1
vote
0answers
7 views

Help understanding proof on Jensen's Inequality

I need help understanding the proof for Jensen's inequality in "Real and Complex Analysis" by Rudin. 3.3 Theorem (Jensen's Inequality) Let $\mu$ be a positive measure on a $\sigma$-algebra ...
0
votes
0answers
7 views

Collision between moving circular discs

I am trying to figure out how to detect collision between two moving circular discs that move along a pretedermined path with a known speed. Example: Circular disk A moves along the path AB with ...
0
votes
1answer
20 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$

Express the number $4$ and $5$ and $6$ and $7$ and $8$ with trigonometric identities or series or equations. example: Express the number $1$, $$cos^2 x + sin^2 x=1$$ Express the number $2$, ...
0
votes
0answers
14 views

Looking for help to understand example of Group

I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra". It says, Let G be the set of all 2x2 matrices $$\begin{pmatrix} a ...
0
votes
0answers
6 views

The number of ways to draw boundaries of constituencies, subject to constraints

A state comprises 45 counties arranged as 5 rows, running east and west, of 9 counties each, the nine colums of 5 running north and south. They're connected horizontally and vertically, i.e. ...
0
votes
0answers
8 views

Zeros of an analytic function

How to prove zeros of an analytical function (non-zero function) is always countable?
0
votes
0answers
6 views

Infinite sub-sequences that make up a sequence

A sequence $\{a_n\}$ can be broken into sub-sequences, $\{a_n\}^1_{k_1}, \{a_n\}^2_{k_2}, \dots,\{a_n\}^m_{k_m}$, if every element in $\{a_n\}$ belongs to at least one of the sub-sequences. I had to ...
1
vote
0answers
23 views

Rigorously proving $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx= \frac{\pi}{2}$

I want to prove the famous formula: $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx = \frac{\pi}{2}.$ There are many ways to do it, for example, by some Fourier analysis. But how about a simple ...
0
votes
2answers
10 views

How to prove that E's limit point must be in E? (rudin)

How to prove that E's limit point must be in E? Thm 2.23 E's open iff E_c is closed. First, suppose E_c is closed. Choose x belongs to E. Then x doesn't belongs to E_c, and x is not a limit point ...
0
votes
0answers
5 views

Inverting change of basis matrices to get back the original coordinate vector

Let $B=\{b_{1},b_{2}\}$ and $C=\{c_{1},c_{2}\}$ be bases for a vector space $V$, and suppose $b_{1}=-2c_{1}+4c_{2}$ and $b_{2}=3c_{1}-6c_{2}$. a. Find the change of coordinates matrix from $B$ to ...
0
votes
2answers
12 views

Confused about proof of division

I thought I was familiar with the regular euclidian algorithm, but I am having trouble understanding a step in this proof from my notes, I am looking for any clarification. $\mathbf{Thereom:}$ Let ...
0
votes
0answers
3 views

Cubic spline interpolation

The question is "Determine the natural cubic spline that interpolates the function f(x) = x^6 over the interval [0,2] using knots 0, 1, 2."
1
vote
0answers
6 views

Bayes theorem with multiple conditions

how to calculate P(a|b,c,d). knowing that b, c and d are NOT independent from each others ? i know how to solve it if there is independency assumption. however, i am just wondering if there is any ...
0
votes
0answers
14 views

$\|P_n\| \to 0$ and $\|Q_n\| \to 0$, but such that $\lim_n S(f; P_n) \neq \lim_n S(f; Q_n)$ · TO show that $f$ is not Riemann Integrable.

Suppose that $f$ is bounded on $[a, b]$ and that there exists two sequences of tagged partitions of $[a, b]$ such that $\|P_n\| \to 0$ and $\|Q_n\| \to 0$, but such that $\lim_n S(f; P_n) \neq \lim_n ...
2
votes
3answers
49 views

Is :$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$ irrational or transcendental or real number?

Is there someone who can show me if :$$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$$ is irrational or real or transcendental Thank you for any help
1
vote
1answer
13 views

Show that $|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$

Consider the random variables $X$ and $Y$ defined in the same space $\Omega$. Show that $$|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$$ This question comes from an old test, I know that ...
0
votes
1answer
17 views

solving equation in terms of $w_1$ and $w_2$

I have a a physics problem involves the following equation $$\tan(\alpha) = \frac{(w_1 + w_2)^{1/2}}{w_3}$$ from a certain set of equations that I use I derive the following equation: ...
0
votes
1answer
17 views

Can two irreducible polynomials have different powers of the same real number as roots?

