$ but in the latter case :
Hint: you have to group terms by two : $\sum \frac{1}{\ln(2n)} - \frac{1}{\ln(2n+1)}$.
Then use the $O()$ notation and express $\ln(2n+1)$ in terms of $\ln(2n)$ and $O(\frac{1}{n})$
This should lead you to $\sum O(\frac{1}{n\ln(n)})$. If you a... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $n(n-1)<3^n$ for all $n≥2$ By induction Prove that $n(n-1)<3^n$ for all $n≥2$. By induction.
What I did:
Step 1- Base case:
Keep n=2
$2(2-1)<3^2$
$2<9$ Thus it holds.
Step 2- Hypothesis:
Assume: $k(k-1)<3^k$
Step 3- Induction:
We wish to prove that:
$(k+1)(k)$<$3^k.3^1$
We know that $k≥2$, so $k+1≥3$
Th... | I think that your solution is fine. However, I would phrase it slightly different.
Step-2. To be completely formal, I would say: Let $k>2$ and assume $k(k-1)<3^k$.
Step 3. We need to show $k(k+1)<3^{k+1}$. We have $$k(k+1)=k(k-1)+2k<3^k+2k<3^k+3^k+3^k=3^{k+1}$$
Where we have used the inductive hypothesis and the fact t... | {
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$A=\{1,2,3,4,5\}$ and $B=\{1,2,3,4,5,6,7\}$ How many function $f:A\to B\text{ where (x+f(x) is odd)}$ Let A={1,2,3,4,5}
And B={1,2,3,4,5,6,7}
How many function $f$ from $A$ to $B$ are, $f:A\to B\text{ where (x+f(x) is odd)}$
What I tried to do was:
I know that odd number + even number = odd number, and that
even num... | Nearly right. For every one of the $27$ ways to treat the odd numbers, there are sixteen ways to treat the even numbers. That gives $27\times16$, rather than $27+16$ different functions.
The function $$f(1)=2,f(3)=2,f(5)=2,f(2)=1,f(4)=1$$ is different from the function $$g(1)=2,g(3)=2,g(5)=2,g(2)=2,g(4)=2$$
| {
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Can we prove the existence of a gcd in $\mathbb Z$ without using division with remainder? For $a,b\in\mathbb Z$ not both $0$, we say $d$ is a gcd of $a$ and $b$ if $d$ is a common divisor of $a$ and $b$ and if every common divisor of $a$ and $b$ divides $d$. With this definition, can we prove the existence of a gcd of ... | For the greatest common divisor to exist and be well defined, we need a unique factorization domain. So if you do not allow the use of unique prime factorization, it is probably not possible. Of course we could come up with a proof, which implicitly shows unique prime factorization on its way, but this would be cheatin... | {
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Does there exist a Benny number? For positive integers $x$, let $S(x)$ denote the sum of the digits of $x$, and let $L(x)$ denote the number of digits of $x$. It can be shown that there are infinitely many numbers that cannot be expressed as $x+S(x)$ or $x+L(x)$ or $x+S(x)+L(x)$ individually or any method of those thr... | For all $n$, we either have $L(n)=L(n+L(n))$ or $L(n)=L(n-L(n))$ (or both). If $L(n)=L(n+L(n))$, let $x=n+L(n)$. Then
$$x-L(x)=(n+L(n))-L(n+L(n))=(n+L(n))-L(n)=n$$
Likewise, if $L(n)=L(n-L(n))$, let $x=n-L(n)$, in which case
$$x+L(x)=(n-L(n))+L(n-L(n))=(n-L(n))+L(n)=n$$
| {
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Writing Corollaries into Proofs I'm taking Discrete Math and one of my homework problems from Epp's Discrete Mathematics with Applications asks me to prove the following:
If $r$ and $s$ are any two rational numbers, then $\frac{r+s}{2}$ is rational.
It's pretty basic, and here is my proof:
We will use the direct met... | I really liked your question and I think I can help you. In fact, I am a discrete math grader for a class that used that text book. A corollary is just another word for a theorem. However, it is a theorem that mainly uses the results of the another, more general(some would argue more important) theorem. Thus, as a grad... | {
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Dividing the interval into $\rm\,n\,$ equal pieces. [Spivak - Calculus, Exercise 20] I was doing exercise 20 of Spivak Calculus, it says
(a) Find a function $\rm\,f\,$, other than a constant function such that $$\rm\,|f(x)-f(y)|\le|y-x|\,$$
(b) Suppose that $\rm\,f(y)-f(x)\le(y-x)^2\,$ for all $\rm\,x\,$ and $\rm\,y... | The intuition is that the sum is telescoping: Write out the first few terms:
$$f(x+[y-x]/n) - f(x) + f(x+2[y-x]/n) - f(x+[y-x]/n)+f(x+3[y-x]/n-f(x+2[y-x]/n)+\cdots +f(y)-f(x+(n-1)[y-x]),$$
and notice how all terms cancel except $f(x)$ and $f(y)$.
Equivalently, if you divide the assumption to get: $|f(x)-f(y)| / |x-y| ... | {
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How to find the polynomial expansion of $S=\prod_{i=1}^n (1+\rho ^ {i-1} \theta)$? Suppose I have a product $S=\prod_{i=1}^n (1+\rho ^ {i-1} \theta)$. How do I find a general formula for the coefficients $\alpha_i$ such that $S=\sum_{i=0}^{n} \alpha_i \theta^i$ ?
Thanks.
| Being the constant term of $S$, $\alpha_0 = 1$, the product of the constants in each factor $1 + \rho^{i-1}\theta$. If $1 < k \le n$, $\alpha_k$ is the sum of the products $\rho^{i_1 -1}\rho^{i_2 - 1}\cdots \rho^{i_k - 1}$, as $1 \le i_1 < i_2 < \cdots < i_k \le n$. So \begin{align}\alpha_k &= \sum_{1 \le i_1 < i_2 <\c... | {
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proving my induction in game theory doubt Highly connected website problem
Suppose we have n websites such that for every pair of websites A and B, either A has a link to B or B has a link to A. Prove by induction that there exists a website that is reachable from every other website by clicking at most 2 links.
I am n... | The base case $n=1$ is trivial (there's only one website)
For the induction step assume we have $n$ pages linked such that page $P_i$ is the "central" page (The page wich can be reached from all other $n-1$ pages within at most two clicks). Now add another page $P_{n+1}$. By assumption, either there is a link $P_i\left... | {
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Need Help with Propositional Logic I am stuck with this proof. I am trying to use deduction (or induction I think) to prove for a tautology with logic laws like De Morgan's, distributive , and implication law etc.
Note: I am not allowed to use truth tables.
Here it is:
$((p \vee q) \wedge (p \rightarrow r) \wedge (q \... | We can use these Rules of inference.
Starting wtih :
$$[((p∨q)∧(p→r)∧(q→r))→r] \equiv$$
we can apply Material implication :
$$\equiv \lnot [(p \lor q)∧(\lnot p \lor r)∧(\lnot q \lor r)] \lor r \equiv$$
followed by De Morgan to get :
$$\equiv [\lnot (p \lor q) \lor \lnot [(\lnot p \lor r)∧(\lnot q \lor r)]] \lor r ... | {
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"timestamp": "2023-03-29T00:00:00",
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Is There A Polynomial That Has Infinitely Many Roots? Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?
| If $c$ is a root of $P(x)$ then $P(x)=(x-c)Q(x)$ for some polynomial $Q(x)$ of lower degree. The degree can't keep getting lower forever.
[This assumes the degree of $P(x)$ is at least $1$.]
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Maximum of *Absolute Value* of a Random Walk Suppose that $S_{n}$ is a simple random walk started from $S_{0}=0$.
Denote $M_{n}^{*}$ to be the maximum absolute value of the
walk in the first $n$ steps, i.e., $M_{n}^{*}=\max_{k\leq n}\left|S_{k}\right|$.
What is the expected value of $M_{n}^{*}$? Or perhaps a bit easier... | Partial answer to the first question: Using the reflection principle, I obtained
$$ \Bbb{P}(M_n^* \geq k) = \sum_{m=0}^{\infty} (-1)^m \left\{ \Bbb{P}((2m+1)k \leq |S_n|) + \Bbb{P}((2m+1)k < |S_n|) \right\}. $$
(Since $|S_n| \leq n$, this is in fact a finite sum for $k \geq 1$ and there is no convergence issue.) Now su... | {
"language": "en",
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How can I know whether to round a quotient up or down (based on whether the number after the decimal point is 5+ or not) with ONLY this information? Say I have a special calculator that, when it divides one number by another, it gives you the answer in the form of, "quotient r remainder." For example, if you divide 5 b... | To exactly answer your question, the criteria becomes:
Is the remainder at least half of the divisor? If so, round up, otherwise round down.
