1
vote
1answer
51 views

Convergent series, is this telescoping?

Let $\left(x_{n}\right)_{n}\subset\mathbb{R}$ and $v_{n}:=\sum_{k=1}^{n}|x_{k+1}-x_{k}|$. If $\left(v_{n}\right)_{n}$ is bounded, prove that both $\left(v_{n}\right)_{n}$ and $\left(x_{n}\right)_{n}$ ...
4
votes
2answers
105 views

Is $a \le b$ a true statement if $a < b$? [duplicate]

My question is: Is $a \le b$ true if $a < b$? For instance: Is $3 \le 4$ a true statement? I think yes, because $a \le b$ is defined as $a < b\vee a = b$ and this should be true, even if $a = ...
0
votes
2answers
65 views

Simple equation need explain

Can anyone please explain me why this equation is true: $$f(x) = \ln(x-(x^3-1)^{1/3}) = \ln\left(\frac{1}{x^2+x\cdot(x^3-1)^{1/3}+(x^3-1)^{2/3}}\right)$$
1
vote
1answer
55 views

Find a subbasis for metric topology of $\mathbb R^2$ such that it's not a basis itself

Find a subbasis for metric topology of $\mathbb R^2$ such that it's not a basis itself . I know that metric topology is the collection of open balls in $\mathbb R^2$ I also know how to use the ...
3
votes
1answer
110 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
0
votes
1answer
76 views

suggestion about elliptic curves

I have read a little bit of number theory and covered upto Kummer's proof of Fermat's Last Theorem for regular primes. I am familiar with the concepts like disciminant, class number. Could anyone tell ...
0
votes
0answers
51 views

About some special kinds of group automorphisms

let $G$ be a finite group with $1\neq Z(G) \lneqq G$. Also let $H=\{x_1,...,x_n\}$ be the set of all disjoint representative elements of right cosets of $Z(G)$ in $G$. Is there any non-trivial ...
0
votes
3answers
44 views

Adding square roots

I forgot how to add square roots, can someone please show me what I'm supposed to do? Example: Add $\sqrt{8} + \sqrt{32}$ I did: $\sqrt{8} = \sqrt{4}\sqrt{2}$ $\sqrt{32} = \sqrt{4}\sqrt{8}$ ...
1
vote
1answer
93 views

product of different order Bessel function integral

$\displaystyle w = \int_0^\infty r\; J_\mu(ar)\;J_\theta(br)\; \text{d}r $ I'd like to solve this integral ,where a and b are real and positive constant. any information regarding this integral help ...
0
votes
3answers
1k views

Find a second degree polynomial that goes through 3 points

I am having trouble calculating the quadratic curve $f(x)$ that goes through 3 points; for example a curve that goes through $A(1,3), B(-1,-5), and C(-2,12)$. I can only guess that the curve is ...
2
votes
1answer
84 views

Finding when the distances to three cities again have different digits

Very confused on this question. How would you solve it, and what would be the answer(s). Recently I was driving down the freeway and spotted the following freeway sign with the distances to three ...
2
votes
2answers
92 views

My homework about algebra

Please help Find the fraction of $$\frac{3^{2^{2555}}+3^{2^{2554}}+3^{2^{2553}}+\cdots+3^{2^{2012}}}{2^{941}}$$ Working method show is really appreciate Thank you Poom
0
votes
0answers
36 views

This shows all double integrals are zero; spot the mistake

Say we have some integral $\int_0^{\pi}d\phi_1\int_0^{\pi}d\phi_2f(\phi_1,\phi_2)\ .$ If I make the substitution $\psi_1=\phi_1-\phi_2$ and $\psi_2=\phi_1+\phi_2$, using the determinant of the ...
2
votes
0answers
112 views

Multiple Hypergeometric Distributions

I need to figure out a problem which involves multiple hypergeometric distributions. Referring to the Urn problem, the problem can be described like the following: We have $n$ urns $u_1,…,u_n$. Urn ...
1
vote
0answers
77 views

Describing the sequence A224239.