Say we have two irreducible polynomials in $Q [x] $. We call them $f, g$. Say one of the roots of $f $ is $a$. Is it possible that $g$ satisfies a root of the form $a^n$ for some natural number $n $? ...
0
votes
1answer
19 views

Looking to understand proposition related to the fundamental theorem of algebra

I am having some problem understanding exactly what the following proposition is saying. Also, is this result have a common name? How important it is, etc. It is $\mathbf{Proposition:}$ Let ...
0
votes
1answer
9 views

Numerical Integration by Undetermined coefficients

The important part of my question is after the bold "Now" The method of undetermined coefficients is defined so that the error of a function $f(x)$ to be integrated is zero. I.e. $E=\int_{a}^{b} ...
0
votes
4answers
33 views

Find $y'$ and $y''$ : $ y=x^2\ln(2x)$

for $x> 0$ : $ y=x^2\ln(2x)$ Product rule: $$(x^2)\cdot[\ln (2x)]'+ (\ln (2x))\cdot[x^2]' $$ $$y'= x^2\frac{1}{2x}\cdot (2)+\ln(2x)\cdot(2x) =2x\ln(2x)$$ ...
0
votes
0answers
17 views

Is there a function that shows n!-lcm(1,2,3…n)?

1!-lcm(1)=0 2!-lcm(1,2)=0 3!-lcm(1,2,3)=0 4!-lcm(1,2,3,4)=12 5!-lcm(1,2,3,4,5)=60 I have not looked into this incredibly deeply, but I found that there may be some sort of connection with primes, ...
0
votes
0answers
25 views

How can I raise a Taylor Series to a power?

I have recently been undertaking the challenge of finding the antiderivative of $x^x$. In doing so, I have come across the idea of raising a Taylor series to a variable exponent. I came to the ...
2
votes
1answer
34 views

Function that decays faster than any polynomial, but not in the Schwartz space?

Motivated by the very restrictive condition imposed in the definition of the Schwartz space, I was wondering about the following question. Is there a $C^\infty$ function that decays faster than ...
1
vote
0answers
6 views

Proof by reflection and Homotopy Type Theory

I have been looking into various proof assistants, and came across the method of proof by reflection in Coq, which (from what I understand) allows one to verify a program provides the correct "answer" ...
0
votes
1answer
35 views

Why does the following limit give two answers?

I want to calculate $$ \lim_{t \to 0} \frac{t^2}{sin^2(t)}$$ and I proceed as follows $$\stackrel{H}{=} \lim_{t \to 0} \frac{2t}{2sin(t)cos(t)} \implies \lim_{t \to 0} \frac{2t}{sin(2t)}$$ and ...
1
vote
1answer
8 views

Numbers written into a square grid

I was working on a problem from The Art and Craft of Problem Solving by Zietz, in the chapter called 'The extreme principle.' Here is the problem: "The integers from 1 to $n^2$ are written into a ...
1
vote
0answers
9 views

a differential geometry question, is $k_\alpha(s) \ge |k_\beta(\alpha(s))|{d\sigma\over ds}$ true or wrong?

let $ \alpha(s)$(s is the arc-length of $\alpha$) be a closed curve in three dimensional space, let $\beta$ be orthogonal projection of $\alpha(s)$ in xy plane. if so, we have ...
0
votes
3answers
24 views

Calculate roots from $\frac{x \cosh(x) - \sinh(x)}{x^2}$

I want to solve the following equation $$f(x) = \frac{x \cosh(x) - \sinh(x)}{x^2} = 0$$ Because the term above is undefined for $x = 0$ I calcuted $$\lim_{x \rightarrow 0}\frac{x \cosh(x) - ...
2
votes
1answer
22 views

Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
0
votes
1answer
18 views

Unique factorization domain: $\mathbb{Z}_{n}[x]$

How to determine all $n\in\mathbb{N}$ such that $\mathbb{Z}_{n}[x]$ is a unique factorization domain? I am guessing that this would be true for all primes, since $\mathbb{Z}_n$ is a UFD when $n$ is ...
0
votes
2answers
33 views

Find six triples of positive integers $(a, b, c)$ such that in $ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$.