That is to say, if you divide $X$ by $Y$ and get "$A$ remainder $B$", then if $2B\geq Y$, you should round up, and otherwise round down. You can check that this works on all you... | {
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Solids of revolutions and their volumes? I am currently self-teaching myself some calculus stuff and I am a bit confused about all these methods to find the volumes given a function rotated along the $y$- or $x$-axis? So far I have come across so many videos with different method names which is what confuses me.
Is the... | Each method ought to provide you with the same answer, but each is different. The disk method uses disks to fit the volume of the bounds that you've been given. The Washer Method uses washer-shaped shapes within the solid to establish volume. Last, the Shell Method uses a bushing-shaped representation to establish area... | {
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Subextensions of cyclotomic field Let $p$ be a prime and $\zeta_p$ be a $p^{th}$ primitive root of unity. Let $G=\operatorname{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, it is well known that every sub-extension of $\mathbb{Q}(\zeta_p)$ can be written as $\mathbb{Q}(\alpha_H)$, where $H\le G$ and
$$\alpha_H=\sum_{\sigma\i... | The trick with primes is that the Galois group is cyclic, hence summing over the elements of the group produces that result just because of the group structure. In fact, since clearly $\alpha_H$ is fixed by all of $H$, the only way this can go wrong is if $\alpha_H$ has degree lower than $[G:H]$.
But this is easy, say ... | {
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How do we know that we found all solutions of a differential equation? I hope that's not an extremely stupid question, but it' been in my mind since I was taught how to solve differential equations in secondary school, and I've never been able to find an answer.
For example, take the simple differential equation $$y'=y... | In the case of y'=y...
Move along the x axis, follow the ordinate line up to the curve where the slope ='s the y-ordinate. In this way you trace out the curve, being only one curve save for translation.
| {
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Good examples of mathematical writing (structural organization, style, typesetting, and so on) A very famous question on MathOveflow asks for examples of good mathematical writing. Here, I'd like to narrow down the topic and ask:
$\color{#c00}{\text{Question:}}$
Could you point out some examples of good mathematical... | Here's a couple review papers that (in my opinion) are written and organized excellently with respect to the categories mentioned:
*
*Boyd, Stephen, et al. "Distributed optimization and statistical learning via the alternating direction method of multipliers." Foundations and Trends® in Machine Learning 3.1 (2011): ... | {
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"timestamp": "2023-03-29T00:00:00",
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How to integrate $\frac{1}{x\sqrt{1+x^2}}$ using substitution? How you integrate
$$\frac{1}{x\sqrt{1+x^2}}$$
using following substitution? $u=\sqrt{1+x^2} \implies du=\dfrac{x}{\sqrt{1+x^2}}\, dx$
And now I don't know how to proceed using substitution rule.
| Use $x=\tan\theta$, $dx=\sec^2\theta\,d\theta$
$\tan^2\theta+1=\sec^2\theta$
$$\int\dfrac{\sec^2\theta\,d\theta}{\tan\theta\sec\theta}=\int\dfrac{\sec\theta\,d\theta}{\tan\theta}=\int\dfrac{d\theta}{\sin\theta}=-\ln|\csc\theta+\cot\theta|+C$$
| {
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"timestamp": "2023-03-29T00:00:00",
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CPU Performance Please, help me to understand the mathematics behind the following formula of CPI. Why do we calculate CPI the way it's done on the pic? The formula reminds me the expected value from stochastic, but do we have a random value here?
| This seems to be calculating the average number of CPU cycles per operation. There are a variety of operations that occur in different relative amounts, so you must weight the cost of an operation with its relative frequency of occurrence.
If all operations occurred with equal frequency, you would just average the cycl... | {
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Sigma of Permutations Given a permutation p of numbers 1, 2, ..., n. Let's define $f(p)$ as the following sum:
$$\large f(p)=\sum_{i=1}^n\sum_{j=i}^n\min({\rm p}_i,{\rm p}_{i+1},...,{\rm p}_j)$$
What is the exact job of this sigma I can't understand it, what is the result of it. Would you please ex... |
The sum is equal to $$\sum_{k=1}^{n}kW_k$$
where $W_n$ is the width of the interval around $n$ in the permutation that contains only numbers larger than or equal to $k$.
For example:
For the permutation $(1,4,3,2)$ we have
$$\begin{align}W_1=4\qquad\text{ because }\qquad(\overline{1,4,3,2})\\W_2=3\qquad\text{ be... | {
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Integrating $\int x\sin x dx$ Could someone outline the step-by-step approach for the following indefinite integral?
$$\int x\sin x \ dx$$
I know the solution is $\sin(x)-x\cos(x)$, but cannot see how one would go about step-wise solving for it in a logical manner that does not include arbitrary guessing (even with int... | To find
(abusing notation viciously)
$F(x) = \int x f(x) dx$,
look at
$g(x) = x\int f(x) dx
$.
By the product rule,
$g'(x)
=xf(x)+\int f(x) dx$,
so
$xf(x) = g'(x)-\int f(x) dx$.
Integrating,
$\int xf(x) dx
=g(x)-\int \int f(x)dx
$.
If $f(x) = \sin(x)$,
then
$\int f(x) dx = -\cos(x)
$
and
$g(x) = -x\cos(x)
$
and
$\int \... | {
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$f(A\cap B)=f(A)\cap f(B)$ $\iff$ $f$ is injective. Let $f:X\to Y$ where $X$ and $Y$ are nonempty. Prove that a sufficient and essential condition for any two subsets $A,B\subseteq X$ to fulfill $f(A\cap B)=f(A)\cap f(B)$ is that $f$ is injective. I sense there is some problem in my proof. I would be glad if you assist... | The proof that the condition (*) implies that $f$ is injective is fine. Here
(*) For all $A, B \subseteq X$ we have that $f[A] \cap f[B] = f[A \cap B]$
Suppose now that $f$ is injective. We need to show (*).
For any function, $A \cap B \subseteq A$, so $f[A \cap B] \subseteq f[A]$, and also, $A \cap B \subseteq B$ ... | {
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logical problem (how long did you walk?) My wife is very kind, she always picks me up at work by car and drives me home. Today, I finished at work 30 minutes earlier! So I decided to walk home... on the way I met my wife. She was on her way to pick me up, so I sat in the car and she drove me home. Today I was home 10 m... | Hint: She drove 5 minutes less in each direction than usual.
| {
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"timestamp": "2023-03-29T00:00:00",
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Inverse of laplacian operator I recently read a paper, the author treats
$$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$
up to a constant in $\mathbb{R}^d$.
I am not familiar with unbounded operator, so my question is: Under what condition can one take the inverse of an unbounded oper... | To make sense of this sort of problem, it's best to work with Distributions also known as generalized functions. The sort of solution you gave above is sometimes called Greens function or a fundamental solution.
See for example Friedlander--Joshi Theory of Distributions.
| {
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Eigenvalue and Characteristic Root I confused with two terminologies in my lecture, due to an expert. Statement on board I wrote is
Roots of characteristic polynomial of real orthogonal matrix have modulus $1$.
But the expert said, please (in fact, "you should") write
If $\lambda$ is an eigenvalue of a real orthogon... | "$\lambda$ is an eigenvalue" means that there is a non-zero vector $v$ such that $Mv=\lambda v$. It is equivalent to saying that $M-\lambda I$ has determinant zero, which is equivalent to saying that $\lambda$ is a root of the characteristic polynomial.
The two statements you have given say the exact same thing, just t... | {
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first 3 terms of $1/(z\sin z)$ in laurent series How could I calculate the first $3$ terms of Laurent series on $1/ (z \sin z)$ around $z=0$?
For $z=0$ it is a pole of $n= 2$.
$C_n$ are $0$ for $n < -2$, therefore the first $3$ terms are $C_{-2}, C_{-1}, C_0$. I failed to integrate to get the coefficients.
Thanks for y... | Say $$\frac{1}{z \sin z}=c_{-2}z^{-2}+c_{-1}z^{-1}+c_0+\dots. $$
It follows that
$$1=(c_{-2}z^{-2}+c_{-1}z^{-1}+c_0+\dots)z \sin z=(c_{-2}z^{-1}+c_{-1}+c_0z+\dots)(z-\frac{z^3}{3!}+- \dots). $$
Try to expand the RHS, collect the coefficients and compare with the LHS. This should give you a sufficient amount of equation... | {
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Sum of sequence of $\binom{n}{r}$ How can we find the sum of $ \binom{21}{1}+ 5\binom{21}{5}+ 9\binom{21}{9}....+17\binom{21}{17}+ 21\binom{21}{21}$?
I have no clue how to begin. I guess complex numbers might help.