I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, ...
0
votes
1answer
20 views

Regular expression for a particular language

Several years ago I came across a paper that defined a regular expression (or collection of regular expressions?) for a specific language. The language is the language of set partitions enumerated by ...
1
vote
2answers
40 views

Estimating two constants from fitted equation

I have an equation $A=\alpha[1-\exp(-\beta\cdot B)]$ where $A$ and $B$ are known 20x1 column vectors and $\alpha$ and $\beta$ are unknown constants. I'm probably missing something really simple but ...
1
vote
0answers
77 views

Stationary action functional

The last part in the derivation of the Euler-Lagrange equations for a stationary action has me confused. It's about the order of differentiation and evaluation, and whichever comes first. I'll ...
1
vote
1answer
67 views

Is there a name for a topological space $X$ in which Every closed subset $A\subsetneq X$ is contained in a countable union of compact sets

As was recommended for me in here I would like to share the following question with you: Is there a name for a topological space $X$ which satisfies the following condition: Every closed subset ...
1
vote
2answers
625 views

find the solution of differential equation that passes through the indicating points

Find the solution of differential equation that passes through the indicating points $dy/dx-y^2 = -9$ (0,3) I have tried to solve it $dy/dx = -9 + y^2$ $dy/dx = (y)^2 - (3)^2$ $dy/dx= ...
0
votes
1answer
82 views

Why do level curves of a function and its harmonic conjugate intersect each other orthogonally?

So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally. The proof itself wasn't that difficult, ...
0
votes
1answer
81 views

Column math please help

If I'm adding the sum of £100 +£320+£220+£20+£6+50p+20p+10p+5p+2p, how would I write that using column math. Thanks
1
vote
1answer
44 views

Poisson distribution for rare event

I was taught in school that Poisson distribution is usually used to model rare events. And I understand the Poisson process is such that the probability of an event in one interval is independent of ...
-1
votes
1answer
41 views

What is present value of the carbon expense for five years?

Suppose I have computed the cost of carbon per mile for my car at 0.009 per mile. Assume that the interest rate is 5% and that I drive the car 20,000 miles per year. What is present value of the ...
0
votes
0answers
41 views

Basis and their orientation

let V be a vectorspace with $v_1 = (3,2,1), v_2 = (2,2,1), v_3 = (1,1,1)$. Do the two basis $A = (v_1, v_2, v_3)$ and $B = (v_2, v_3, v_1)$ have the same orientation? Since this is a new thematic for ...
1
vote
2answers
94 views

Are there hidden events?

Consider the sample space S = {a, b, c, d} and a probability function Pr : S ->R on S. De fine the events A = {a}, B = {a, b}, C = {a, b, c}, and D = {b, d}. You are given that Pr(A) = 1/10, Pr(B) = ...
1
vote
1answer
35 views

Largest Singular value of a Matrix

Prove that if $A\in \mathbb{R}^{m\times n}$, then $$\sigma_{\text{max}} (A) = \underset{y\in\mathbb{R}^m\\x\in \mathbb{R}^n}{\text{max}}\frac{y^TAx}{\Vert{x}\Vert_2\Vert y\Vert_2}.$$
1
vote
1answer
52 views

Geometrical Calculus - Mini-Max Problem

Two vehicles are heading for a crossroad (point C) and intend to pass straight through. Vehicle A is $100$ km due North travelling at $80$ km/hr towards C Vehicle B is $150$ km due East travelling at ...
0
votes
1answer
80 views

weak-*-convergence of measures ==> convergence of the total mass?

Let $X = [0,1]$. Let $\mu_n$ be a sequence of regular signed Borel measures on $X$, which converges to a measure $\mu$ on $X$ in weak-star, i.e. for any $f\in C_0(X)$, we have $\int_X f \mu_n(dx) \to ...
1
vote
1answer
99 views

When is a minimum distance decoder also a maximum likelihood decoder?

It is well known that if we have a binary symmetric channel with crossover probability $\epsilon<0.5$ and we send a word $x$ through it, the most likely word is the one with minimum hamming ...
1
vote
1answer
54 views

Checking some Regular Expression problems

I'm given the alphabet $$ \Sigma = {\{a,b}\} $$ I tried to write a regular expressions for presenting the following sets: All strings in $$\Sigma ^ *$$ with: a-) number of 2s divisible by 4 b-) ...
1
vote
0answers
27 views

Single-Digit Errors

I've been assigned the following homework problem: Given an eight digit number $a_1a_2...a_8$ and a check digit $a_9$, $7a_1+3a_2+9a_3+7a_4+3a_5+9a_6+7a_7+3a_8+9a_9 \equiv 0 \mod{10}$ ...
2
votes
2answers
156 views

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
0
votes
1answer
45 views

Show that $T$ is a linear transformation given Orthonormal basis

Suppose that $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and suppose that $\{v_1,v_2,\cdots,v_n\}$ and $\{Tv_1,Tv_2,\cdots,Tv_n\}$ are orthonormal basis of $\mathbb{R}^n$. Prove that $T$ is a linear ...
0
votes
3answers
319 views

