-1
votes
1answer
159 views

General term of sequence 3

Let be $$6,8,12,15,20,24,30,35,42,… $$ a sequence of natural numbers. Guessing the recurrence then using generating functions I can prove that general term of sequence is ...
6
votes
3answers
661 views

How to prove cardinality equality ($\mathfrak c^\mathfrak c=2^\mathfrak c$)

How do I prove this cardinality equality:$\mathfrak c^\mathfrak c=2^\mathfrak c$ I have failed to prove this after lots of trail - but I am certain it's true How can I prove this?
4
votes
2answers
283 views

Gradient descent in a distributed manner

Let $p(x_1,x_2,x_3)$ be a scalar function. The goal is to find $x_1,x_2,x_3$ to minimize $p(x_1,x_2,x_3)$. Now consider the gradient descent method: $$ \left( \begin{array}{c} x_1 \\ x_2 ...
0
votes
1answer
94 views

Quick check — Differential equation

Just want to quickly check that I have got this right: Am I right in thinking that the solution of $\dfrac{d^2y}{dx^2} = \sin(y)$ is $y(x) = 2 ...
3
votes
2answers
652 views

Homology of $\mathbb{R}^3 - S^1$

I've been looking for a space on the internet for which I cannot write down the homology groups off the top of my head so I came across this: Compute the homology of $X : = \mathbb{R}^3 - S^1$. I ...
3
votes
1answer
75 views

Each open set is cofinite in a regular space implies the space is finite

Let $X$ be a regular ($T_{1}$) space such that for each non-empty open subset $U$ of $X$ the complement $X \setminus U$ is a finite set. Why this implies $X$ is finite?
1
vote
1answer
236 views

$ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$ and $ b_1 = 1$, show that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}{n^2}$ [closed]

In the recursion $ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$, with $ b_1 = 1,$ how can one prove that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}{n^2}$?
5
votes
3answers
477 views

$\mathbb R = X^2$ as a Cartesian product

I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ ...
4
votes
1answer
203 views

Prove that $|S|< \frac{2p}{k+1}$

Given a prime number $p$. Let $a_1,a_2 \cdots a_k$ ($k \geq 3$) be integers not divisible by $p$ and having different residuals when divided by $p$. Let $$ S= \{ n \mid 1 \leq n \leq p-1, (na_1)_p ...
1
vote
1answer
292 views

Wikipedia Article — Legendre Transform

I was reading the wiki article on Legendre Transform. I would be grateful if someone could explain the section at http://en.wikipedia.org/wiki/Legendre_transformation#Examples ie how they arrived at ...
5
votes
2answers
467 views

Finding a primitive explicitly

Let $A=\{z\in\mathbb{C}~:~|z|>4\}$. Let $f(z)=\frac{z}{(z-1)(z-2)(z-3)}$ and $g(z)=\frac{z^2}{(z-1)(z-2)(z-3)}$. I have been asked whether or not $f$ and $g$ have global primitives in $A$ and if ...
3
votes
1answer
341 views

A question on limit of a Dirichlet's integral

I want to know the idea/intuition behind the Dirichlet integral. For example consider $$I(\alpha) = \int_0^{\delta} g(t) \frac{\sin(\alpha t)}{t} dt$$ . Why would $I(\alpha)$ tend to some constant ...
4
votes
1answer
419 views

Is it possible to efficiently factor a semiprime given a bit-permutation relating the factors?

Is it possible to efficiently factor a semiprime given a bit-permutation relating the factors? For example, suppose we have $n = p * q = 167653$; in this case, $p = 359 = 101100111_2$ and $q = 467 = ...
1
vote
1answer
1k views

Euler's Formula Conversion with coefficients

If I have an equation such as $x(t) = \displaystyle \sum_{n=1}^N \left( a_n \cos(\omega_nt) + b_n \sin(\omega_n t) \right)$, how do I convert it to a sum of complex exponentials? In other words what ...
4
votes
1answer
153 views

Need help proving/understanding an inequality

I have been asked to prove the following inequality, given that $f:\mathbb{D}\to\mathbb{C}$ is an analytic function on the open unit disc $\mathbb{D}\subset\mathbb{C}$. ...
4
votes
2answers
404 views

Can $\int_0^{\infty} e^{-a^2 t^2 - b t} \sin c t \mathrm dt$ be done?