EDIT: Actually the real question was that the sum above was k and we had to find its prime factors. And ... | Your sum is $$S=\sum_{k=0}^m(4k+1)\binom{n}{1+4k}$$ with $n=21, m=5$. To find this sum we proceed as below $$(1+x)^n=\sum_{k=0}^n \binom{n}{k}x^k\\ \implies n(1+x)^{n-1}=\sum_{k=0}^n k\binom{n}{k}x^{k-1}$$ Let $\alpha=e^{j2\pi/r}$ be the $r$th root of unity. Then, $$n(1+\alpha^s)^{n-1}=\sum_{k=0}^n k\binom{n}{k}\alpha... | {
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supremum and infimum: $\frac{n^n}{n!^2}$ So I have this set and I need to find a sup and inf.
$$A=\{\frac{n^n}{n!^2}:n=1,2,3...\}$$
I'd like to know if the part of proof that I have already done is good and I need help with the other part.
I want to check if the series $\frac{n^n}{n!^2}$ is monotonic.
$$\frac{(n+1)^{n+... | Here is an elementary way: it only uses a refined version of Bernoulli's inequality.
Let $u_n=\dfrac{n^n}{(n!)^2}$. We first show $(u_n)$ is a decreasing sequence:
$$\frac{u_{n+1}}{u_n}=\frac{(n+1)^{n+1}}{\bigl((n+1)!\bigr)^2}\cdot\frac{(n!)^2}{n^n}=\Bigl(\frac{n+1}{n}\Bigr)^n\cdot\frac1{n+1}$$
Now it is well known tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1139000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Recursive sequence. Need help finding limit. This is my recursive sequence:
$a_1=\frac{1}{4};\space a_{n+1}=a_n^2+\frac{1}{4}$
for $n\ge 1$
In order to check if this converges I think I have to show that
1) The sequence is monotone increasing/decreasing
2) The sequences is bounded by some value
For 1) I am going to use... | $a_{n+1}-a_n=\frac{(2a_n-1)^2}{4}> 0$. So this is a monotone increasing sequence. Now to see whether the sequence is bounded or not, observe that the limiting value should satisfy $a=a^2+1/4\implies a=1/2$. So, let the sequence be unbounded. Then $\exists N$ such that $a_{N-1}\le 1/2,\ a_N>1/2$. But $a_{N}>1/2\implies ... | {
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"url": "https://math.stackexchange.com/questions/1139114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate integral with cantor measure
Calculate the integral $$\int_{[0,1]}x^2d\mu_F$$ where F is the cantor function. Use the following hints about the cantor function:
*
*$F(1-x)=1-F(x)$
*$F(\frac x 3)=\frac{F(x)}{2}\quad\forall x\in[0,1]$
*$F(0)=0$
I thought that $$\int_{[0,1]}x^2d\mu_F=\int_{[1,0]}(1-x)^2d\m... | Let $C_1=\left[0,\frac{1}{3}\right]\cup\left[\frac{2}{3},1\right]$, $C_2=\left[0,\frac{1}{9}\right]\cup\left[\frac{2}{9},\frac{3}{9}\right]\cup\left[\frac{6}{9},\frac{7}{9}\right]\cup\left[\frac{8}{9},\frac{9}{9}\right]$ and so on the usual sets used to define the Cantor set. Then $\mu_F$ is the limit as $n\to +\infty$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1139217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving the irreducibility of a specific family of polynomials I want to show that $f(x)=x^{4k} - 3x ^{3k} + 4x^{2k}-2x^k +1$ is irreducible in $\mathbb{Q}$ for all $k\in \mathbb{N}$. When $k=1$, it is easy to show; however I have trouble in proving this while $k\ge 2$. I have tried lots of irreducibility tests, but I ... | Lemma: If $F$ contains a primitive $k$th root of unity then $f(x)=x^k-b$ is irreducible over $F$ if $b$ has not any $n$th root in $F$, $n>1$.
Proof: We know $A=\{\sqrt[k]{b},w\sqrt[k]{b},w^2\sqrt[k]{b},...,w^{k-1}\sqrt[k]{b}\}$ is a subset of $K=F(\sqrt[k]{b})$ so $K/F$ is Galois. Its Galois group is a subgroup of $\ma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find the convolution of $x(t)*h(t)$ I am studying for an exam and have the following question:
$$x(t) = u(t)$$
$$h(t) = [e^{-t}-e^{-2t}]u(t)$$
where u(t) is a unit-step function. I need to find the convolution x(t)*h(t).
So:
$$
x(t)*h(t) = \int_{-\infty}^\infty u(t)[e^{-t}-e^{-2t}]u(t-\tau)d\tau\
$$
$$
x(t)*h(t) = u(t)... | using your notation,
\begin{align*}
(x\ast h)(t) &= \int_\mathbb{R} u(\tau)h(t-\tau)d\tau \overset{\textrm{u is step function}}{=} \int_0^\infty h(t-\tau)d\tau \\ &= \int_0^\infty (e^{-(t-\tau)}-e^{-2(t-\tau)})u(t-\tau) d\tau \\ &= \int_0^t (e^{-(t-\tau)}-e^{-2(t-\tau)}) d\tau, \qquad t\geq 0 \\ &= \left( \frac{1}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1139412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Suppose that you had a machine that could find all four solutions for some given $a$. How could you use this machine to factor $n$? Question: Suppose $n = pq$ with $p$ and $q$ distinct odd primes.
Suppose that you had a machine that could find all four solutions for some given $a$. How could you use this machine to fac... | Hint $\ $ Suppose $\,f(x)\in\Bbb Z_n[x]\,$ has more roots than its degree. Iterating the Factor Theorem we can write $\,f(x) = c(x-r_1)\cdots (x-r_k)\,$ By hypothesis it has at least one more root $\,r\not\equiv r_i\,$ so $\,c(r-r_1)\cdots (r-r_k)\equiv 0\pmod n,\,$ so $\,n\,$ divides that product, but does not divide ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to show that this Cayley Table does not form a group Given the following Cayley Table (where e is the identity element):
How would I go about proving that the table does not form a group?
I have checked closure, identity, inverses, and all 27 combinations of associativity excluding the ones that include the identi... | With the translation $e=0$, $a=1$, $b=3$, and $c=2$, we can recognize that our table is the addition table modulo $4$. More formally, the structure $M$ with the given multiplication table is isomorphic to the additive group $\mathbb{Z}_4$, via the mapping $\varphi$ that takes $e$ to $0$, $a$ to $1$, $b$ to $3$, and $c$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$⊢p \land q \to (p\to q)$ - Natural deduction proof confusion I have the following:
$$⊢p \land q \to (p\to q)$$
I'm having a difficult time trying to figure out where to begin. I believe that I am supposed to assume p and q and then somehow use the copy rule to construct the equation, however I am not quite sure.
Can s... | Assuming the conjunction operator has higher precedence than the conditional operator, what needs to be proved is the same as this:
$$⊢(p∧q)→(p→q)$$
Here is a proof:
The OP made the following attempt:
this is my attempt... first assume p, assume q, copy p, copy q, introduce →, introduce ∧, then introduce → between bo... | {
"language": "en",
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Order and Least Common Multiple Abelian Question \item Let $G$ be an abelian group and let $x, y\in G$ be elements so
that $o(x)=m$ and $o(y)=n$. Show that $o(xy)=\frac{mn}{(m,n)}$.
(Note that this is the least common multiple of $m$ and $n$) Is this
true if $G$ is non-abelian? Give an example.
My Solution
Let $r$ be t... | take the non-abelian group on two generators, $x$ and $y$ with $x^2 = y^2 = e$. in this case $xy$ generates an infinite cyclic subgroup
| {
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"timestamp": "2023-03-29T00:00:00",
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How to check continuity of $f(a) = \int_0^1 \frac{\sin(ax + x^2)}{x}\, dx$ on $[0,1]$? How can I see the continuity of $f(a) = \int_0^1 \frac{\sin(ax + x^2)}{x}\, dx$ on $[0,1]$?
I have no idea how to approach.