Why Taylor series does not converge for all x in the domain of the function

Example: $$ f(x)=\frac{1}{1+x} \qquad x\neq-1 $$ $$ f(x)=1-x+x^2-x^3+x^4-x^5+\;... \qquad |x| < 1 $$ Why Taylor series does not converge for all x in the domain of the function?
1
vote
2answers
85 views

Regarding Irreducibility of two variable polynomial

Following is an example taken from Dummit Foote - Abstract Algebra after Proposition $9.4.12$ The idea of reducing modulo an ideal to determine irreducibility can be used also in several ...
0
votes
1answer
248 views

Game Theory - trying to find game name by description

My hobby AI research have led me to a thorethical game of particular design. As design is pretty simple, I'm almost sure that such game has well-known name and tons of research already done around it. ...
5
votes
1answer
504 views

Interpolation inequality on Holder space

Let $0< \beta < \gamma <1$. Show that the interpolation inequality holds. $$||U||_{C^{0,\gamma}(U)} \le ||U||^{\frac{1-\gamma}{1-\beta}}_{C^{0,\beta}(U)} ...
2
votes
2answers
46 views

Find this ODE solution $xy''+2y'-xy=e^x$

Find this following ODE solution $$xy''+2y'-xy=e^x$$ my try: since $$xy''+y'+y'-xy=e^x$$ then $$(xy')'-xy+y'=e^x$$ then I can't
1
vote
2answers
77 views

GCD of pairs of integers

So I think I am just psyching myself out right now and this is way to easy but I am running on no sleep in the past few days so forgive me please. The question is what are the greatest common divisors ...
2
votes
0answers
71 views

Human accessible mathematics : Objects defined by a finite number of steps

This is about the uncountable spaces like real numbers : When i was a student, I was proud to talk about them as if they were little toys I could play with. But ... I once realized that "most" of ...
0
votes
3answers
67 views

A difficult equation containing exponent 2 and 3

I couldn't solve this equation: $$ \frac{2}{x^2} + \frac{2}{2x} - \frac{(x+1)^2}{x^3} = \frac{1}{27} $$ Do I have to multiply everything by $x^3$ and also the righthand side $1/27$? $1 \cdot x^3/27 ...
2
votes
0answers
67 views

Weak convergence in $L^1$

Does anyone have a reference for the following statement or similar ones? Let $U$ be an open bounded set in $\mathbb R^n$ and let $f\in C^0(U\times S^1)$. Then the sequence $f_m (x):=f(x,mx_i)$ ...
0
votes
1answer
80 views

Tangent to the circle given a point it must pass through a point on another circle

I need to find the angle at origin caused by two lines (one is the radius of one circle, and the other is tangent to the other circle). Please see image below: The Point A on the green circle is ...
3
votes
2answers
114 views

Radius of Curvature

I was asked to show that the expression is constant in a circle : $\dfrac{\left[1+\left(\dfrac{\operatorname d \!y}{\operatorname d \!x}\right)^2\right]^{\frac 3 2}}{\dfrac{\operatorname d ...
0
votes
1answer
203 views

Obtaining useful information from graph obtained via Monte-Carlo Simulations

I've been running Monte Carlo Simulations on some Matlab code and then plot the graph shown below. I was just wondering what useful information I could collect from this graph? Edit: fit ...
2
votes
5answers
115 views

How find this sum $f(n)=\sum_{i=1}^{n}\dfrac{\binom{n}{i}}{i}$

Find this sum closed form $$f(n)=\sum_{i=1}^{n}\dfrac{\binom{n}{i}}{i}$$ My idea: since $$\dfrac{1}{i}=\int_{0}^{1}x^{i-1}dx$$ so ...
0
votes
0answers
63 views

Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
1
vote
1answer
91 views

Biholomorphic maps from unit disc

Let $f$ be biholomorphic map from the unit disc onto some $D \subset \overline{\mathbb{C}}$ (considered as a Riemann sphere, so it is holomorphic) with $$f(z)=\frac{1}{z}+c_1z+c_2z^2+\cdots$$ What ...
0
votes
1answer
48 views

Computing $E\left[X\,\middle|\,X \leq \dfrac{Y}{2}\right]$

Let X be a uniformly distributed over $[0,2]$, and $Y$ to take values from $[0,\infty]$, how do we compute $E\left[X\,\middle|\,X \leq \dfrac{Y}{2}\right]$. My attempt: $$ E\left[X\,\middle|\, X ...

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