Solve for $a, b$ and $c$ in the following equation such that Find six triples of positive integers (a, b, c) such that $$ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$$ I have tried various ...
-1
votes
0answers
15 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides (a, a, b) has the same area and the same perimeter as a triangle with sides (c, c, d) where a, b, c and d are positive integers ...
0
votes
2answers
12 views

How to prove this claim about this function?

I thnk first I need to show the that the function is monotonic. Or maybe use it for the proof of this claim : for all $$c > 0 % MathType!MTEF!2!1!+- % ...
0
votes
1answer
15 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
0
votes
1answer
32 views

Six variables. System of equations.

$$ \begin{align} x & =\frac{R+\frac{G+B}{-2}}{R+G+B} \\[10pt] y & =\frac{\frac{(G-B) \sqrt{3}}{2}}{R+G+B} \\[10pt] z & =R+G+B \end{align} $$ How do I get the formula for ...
0
votes
0answers
11 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
0
votes
0answers
10 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
2
votes
4answers
38 views

What does $f^{-1}(B)= \{ x \in X \mid f(x) \in B\}$ mean?

I have encountered the expression $$f^{-1}(B) = \{ x \in X \mid f(x) \in B\}$$ My questions are: 1) What does the $-1$ exponent mean in this context? 2) Is it right to say "if the set $X$ ...
0
votes
0answers
10 views

Difference between stochastic process and chaotic system

Can anyone please point out some difference and similarity between stochastic system and chaotic system?
0
votes
0answers
15 views

Determining parity of the multiplicative inverse?

Let $\mathbb{F}_p$ be a finite field of characteristic $p > 2$. Represent the $p$ elements of the field as integers $\{0,1,\ldots p-1\}$. Now, lets divide the elements of the field in two groups by ...
1
vote
0answers
77 views

Can this we define the zeta function like this?

Background We define a smooth continuous function function where: $$ p(i) = p_i $$ where p_i is the i'th prime. We also define the following series: $$ \alpha(s) = (\ln(2))^s + (\ln(3))^s + ...
0
votes
0answers
12 views

Derivative of indicator function and summation

I would like to take the derivative of $G = \sum_{i=1}^{n}\Big(\mathbb{1}\{i \geq x + k\} v(x) + \mathbb{1}\{i < x + k\} v(i)\Big)$ with respect to $x$, where $\mathbb{1}\{\cdot\}$ is the indicator ...
0
votes
2answers
17 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
0
votes
1answer
47 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
0
votes
1answer
29 views

Show that $\sup (A\cdot B)=\max\{\sup A\cdot\sup B, \sup A\cdot\inf B,\inf A\cdot\sup B,\inf A\cdot\inf B\}$

Given nonempty subsets $A$ and $B$ of positive real numbers, define $$A\cdot B=\{z=x\cdot y:x\in A,\,y\in B \}$$ show that if $A$ and $B$ are bounded sets of real numbers, then $$\sup(A\cdot ...
0
votes
0answers
21 views

How to solve this kind of recurrence relation in closed form? $F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$

How to solve this recurrence relation in closed form? $$F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$$ I know how to solve recurrence relations for less than four calls by solving the ...
2
votes
2answers
21 views

Special case on counting in a string of 7 letters

I have the following question: Suppose $S_7$ is the set of all strings of length seven that can be formed with the letters $A, B, C, D, E, F$ and $G$ when repetitions are allowed. How many strings ...
2
votes
0answers
41 views

What is the maximum amount of solutions to $f(x+1)f(x)= ax^2+bx+c$.

$f(x)$ is going to be in the form $mx+h$ thus, $(mx+m+h)(mx+h) = ax^2+bx+c$. With basic algebra $m= \pm \sqrt{a}$. Also $(m+h)(h)=c$. I would guess that because $(m+h)h=c$ has two solutions max if $m$ ...
-2
votes
1answer
11 views

Central vertex of a cyclic graph and of a complete graph

What is the central vertex for a cyclic graph $C_n$? and for complete graph $K_n$? The eccentricity is the same for all vertices!

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