Any comment would be very appreciated.
| Let $a, b\in \Bbb R$. For fixed $x\in [0,1]$, the mean value theorem gives $$\sin(ax + x^2) - \sin(bx + x^2) = x\cos(cx + x^2)(a - b),$$ where $c$ is a number between $a$ and $b$. Thus $$|\sin(ax + x^2) - \sin(bx + x^2)| \le x|a - b|.$$ Since this holds for every $x\in [0,1]$, we have $$|f(a) - f(b)| \le \int_0^1 |\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1140153",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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$p\land\neg q\to r, \neg r, p ⊢ q$ -natural deduction I have the following:
$$p\land\neg q\to r, \neg r, p ⊢ q$$
I know that my attempt is incorrect, but I will show it anyways:
Step 1) $p\land\neg q\to r$ ----premise
Step 2) $\neg r$ -----premise
Step 3) $p$ -----premise
Step 4) $\neg q\to r$ ---- e1
Step 5) $\neg \... | $$¬r \Rightarrow ¬(p \land ¬q) \mbox{ by modus tollens}$$
$$¬(p \land ¬q) \iff ¬p \lor ¬¬q \iff ¬p \lor q$$
$$( ¬p \lor q) \land p \Rightarrow q \mbox{ by definition of the disjunction operator.}$$
$$\therefore p\land\neg q\to r, \neg r, p ⊢ q$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Sum $\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n(n^4+4)}$ I want to evaluate the sum $$\large\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n(n^4+4)}.$$ I did partial fraction decomposition to get $$\frac{1}{2^n}\left(\frac{-1}{n^2+2n+2}+\frac{4}{n^2-2n+2}+1\right)$$ I am absolutely stuck after this.
| Note that
$$\dfrac{n^4+3n^2+10n+10}{2^n(n^4+4)}=\dfrac{1}{2^n}+\dfrac{3n^2+10n+6}{2^n[(n^2+2)^2-(2n)^2]}$$
Then let's find constants $A,B$ suct that
$$\dfrac{3n^2+10n+6}{(n^2+2n+2)(n^2-2n+2)}=\dfrac{A(n+1)+B}{(n+1)^2+1}-4\Big[\dfrac{A(n-1)+B}{(n-1)^2+1}\Big]$$ to obtain the form $$f(n+1)-f(n-1).$$
For $n=-1,$ we hav... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove that process is uniformly integrable
Let $(X_t)_{t\ge 0}$ be a stochastic process, and let $Y$ be an
integrable random variable, such that $|X_t|\le Y$ for $t\ge0$. Prove
that $(X_t)_{t\ge 0}$ is uniformly integrable.
From definition, we have that $(X_t)_{t\ge 0}$ is uniformly integrable if
$$\sup_{t\in [0... | The problem is that you showed the result for a fixed $t$, but not that it holds uniformly in $t$.
Hint: note that for each $t$ and $\varepsilon$, the inequality
$$|X_t|\mathbb 1_{\{ |X_t|\gt\varepsilon\}} \leqslant Y\mathbb 1_{\{ Y\gt\varepsilon\}} $$
holds. Integrate and conclude by monotone convergence.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to get an open ball in $[0,1]$ that contains $[0,1]$? The definition of bounded we have is that if $X$ is a metric space, $z \in X$, and $X \subseteq X$, then there exists an open ball $B_z(R)$ with finite radius $R$ of $X$ centered at $z$ such that $X \subseteq B_z(R)$.
If $X = [0,1]$, it seems like the largest op... |
If $X=[0,1]$, it seems like the largest open ball possible is $(0,1)$ …
And this is what is false. Any open ball of any radius in a metric space $X$ is by definition considered, well, as an open ball in it, regardless of how the ball or $X$ itself look like to us.
The definition of an open ball $B_z(R)$ in a metric s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it true that P(x|y,z)=P(x|y) if x and z be independent? Why? I know that if x and z be independent, P(xz) = P(x)P(z).
I want to know if x and z be independent, can I cancel z from P(x|yz)? Why?
Thanks.
| Although other responses are useful, but here is another solution with different point of view:
$$
P(x|y,z)=\frac{P(x,y,z)}{P(y,z)}=\frac{P(x,y,z)}{\int_xP(x,y,z)dx}=\frac{P(y|x,z)P(x)P(z)}{\int_xP(y|x,z)P(x)P(z)dx}=\frac{P(y|x,z)P(x)}{\int_xP(y|x,z)P(x)dx}
$$
and the last term is obviously related to $z$, unless $y$ a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integer Linear Programming Without using a computer, I have to solve the following integer linear programming:$$\min \quad x_1+x_2+x_3$$
$$\operatorname{sub} :\begin{cases}x_1\le9\\x_2\le7\\x_3\le5\\3x_1+6x_2+8x_3=80\\x_1,x_2,x_3\in\mathbb{N}\end{cases}$$
Is there any algebraic method to compute the solution (I can't u... | The problem is beyond the typical 11-year-old but if he is bright at math you should be able to explain how to solve it.
Does he have simple algebra?
The first thing to notice is that $3x_1+6x_2$ is divisible by 3, so $80-8x_3$ must also be divisible by 3. The only two allowable $x_3$ that satisfy this are $1$ and $4$.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Doubt in Rudin's Proof: Once I go through the proof of the below theorem, I could encounter that he used dominated convergence theorem to prove $(f)$, in that how they claim that
$$\frac{e^{-ix(s-t)}-1}{s-t}\leq |x|$$
Kindly explain.
| $\phi(x,u)=(e^{-ixu}-1)/u$ so that
$$
|\phi(x,u)|=2\,\frac{|\sin(ux/2)|}{|u|}\le2\,\frac{\min(1,|\tfrac12xu|)}{|u|}=\min\left(\frac{2}{|u|},|x|\right)
$$
which implies the inequality.
The critical step is
$$e^{iy}-1=e^{iy/2}(e^{iy/2}-e^{-iy/2})=2ie^{iy/2}\sin(y/2)$$ and $|ie^{iy/2}|=1$
| {
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It's possible to calculate the frequency of distribution of digits of $\pi$? It's possible using mathematical formula to calculate frequency of distribution of digits of $\pi$ or other constant?
I know that there are already plenty of data available with statistics and you can extract that information, but it's actuall... | From Wolfram:
It is not known if $\pi$ is normal (Wagon 1985, Bailey and Crandall 2001), although the first 30 million digits are very uniformly distributed (Bailey 1988).
In other terms, it appears that the distribution of the digits of $\pi$ (in its decimal expansion) is still unknown.
| {
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"url": "https://math.stackexchange.com/questions/1140980",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Integrate $\int\frac{\sqrt{\tan(x)}}{\cos^2x}dx$ I need help with this integral: $$\int\frac{\sqrt{\tan x}}{\cos^2x}dx$$ I tried substitution and other methods, but all have lead me to this expression: $$2\int\sqrt{\tan x}(1+\tan^2 x)dx$$ where I can't calculate anything... Any suggestions? Thanks!
| As you have noted, your integral simplifies to
$$2\int\sqrt{\tan x}\ \sec^2x\ dx$$
If one makes the substitution $u=\tan x$, one gets $du=\sec^2x dx$, which reduces our integral to
$$2\int u^{1/2}du$$
$$=2\frac{u^{3/2}}{3/2}+C$$
$$=\frac{4u^{3/2}}{3}+C$$
$$=\frac{4\tan^{3/2}x}{3}+C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1141074",
"timestamp": "2023-03-29T00:00:00",
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Let $f(x)$ be continuous on $[0,2]$, and differentiable on $(0,2)$ such that $0
Let $f(x)$ be continuous on $[0,2]$, and differentiable on $(0,2)$ such that $0<f(1)<f(0)<f(2)$. Prove that $f'$ has a solution on $(0,2)$.
Here's a little crappy sketch:
My attempt:
From $f(1)<f(0)<f(2)$ and continuity, there's a point $c... | Since $f$ is continuous and $[0,2]$ is compact, $f$ attains its global minimum at some point $x_0\in[0,2]$. As $f(1)<f(0)$ and $f(1)<f(2)$, we see that in fact $x_0\in(0,2)$. As we have a minimum in an open intervall, we conclude $f'(x_0)=0$.
| {
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"url": "https://math.stackexchange.com/questions/1141194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Splitting the dollar Nash equilibrium I'm working on a game theory problem I can't seem to figure out.
Players 1 and 2 are bargaining over how to split $\$10$. Each player names an amount $s_i$, between 0 and 10 for herself. These numbers do not have to be in whole dollar units. The choices are made simultaneously. Ea... | Here, in case b), given the sum >10, a player would always try to choose a pay off that would maximize his utility. He would choose 10, and hence the rest of the amount that is 0 should go to the player 2, provided he had chosen an amount less than 10. However, player 2 also would try to maximize his utility and choose... | {
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"timestamp": "2023-03-29T00:00:00",
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How to find the pdf of difference of r.v How do I calculate the pdf for the following case? In general, if we have 2 r.v. $x,y$ which are normal, then the pdf of the difference of 2 r.v. which are Gaussian will also be Gaussian, I think with mean $\mu_Z = \mu_x - \mu_y$ and variance $\sigma^2_Z = \sigma^2_x + \sigma^2... |
How do I calculate the pdf for the following case? The pdf of the difference of 2 r.v. which are Gaussian will also be Gaussian, I think with mean $\mu_D =\mu_X −\mu_Y$ and variance $\sigma^2_D =\sigma^2_X +\sigma^2_Y\;$.