I'm a physicist and I've been encountering integrals like $\int_0^{\infty} e^{-a^2 t^2 - b t} \sin c t \;\mathrm dt$, where everything is real. Mathematica could not solve it and I could not find it ...
6
votes
1answer
605 views

For complex matrices, if $\langle Ax,x\rangle=\langle Bx,x\rangle$ for all $x$, then $\langle Ax,y\rangle=\langle Bx,y\rangle$ for all $x$ and $y$?

Given $A$ and $B$, $n\times n$ complex matrices. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n}$, then the following are equivalent: (1) $\langle Ax,y\rangle=\langle Bx,y\rangle$, for ...
3
votes
1answer
1k views

Hausdorff locally compact and second countable is $\sigma$-compact

I want to show that every Hausdorff, locally compact and second countable is $\sigma$-compact. I'm having trouble writing this rigorously. Can we proceed as follows: Let $X$ be a locally compact ...
4
votes
2answers
296 views

Does an infinite matrix exist with each row converging to 0 and each column to 1?

Is there an infinite matrix $A_{mn}$ such that $\lim\limits_{n \to \infty }A_{mn}=0 $ for every $m$ and $\lim\limits_{m \to \infty }A_{mn}=1 $ for every $n$ ? Any clue as to how to start on this?
22
votes
1answer
830 views

How far can probability intransitivity be stretched?

Once upon a time I read about nontransitive dice - sets of dice where "is more likely to roll a higher number than" is not a transitive relation. After the surprise wore off, I wondered - just how far ...
1
vote
2answers
88 views

Question about a Question: Simplifying Fractions

In a question I asked several weeks ago an interim step reached was a.): $$\frac{1}{(x-6)!6!}=\frac{1}{(x-4)!4!}$$ hence b.): $$ \frac{(x-4)!}{(x-6)!}=\frac{6!}{4!}$$ I'm not following how we got ...
10
votes
2answers
382 views

What's the history of the result that $p_{n+1} < p_n^2$, and how difficult is the proof?

In Edsger Dijkstra's monograph "Notes on Structured Programming", he describes a simple imperative program for generating an array of the first $n$ primes. For each prime $p_n$, it finds the next ...
7
votes
3answers
716 views

Definition of manifold

From Wikipedia: The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. A topological manifold is a topological ...
0
votes
1answer
172 views

How to prove an inequality involving averages of different order statistics?

Let $X$ be a continuous random variable with values ranging from 0 to 1. Let $X_{kn}$ be the random variable representing the $k$th smallest order statistic of $n$ draws from $X$. Note that ...
2
votes
2answers
124 views

Can the following be integrated?

Can this integral be calculated analytically? $$ \int_{-\pi}^\pi \frac{\mathrm dx}{2\pi} \frac{(e^{i y t}+e^{i x t})(e^{ix}+e^{-i(y-z+x)})}{\cos (y-z+x)- \cos x} \left(\frac{1}{1+e^{2\beta(a-b\cos ...
2
votes
1answer
92 views

Looking for “average” of two permutations

I am a computer programmer and I am building a search engine for a client. Right now I am puzzling myself about the order in which I should return search results. There are two obvious orderings: ...
5
votes
5answers
955 views

How might I refresh high school level mathematics and extend upon it much further?

I did well at Mathematics at school (top 0.1% in the country, approximately), however I stopped studying it when I was 16. Since then I've studied a couple of highly specific mathematics modules in ...
31
votes
4answers
5k views

Construction of a Borel set with positive but not full measure in each interval

I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere. To be precise, if $\mu$ denotes Lebesgue ...
6
votes
0answers
489 views

Positive definite function zoo

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$. For a definition and discussion of ...
1
vote
1answer
246 views

Completely regular space and existence of a continuous function

This is problem $1$, Section $3$, page $252$ of Dugundji's book. Let $X$ be a completely regular space, $C \subset X$ compact and $U$ an open set containing $C$. Prove there exists a continuous map ...
1
vote
1answer
1k views

what is the cumulative distribution function of a logistic function?