Yes. Now you have a Gaussian random variable with given mean and variance.
$$D\sim \mathcal{... | {
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"timestamp": "2023-03-29T00:00:00",
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Examples of a cayley table that represents a structure that satisfies all group axioms except associativity I'm curious if there are any cayley tables on a finite amount of elements that satisfy the axioms of a) closure, b) identity, and c) inverse, but that for at least one triple of elements do not satisfy the associ... | For the table
$$\matrix{e&a&b\cr a&e&e\cr b&b&e\cr}$$
we have
$$(ab)a=ea=a\quad\hbox{but}\quad a(ba)=ab=e\ .$$
Another example:
$$\matrix{e&a&b\cr a&e&a\cr b&b&e\cr}\ ,\qquad (ab)a=e\ ,\qquad a(ba)=a\ .$$
| {
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If $f$ has a pole, does $f^2$ has a pole? I don't understand something in the exercise 2.17 of Algebraic Curves of Fulton.
Let $k = \overline{k}$ a field and $V$ be the variety defined by the zero of $ I = ( y^2 - x^2(x-1) ) \subset k[x,y]$.
Let $\overline{x}, \overline{y}$ be the coordinate functions. Then $z = \fra... | The curve $V$ is not smooth at $(0,0)$. Around that point, your curve looks like $y^2=x^2$, which has two branches, one on which $y/x = +1$ and one on which $y/x = -1$. One way to think of what is happening is that the "function" $y/x$ has no limit as $(x,y) \to (0,0)$, but $(y/x)^2$ does. The behavior of $y/x$ around ... | {
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What is meant by the form of a polynomial in $A_n$ deduced from a polynomial $f$ over $\mathbb{Z}_p$? I am reading Serre's A Course in Arithmetic and am having trouble understanding what he means by a polynomial deduced from a polynomial over $\mathbb{Z}_p$.
Specifically Serre writes,
Notation.-- If $f\in\mathbb{Z}_p... | From the definition of the $p$-adic numbers, you get a natural map $\mathbb Z_p \to \mathbb Z/p^n\mathbb Z$ for all $n$. We can call this map modulo $p^n$.
For example, if you have $a=(1,1,10,64,...)$ in $\mathbb Z_3$, then $a \mod 3=1$, $a \mod 9=1$, $a \mod 27=10$, $a \mod 81=64$.
| {
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Looking for a formula to represent the sequence $2,4,2,8,2,4,2,16,2,4,2,8,\dots$ Is there a formula with which I can represent the sequence $2,4,2,8,2,4,2,16,2,4,2,8,\dots$?
| Let $P_n$ denote the number of zeros at the end of the binary representation of $n$.
Note that $P_n$ also gives the number of times that $n$ is divisible by $2$.
Your sequence can be represented as: $$a_n=2^{P_n+1}$$
| {
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"question_score": "1",
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"answer_id": 0
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finding certain sequences that satisfy a requirement I need to find sequence $(z_n) $ and $(w_n)$ such that $|z_n| \to 1 $ and $|w_n | \to 1 $ but
$$ \Big| \frac{ w_n - z_n }{1- \overline{w}_n z_n } \Big | \; \; \text{doesn't converge to} \; \; 1 $$
My try
Put $z_n = 1 + \frac{1}{n}$ and $w_n = 1 - \frac{1}{n} $, then... | Any positive number can be a solution to your problem.
To prove this take $L\ge 0$.
If $L=1$ then, for $w_n=1-\frac{1}{n}$ and $z_n=1-\frac{1}{n^2}$, it holds
$$
\Bigl|\frac{w_n-z_n}{1-\bar{w_n}z_n}\Bigr|=\frac{n^2-n}{n^2+n-1}\to 1.
$$
If $L\ne 1$ then, for $w_n=1-\frac{1+L}{1-L}\frac{1}{n}$ and $z_n=1-\frac{1}{n}$, i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Variance is the squared difference - why not to the 3, or 4 instead? So there is this question about why variance is squared.
And the answer seems to be "because we get to do groovy maths when it is squared". Ok, that's cool, I can dig.
However, I'm sitting reading some financial maths stuff, and a lot of the equati... | In principle, decisions involving large amounts of money should be made using the nonlinear utility of money. However, that is subjective and hard to quantify.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Sequence $a_{n+1}=\sqrt{1+\frac{1}{2}a_n^2}$ I am trying but cant figure out anything.
$a_{n+1}=\sqrt{1+\frac{1}{2}a_n^2}$
I am trying to proove that $a_n^2-2<0$.
Getting $$a_{n+1} -a_n=\dots=\frac{2-a_n^2}{2\left(\sqrt{1+\frac{1}{2}a_n^2} +a_n\right)}$$
Then I have no clue how to proove it since I am not given $a_1$.I... | If you want a proof by contradiction: If $a_{n+1}^2 - 2 \geq 0$, then $1 + {1 \over 2} a_n^2 \geq 2$, which after a little algebra is the same as $a_n^2 - 2 \geq 0$. And a direct proof is obtained by doing these steps in the opposite direction, with the $\geq$ sign replaced by $<$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
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A measurable function (with complete measure) is sum of two other functions Let $(\Omega, \mathcal A, \mu)$ be a measure space and let $\overline{\mu}$ denote the completion of $\mu$. I have to show that if $f \colon \Omega \to \mathbb R$ is $\overline{\mu}$-measurable then $f = f_1 +f_2$, where $f_1$ is $\mu$-measurab... | Hint:
*
*Prove the claim for indicator functions, i.e. $f=1_A$ with $A \in \bar{\mathcal{A}}$.
*Extend it to simple functions.
*Let $f \geq 0$ be $\bar{\mu}$-measurable. Then there exists a sequence $(f_n)_{n \in \mathbb{N}}$ of simple functions which are $\bar{\mathcal{A}}$-measurable and satisfy $f_n \to f$ as $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How do you express the Frobenius norm of a Matrix as the squared norm of its singular values? Let the Frobenius norm of an m by n ($m \times n$) matrix M be:
$$|| M ||_{F} = \sqrt{\sum_{i,j} M^2_{i,j}}$$
I was told that it can be proved that, if M can be expressed as follows (which we can because of SVD):
$$ M = \sum^{... | $\sum_{i}\sigma_i^2=Trace(\Lambda \Lambda^T)$ where $M=U\Lambda V^T$. Then, $$\|M\|_F^2=Trace(MM^T)=Trace(U\Lambda V^TV\Lambda^T U^T)=Trace(U\Lambda \Lambda^TU^T)=Trace(\Lambda\Lambda^T U^T U)=Trace(\Lambda\Lambda^T)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Can a curve's unit normal vector be determined using the second derivative? Because $$T = \frac {r'}{|r'|}$$ I was wondering whether or not it was also valid to solve for the unit normal vector with the second derivative without first solving for T: $$N = \frac {r''}{|r''|}$$
| No, it's not. The problem is that if you differentiate $T$, which leads to a multiple of $N$, you have to use the quotient rule, not just differentiate the top and bottom. You might want to try this with a simple curve like
$
( x(t), y(t) ) = (t, t^2)
$
at $t = 1$ to see that it doesn't work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove that the Interior of the Boundary is Empty Suppose X is a Metric Space
Let S $\subset X$
Prove that if S is Closed then, the Interior of the Boundary of S is Empty
Totally stuck on how to solve this.
| It is true in general topology that the boundary of an open set has empty interior, and the same is true for a closed set.
Lemma: A set $U$ is open iff $\partial U = \bar{U}\setminus U$.
Let $U$ be an open set. Then $\partial U$ is disjoint from $U$. Suppose for contradiction that $\partial U$ contains an non-empty ope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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When writing the integral sign $\int$, how does one know what integral is being discussed? We have the Lebesgue integral and the Riemann Integral. Generally, must the integral sign $\int$ refer to one or the other exclusively or does it depend on the integrand? Can someone provide intuition to this concept?
For example... | If a function is continuous on some closed interval then the two integrals will agree, hence a distinction is not necessary. Otherwise I believe the context should be enough.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Derivative of $\frac { y }{ x } +\frac { x }{ y } =2y$ with respect to $x$ $$\frac { y }{ x } +\frac { x }{ y } =2y$$
Steps I took:
$$\frac { d }{ dx } \left[yx^{ -1 }1+xy^{ -1 }\right]=\frac { d }{ dx } [2y]$$
$$\frac { dy }{ dx } \left(\frac { 1 }{ x } \right)+(y)\left(-\frac { 1 }{ x^{ 2 } } \right)+(1)\left(\frac {... | $\text{ Assuming your y' is correct... } \\ \text{ then we should get rid of the compound fractions.. } \\ y'=\frac{\frac{-y}{x^2}+\frac{1}{y}}{2-\frac{1}{x}+\frac{x}{y^2}} \\ \text{ now we need to multiply top and bottom by } \\ x^2y^2 \text{ this is the lcm of the bottoms of the mini-fractions } \\ y'=\frac{-y(y^2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving if $ \Gamma_{2}(R)\smallsetminus J(R) $ is a forest then it is either totally disconnected or a star graph These days I am reading the research paper Graphs associated to co-maximal ideals
of commutative rings by Hsin-Ju Wang.
In this paper, $ R $ denotes a commutative ring with the identity element. $ \Gamma(R... | $(i)\Rightarrow(ii)$ If $Ra+Rb=R$ then $Ra^i+Rb^j=R$ for all $i,j\ge 1$.
Next $(R_1\times R_2)(1,0)+(R_1\times R_2)(a,1)=R_1\times R_2$, $(R_1\times R_2)(a,1)+(R_1\times R_2)(b,1)=R_1\times R_2$ since $a+b=1_{R_1}$, and $(R_1\times R_2)(b,1)+(R_1\times R_2)(1,0)=R_1\times R_2$. (For checking all these try to write $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that boundary of a closed set is nowhere dense Let $H$ be a closed set then, $Cl(H) =H$ and hence the $\partial H \subset H$.
Now to show that the boundary is nowhere dense, it would suffice to show that $Int(Cl(\partial H)) =\emptyset$,
i.e., $Int(\partial H) = \emptyset$, but how do I proceed further in order to... | Let $U$ be an open set such that $U\subset\partial H$. We'll show that $U=\emptyset$:
Since $\partial H\subset H$ (since $H$ is closed), we must have $U\subset H$. Since $U$ is open, this implies that $U\subset\operatorname{Int}(H)$.
Hence $U\subset\partial H\cap\operatorname{Int}(H)=\emptyset$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Cauchy product associativity proof Great. I need a proof that the Cauchy product is an associative operation. I can easily proof that it is a commutative operation, find identity series and find invertible and inverse series, BUT for some reason I fail to proof this damn associativity.
The proof should not use any fanc... | Here’s a very elementary approach, albeit one that’s not the kind of algebraic manipulation that you probably had in mind. Instead of just manipulating the expression, we identify the set of triples of indices that appear in terms of the two double summations. Let
$$I=\{\langle p,q,r\rangle\in\Bbb Z^3:p+q+r=n\text{ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Is matrix transpose a linear transformation? This was the question posed to me. Does there exist a matrix $A$ for which $AM$ = $M^T$ for every $M$. The answer to this is obviously no as I can vary the dimension of $M$. But now this lead me to think , if I take , lets say only $2\times2$ matrix into consideration. Now ... | The operation that transposes "all" matrices is, itself, not a linear transformation, because linear transformations are only defined on vector spaces.
Also, I do not understand what the matrix $A=M^TM^{-1}$ is supposed to be, especially since $M$ need not be invertible. Your understanding here seems to be lacking...
H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 10,
"answer_id": 4
} |
If $f$ is differentiable and $f'\geq m\geq0$, $|\int_a^b\cos{f(x)}dx|\leq2/m$
Suppose $f:[a,b]\to\mathbb R$ is a differentiable function such that its derivative is monotonically decreasing and $f'(x)\geq m>0$ for all $x\in[a,b]$. Prove that $$|\int_a^b\cos f(x)dx|\leq\dfrac{2}{m}$$
I am having some problem with mani... | Since $f'(x) \ge m > 0$, we have $$\int_a^b \cos f(x)\, dx = \int_a^b \frac{1}{f'(x)}f'(x)\cos f(x)\, dx = \int_a^b \frac{1}{f'(x)}\frac{d}{dx}(\sin f(x))\, dx.$$ Since $f'$ is monotonic decreasing and positive, by the Bonnet mean value theorem for integrals, $$\int_a^b \frac{1}{f'(x)} \frac{d}{dx}(\sin f(x))\, dx = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simplifying a binomial expression I am interested in counting the number of hyperedges in the complete, $t$-uniform hypergraph on $n$ vertices which intersect a specified set of $k$ vertices. This is trivial, the answer is:
$$\sum_{i=1}^t {k \choose i}{n-k \choose t-i}.$$
My questions is whether there is a nice simpli... | We can use the Chu-Vandermonde identity (see Equation 7 in linked page):-
$$\sum_{i=0}^t {k \choose i}{n-k \choose t-i} = {n \choose t}$$
so that the sum can be simplified to
$$\sum_{i=1}^t {k \choose i}{n-k \choose t-i} = {n \choose t}-{n-k \choose t}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Computing $\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n}\,dx$? I am trying to compute this integral/limit, I don't feel like I have any good insight...
$$\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx.$$
I have tried to make a change of variable to get rid of the $n^2$, I changed to $X=\fr... | Notice that
$$\int_0^{n^2} n\sin(x/n)e^{-x^2} dx=\int_0^{n^2} \left(n\sin(x/n)-x\right)e^{-x^2} dx + \frac{1-e^{-n^4}}{2}.$$
Moreover $|\sin(t)-t|\leq t^2$ implies $\left|n\sin(x/n)-x\right|\le n(x/n)^2=x^2/n$. Therefore
$$\left|\int_0^{n^2} \left(n\sin(x/n)-x\right)e^{-x^2} dx\right|\leq
\int_0^{n^2} \left|\left(n\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Metric space of infinite binary sequences Let $\Omega = \{0,1\}^{\mathbb{N}}$ be the space of infinite binary sequences. Define a metric on $\Omega$ by setting $d(x,y) = 2^{-n(x,y)}$ where $n(x,y)$ is defined to be the maximum $n$ such that $x_i = y_i$ for all $i\le n$. Show that $(\Omega, d)$ is a compact metric space... | HINT: Give $\{0,1\}$ the discrete topology, and let $\tau$ be the resulting product topology on $\Omega$; $\Omega$ is certainly compact in this topology, since it’s a product of compact spaces. If $\tau_d$ is the topology generated by the metric $d$, show that $\tau_d=\tau$.
Alternatively, let $\sigma=\langle x_n:n\in\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Parametric equation of an arc with given radius and two points so I need the parametric equation of the arc. So, arc is a sector of a circle.
Parametric circle equation is:
$$
c \equiv f(t) = (\cos(t), \sin(t)),\quad 0\le t < 2\pi
$$
So, we just need to find proper domain of the function, actually $t_1$ and $t_2$, star... | Given the two endpoints $P$ and $Q$, the center $C$, and the radius $r$, then
$$
s=2\arctan\left(\frac{P_y-C_y}{P_x-C_x+r}\right)
$$
$$
t=2\arctan\left(\frac{Q_y-C_y}{Q_x-C_x+r}\right)
$$
The equation would be
$$
C+r(\cos(\theta),\sin(\theta))
$$
for $\theta$ between $s$ and $t$.
Beware that there are two circular arc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Need little clarification about Neyman-Pearson Lemma According to my text-book Neyman-Pearson lemma says that the most powerful test of size $\alpha$ for testing point hypotheses $H_0: \theta=\theta_0$ and $H_1: \theta=\theta_1$ is a likelihood ratio test of the form
\begin{align*}
\phi(x)= \left\{
\begin{arr... | $l(X)$ is a function of a random variable so it is a random variable. If you can find the distribution of $l(X)$, then you can calculate the integral as $1-F_l(k)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In how many ways can you distribute 100 lemons between Dana, Sara and Lena so that Lena will get more lemons than Dana? Assume Dana has 0 lemons, so Lena must have 1 lemon. Now all i need to distribute is
$$x_1 + x_2 = 99 \text{ // because Lena already has 1 and Dana has 0}$$
The answer to above is 100.
Now assume Dana... | Alternatively, let $D, S$ and $L$ be the values, and let $L=D+1+L_0$. Then you want non-negative integer solutions to $1+D+S+L_0=100$, or $2D+S+L_0=99$.
You are doing $$\sum_{D=0}^{49} \sum_{S=0}^{99-2D} 1=\sum_{D=0}^{49} (100-2D)$$ which is a correct way to count this value.
A generating function solution would be to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Basic property of a tensor product I think that this might follow from a basic properties of tensor products, but I am q bit stuck...
Let $A$ be a $k$-algebra. Let $l/k$ be a finite field ext. of $k$. Suppose
$A \otimes_k l$ is an integral domain. Does it follow that $A \rightarrow A \otimes_k l$
defined by $a$ to $a ... | If $V$ and $W$ are two $k$-vector spaces, and $w\in W$ is nonzero, then the map $V\to V\otimes_k W$ sending $v$ to $v\otimes w$ is always injective, because vector spaces all have a basis, and are therefore flat.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Uniform convergence on a measurable set implies convergence a.e.? Suppose for each $\epsilon$ there exists a measurable set $F$ such that $\mu(F^c) < \epsilon$ and $f_n$ converges to $f$ uniformly on $F$. Prove that $f_n$ converges to $f$ a.e.
I have been thinking about this question for a while and I am not quite sure... | This can be proved by contradiction. Suppose there is a measurable set $A$ and $\delta > 0$ such that $\mu(A) \ge \delta > 0$ and $f_n$ does not converge to $f$ pointwise on $A$, (such a set exists if $f_n$ does not converge a.e., then $f_n$ does not converge uniformly on any set intersecting $A$.)
Then, for all $\epsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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For what values of $p$ does $\sum_{k = 1}^{\infty} \frac{1}{k\log^p(k+1)}$ converge? Find all $p\geq 0$ such that the following series converges $\sum_{k = 1}^{\infty} \frac{1}{k\log^p(k+1)}$.
Proof: the general term for the series is $\frac{k^p}{k^p\log^p(k+1)^n} = \frac{1}{k\log^p(k+1)^n}$.
By comparison, $\frac{1}{\... | By Cauchy condensation test all gets reduced to the study of
$$\sum_{k = 1}^{\infty} \frac{1}{k^p}$$
whence we get the conclusions according to the values of $p$.
Q.E.D.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Riemann-Stieltjes Integration problem I have two functions $f$ and $g$ and I need to show that $f$ is Riemann-Stieltjes integrable with respect to $g$. I was able to calculate the integral, but I'm not sure how to actually prove why it is Riemann-Stieltjes integrable.
Let
\begin{align*}
f(x) &=x^2 \qquad x \in [0,5]\\
... | Because $f$ is continuous, it is sufficient to show that $g$ is of bounded variation (equivalently: $g$ is the difference of two monotone functions).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find general solution $y''y^3=1$ I'm having this question in my homework assignment in Linear Algebra and diffrential equation class, and trying to find the general solution for this second ODE.
$$y''y^3 = 1$$
Using substitution I said $p = y'$ and $p' = y'' \rightarrow \frac{dp}{dx}= \frac{dp}{dy} \times \frac{dy}{d... | Update This is now a complete solution.
Let $v(x)=y(x)^2$. Then $v'=2yy'$ and so (here we assume that $v\neq 0$)
$$
v''=2(y')^2+2yy''=\frac{1}{2}\frac{(v')^2}{y^2}+\frac{2}{y^2}=\frac{1}{2}\frac{(v')^2}{v}+\frac{2}{v},
$$
or
$$
vv''-\frac{1}{2}(v')^2-2=0
$$
This differential equation can be solved as follows. Different... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Is there a commutative operation for which the inverse of the operation is also commutative? For instance addition is commutative, but the inverse, subtraction, is not.
$$
5+2 = 2+5\\
5-2 \neq 2-5
$$
Same for multiplication/division:
$$
5\times4 = 4\times5\\
5/4 \neq 4/5
$$
So is there a group operation $\circ$ with th... | Note that $-$ (minus) is really just a short way of writing $+$ something negative.
In fact, what you call $\circ^{-1}$ is just the composition $$G \times G \xrightarrow{id \times inv} G \times G \xrightarrow{\cdot (mult)} G$$
So the condition you are asking for $a \circ^{-1} b= b \circ ^{-1} a$ is equivalent to the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1144995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 6,
"answer_id": 3
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Sketch a graph that satisfies the following conditions
Sketch the graph of an example of a function f(x) that satisfies all of the following conditions:
Here is what I have so far:
Am I on the right track? I think the graph satisfies all of the conditions, but the lines cross at about (2,3)- is that acceptable? I... | While, you catches all of the conditions, except #3, which is at $(0,-1)$, which is an isolated point.
But your plot has multiple y values for $x \in (0,2)$, which is not right.
What you should do is:
Connect $(2^+, -\infty)$ to $(+\infty, 3)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1145046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how to find an integral curve in Lie group? Given a Lie group $G$, $e$ is its identity element and $g$ is one element of $G$. I want do find a curve $\gamma(t)$ that satisfies these conditions:
1) passes $g$ and $e$, that is $\gamma(0)=e,\gamma(t_g)=g$;
2) the tagent vector at $e$ is $v\in T_eG$, that is $\gamma'(t)|_{... | In general no for left invariant vector fields, since $\gamma(t_g)=\exp(t_gv)$ may not be equal to the $g$ you want. For general case, of course...say $g=\exp(t_gu)$, then define $\gamma(t)=\exp(tv+t^2(u-v)/t_g)$.
| {
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"url": "https://math.stackexchange.com/questions/1145148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Algorith/ Equation to get the ith element in N x N I am having a difficulty figuring out the equation to get the ith element in $\mathbb{N}\times \mathbb{N}$ ( crossing the set of natural numbers).We have $\mathbb{N}\times \mathbb{N} = \{ (1,1),(1,2),(2,1),(1,3),(2,2)\dots\}$ if we traverse the matrix with a diagonal.C... | I believe I got your question and you can find several solutions here:
https://stackoverflow.com/questions/1779199/traverse-matrix-in-diagonal-strips
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of this function at 0+ Let $$f(x):= x\log\left(\frac{1}{x}\right) \exp((-\log x)^{\alpha}) .$$
where $\alpha$ is fixed.
My question is, what is the limit $\lim_{x \downarrow 0} f(x)$ ?
This is easily shown to be $\infty$ for $\alpha \in [1,\infty)$, and also it's easy to see that this limit is $0$ for $\alpha \... | Set $$y = -\log x = \log\left(\frac{1}{x}\right)$$ to get $$\begin{align}x\log(\frac{1}{x})\exp((-\log x)^\alpha)=y\exp(-y+y^\alpha) \\ = \frac{y}{\exp(y-y^{\alpha})}\end{align}$$ Then since $$\lim_{x \rightarrow 0}y= \lim_{x \rightarrow 0} (-\log(x))= \infty$$ and numerator and denominator tends to infinity you can ap... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Infinite sum of asymptotic expansions I have a question about an infinite sum of asymptotic expansions:
Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$
with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^3}$, $a_{2k}\leq \dfrac{1}{k^4}$, $\cdots$.
Does it follow that $\sum_{k=1}^{\inf... | There are a lot of implicit limits you're juggling at once, so I would suspect that this is not true in general. It does hold though if you have some additional uniformity assumptions. For instance...
Suppose in addition that
*
*you can find a positive sequence $(b_{kn})$ such that, for all integers $n > 0$,
$$
\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1145492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability of going into an absorbing state If I have a random walk Markov chain whose transition probability matrix is given by
$$
\mathbf{P} = \matrix{~ & 0 & 1 & 2 & 3 \\
0 & 1 & 0 & 0 & 0 \\
1 & 0.3 & 0 & 0.7 & 0 \\
2 & 0 & 0.3 & 0 & 0.7\\
3 & 0 & 0 & 0 & 1
... | The probability that we get to state zero immediately is $0.3$.
The next possibility is that we get to state two then we get back to state one and then to state zero, the probability of which event is $0.7\cdot0,3\cdot0.3=0.7\cdot0.3^2$. The probability of the next possibility is $0.7\cdot0.3\cdot0.7\cdot0.3\cdot0.3=0.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
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Prove using a proof sequence and justify each step Prove using a proof sequence that the argument is valid [ A --> (B ∨ C) ] ∧ B' ∧ C' --> A'
I'm having some trouble figuring the proof out here. Here is what I have so far. Is this on the right track?
*
*A-->(B ∨ C) (Given)
*B’ (Given)
*C’ ... | $\color{gray}{\boxed{\color{black}{
\because\quad \text{Assuming } A
\\[2ex]
\qquad A ,\; [ (A\to B\vee C)\wedge B' \wedge C' ]
\\ \quad \vdash (\text{ conjunction elimination: } S\wedge T \vdash S, \text{ and } S\wedge T\vdash T)
\\ \qquad A ,\; (A\to B\vee C) ,\; B' ,\; C'
\\ \quad \vdash (\text{ implication elimina... | {
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Why is an open ball in $\mathbb R^n$ not compact? By definition of compactness, an open cover of an open ball in $\mathbb R^2$ always has a collection of subcovers that cover the ball. But why is a open ball not compact?
| For any open ball $B(x,r)$ where $x \in \mathbb{R}^m$ and $r \in \mathbb{R}$, the cover given by the collection $\{ B(x,r - \tfrac{1}{n}) \}$ where $n \in \mathbb{N}$ is an open cover of $B(x,r)$ but no finite subcover will cover it.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$T$ is linear. Show: $T$ is onto if and only if that $T$ maps spanning sets to spanning sets Prove that $T$ is surjective if and only if for every set of vectors $\{v_1,...v_k\}$ that span $\mathbb{R}^m$, the set $\{T(v_1)...T(v_k)\}$ spans $\mathbb{R}^n$.
How would I prove this?
a surjective function is a function who... | Let $T:\Bbb R^m\to\Bbb R^n$ be a surjective linear map and let $v_1,\dotsc,v_k\in\Bbb R^m$ span $\Bbb R^m$.
To show that $\{T(v_1),\dotsc,T(v_k)\}$ spans $\Bbb R^n$, let $w\in\Bbb R^n$. Since $T$ is surjective there exists a $u\in\Bbb R^m$ such that $T(u)=w$. Since $\{v_1,\dotsc,v_k\}$ spans $\Bbb R^m$ there exist $\l... | {
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"timestamp": "2023-03-29T00:00:00",
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Find an example of a regular triangle-free $4$-chromatic graph Find an example of a regular triangle-free $4$-chromatic graph
I know that for every $k \geq 3$ there exists a triangle-free $k$-chromatic graph.
So if I can find a triangle-free graph $H$ such that $\chi(H)=3$, then I can use the Mycielski construction to ... | Another example is given by Kneser graphs $K(n,k)$ with suitable parameters.
By Lovasz' theorem, the chromatic number of $K(n,k)$ is given by $n-2k+2$.
Moreover, if $n<3k$ we have that $K(n,k)$ is triangle-free, so:
$\color{red}{K(8,3)}$ is a triangle-free, $10$-regular graph on $56$ vertices with chromatic number $\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why do repeated trigonometric operations approach seemingly arbitrary limits? So I was messing around on my iPhone calculator trying to find the the precision of the calculator by finding at what point sin(x) was equal to x. I found myself repeating the sine function sin(sin(sin(....sin(x)...)))). Predictably the limit... | Turns out these values aren't arbitrary. Rather, they are the approximate solutions to $\cos x = x$ in radians and degrees.
(In the last line of your post you mean "...for repeated cosine operations in degrees".)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove this Matrix is Positive Definite Suppose
u $ \neq 0 \in$ $\mathbb{C}^m$
$\alpha \in \mathbb{R}$
For what Values of $\alpha$ is $I + \alpha uu^{*}$ Positive Definite?
Progress so far :
$\forall x \neq 0 \in \mathbb{C}^m$
We Have $x^*Ix$ > 0
I cant determine the values of $\alpha$ which make the overall quantity ... | Hint:
$$
x^*\alpha uu^* x = \alpha (x^*u)(u^*x) = \alpha (x^*u)\overline{(x^*u)} = \alpha |x^*u|^2
$$
then use Cauchy-Schwarz; this will yield an $|x|^2$ to match $x^*Ix = |x|^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How do I find the kernel of a composition of functions? Functions $g$ and $f$ are linear and injective. How do I go about finding the kernel of $g \circ f$?
I'm asking because I want to prove that $\ker(f) = \ker(g \circ f)$.
| If $g$ is injective then $\ker(f) = \ker(gf)$ is certainly true. To prove two sets are equal you show that each is contained in the other. For example if $x \in \ker(f)$ then $f(x) = 0$. But then $gf(x) = g(0) = 0$ so $x \in \ker(gf)$. This proves $\ker(f) \subseteq \ker(gf)$. I'll leave the other direction to you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to derive the solution in quadratic optimization I'm reading the book "Convex Analysis and Optimization" written by Prof. Bertsekas. In Example 2.2.1, there are the following description:
I don't know how to derive the equation 2.2. Could anyone help give a hint, please? Thanks!
UPDATE:
With the references kindly ... | See this:
http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/least-squares-determinants-and-eigenvalues/projections-onto-subspaces/MIT18_06SCF11_Ses2.2sum.pdf
It shows you how to project a vector onto the span of $A$.
Then here:
http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-20... | {
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"url": "https://math.stackexchange.com/questions/1146291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If p is an odd prime, prove that $a^{2p-1} \equiv a \pmod{ 2p}$ Let $m = 2p$
If p is an odd prime, prove that $a^{2p - 1} \equiv a \pmod {2p} \iff a^{m - 1} \equiv a \pmod m$.
I have no idea on how to start. I was trying to find a form such that
$a^{m - 2} \equiv 1 \pmod m$. But I got stuck. Can someone give me a hint ... | Hint: $$\phi(2p)=\phi(p)$$
for all odd primes where $\phi$ is the Euler-phi function.
Edit:
$$a^{\phi(2p)}\equiv a^{\phi(p)}\equiv a^{p-1}\equiv 1 \pmod {2p}$$
Hence $a^p\equiv a$ and $a^{p-1}\equiv 1 \Rightarrow a^{2p-1}\equiv a \pmod {2p}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
$G/Z(G)$ is cyclic then is abelian? my question is if $G/Z(G)$ is order of p then is commutative then is abelian group but if G is abelian then $G=Z(G)$ therefore $G/Z(G)$ is not order of p?Is it contradiction?
| It shows $|G/Z(G)|=$ prime $p$ can't hold. The only way $G/Z(G)$ can be cyclic is in the most obvious of ways: when it is trivial. It is a useful fact that shows up a lot in exercises. E.g., reasoning by contradiction at some step you end up concluding that the index of $Z(G)$ in $G$ is prime, and, voilà, you get that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluation of definite integral in terms of Bessel function Can I express the integral $\int_0^1[\cos (xt)/(1-t^2)]dt$ in terms of Bessel Polynomial? I tried by putting $t=\sin \theta$ and used the integral representation of Bessel's polynomial $J_n(x)=(1/\pi)\int_0^\pi \cos(n\theta-x\sin \theta)d\theta$. I expect the ... | Due to issues relating to convergence, the only values of x for which the integral converges are
of the form $x=(2k+1)~\dfrac\pi2$ , and the result is $I_{2k+1}=\dfrac\pi4\sqrt{|2k+1|}\cdot J^{(1,0)}\bigg(-\dfrac12~,~|2k+1|\dfrac\pi2\bigg)$,
for all $k\in\mathbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve this linear congruence equation and more general cases? Okay so I'm trying to solve $5x \equiv 7 \mod 11$ and this is the particular example that I can't do. Can someone help me learn how to solve these and more general examples $ax \equiv b \mod n$. I believe there is only one solution (well infinitely ma... | Since 11 is a prime number, any number congruence it has an inverse:
$$5x \equiv 7 \mod 11$$
$$5 \times 9 =45= 4\times 11+1$$
$$5^{-1} \equiv 9 \mod 11$$
$$5^{-1} \times 5 x\equiv 5^{-1} \times 7 \mod 11$$
$$x\equiv 9 \times 7 \equiv 63 \equiv 8 \mod 11$$
$$x\equiv 8 \mod 11$$
If 11 was not prime, you had to try 11 c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Real analysis - proof approach help I am taking a course in Real Analysis this semester and thought I would work ahead a little bit.
I am just reading for the moment, but came across an interesting exercise, and was wondering if I could see how one of you would solve it.
Assume that $A$ and $B$ are nonempty, bounded ab... | I have now attempted this by myself and will post my proof here for critique. I welcome the most prudent critiques you can offer.
Firstly, since $A$ and $B$ are nonempty, and bounded above, they satisfy the axiom of completeness and thus we know that $\sup A$ and $\sup B$ exist.
Since $B \subseteq A$, then for all $b \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
The number $90$ is a polite number, what is its politeness?
The number $90$ is a polite number, what is its politeness?
A. $12$
B. $9$
C. $6$
D. $14$
E. $3$
How did you get that answer? I tried Wikipedia to figure out what a polite number was and how to figure out its politeness but I'd like to see it done step by st... | A polite number, it seems, is a positive integer $n$, such that there is a list of consecutive positive integers $a, a+1,\dots, a+r$ with $n = a + (a + 1) + \dots + (a + r)$.
The politeness is the number of representations of a polite number. For example $9$ is polite and its only representations are $2+3+4$ and $4+5$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Calculating the limit: $\lim \limits_{x \to 0}$ $\frac{\ln(\frac{\sin x}{x})}{x^2}. $ How do I calculate $$\lim \limits_{x \to 0} \dfrac{\ln\left(\dfrac{\sin x}{x}\right)}{x^2}\text{?}$$
I thought about using L'Hôpital's rule, applying on "$\frac00$," but then I thought about $\frac{\sin x}{x}$ which is inside the $\... | As $\dfrac{\sin x}{x}=1-\dfrac{x^2}6+o(x^3) $, we have:
$$\frac{\ln\Bigl(\cfrac{\sin x}{x}\Bigr)}{x^2}=\frac{\ln\Bigl(1-\cfrac{x^2}6+o(x^3)\Bigr)}{x^2}=\frac{-\dfrac{x^2}6+o(x^3)}{x^2}=-\frac16+o(x),$$
which proves the limit is $\,-\dfrac16$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
} |
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