I've found on wikipedia for Logistic function (http://en.wikipedia.org/wiki/Logistic_function) they have the formula for a Logistic curve: $P(t) = 1 / (1 + e^{-t})$ and they have a diagram of the ...
2
votes
2answers
584 views

Commutation when minimal and characteristic polynomial agree

Hello I am studying for the qualifying exam in algebra and I am having trouble solving this seemingly easy problem. If A is a matrix whose minimal polynomial and characteristic polynomial agree, and B ...
1
vote
1answer
182 views

How do I do a power calculation where the effect size and values of $\alpha$ and $\beta$ are known?

I am planning a study where the endpoint of the current practice is 40% successful. The null hypothesis will be rejected if the intervention produces a 60% success rate (treatment effect of interest). ...
3
votes
3answers
754 views

Is the language of all strings over the alphabet “a,b,c” with the same number of substrings “ab” & “ba” regular?

Is the language of all strings over the alphabet "a,b,c" with the same number of substrings "ab" & "ba" regular? I believe the answer is NO, but it is hard to make a formal demonstration of it, ...
6
votes
3answers
158 views

Solution to $a\cdot e^{bx} - cx = d$

Similar to the question asked here: Solving $e^x + x = 5$ for $x$ without using a numerical method? How can I get a solution for $a\cdot e^{bx} - cx = d$, where a, b, c, d are constants? Is there a ...
1
vote
1answer
145 views

Convex functions

How does one show that $\phi(x)$ convex and twice differentiable implies that $x\phi(\frac{y}{x})$ is convex on the plane $x>0$? Thanks.
13
votes
3answers
581 views

For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?

True\False? For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$. I know that every complexed matrix has a Jordan form matrix $J$ such that $P^{-1}CP=J$, ...
5
votes
3answers
296 views

Stone Cech compactification of the naturals a cts image of the Cantor set?

Is $\beta \mathbb{N}$, the Stone-Cech compactification of $\mathbb{N}$, a continuous image of the Cantor set?
5
votes
1answer
163 views

Questions on $\overline{\mathbb{F}_p}|\mathbb{F}_p$

I have some questions on the field extension $\overline{\mathbb{F}_p}|\mathbb{F}_p$ for some prime number $p$: Are the any other intermediate fields besides $\mathbb{F}_{p^n}$ for $n \in \mathbb{N}$ ...
3
votes
1answer
153 views

Polygon written as intersection of triangles

Please help me with a reference or a proof for the following: Find $n$ such that any convex polygon with $100$ sides can be obtained as an intersection of $n$ triangles. First, $n \geq 34 $ ...
3
votes
1answer
220 views

Product of sigma compact spaces

Let $\{X_{i}: i \in I\}$ be a family of $\sigma$-compact but not compact topological spaces. How to show that $\prod_{i \in I} X_{i}$ is $\sigma$-compact if and only if $I$ is finite? So here it ...
0
votes
1answer
611 views

Combination of 5 cards given a 7 card set

I haven't done a whole lot of maths since uni and cant remember the equation for this. What I want to know: Given 7 playing cards, how many 5 card combinations are there from this set? This is a ...
8
votes
2answers
584 views

Question about all the homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}$

An exercise in "A first course in Abstract Algebra" asked the following: Describe all ring homomorphisms from the ring $\mathbb{Z},+,\cdot$ to itself. I observed that for any such ring homomorphism ...
5
votes
2answers
686 views

How are TV Texas holdem poker percentages worked out

I watch with some mild interest on TV the poker games when I have little better to do at night. And I notice these guys playing Texas Holdem Poker. My question is, the TV has beside each players name ...
4
votes
1answer
387 views

Is the number of prime ideals of a zero-dimensional ring stable under base change?

Let $A$ be a zero-dimensional ring of finite type over a field $k$ and let $X= \textrm{Spec} \ A$ be its spectrum. Note that $X$ is a finite set. Suppose that $k\subset K$ is a finite field extension ...
2
votes
5answers
3k views

Winning on Prize Bonds

In ireland the goverment issue a special "Savings Bond" called Prize bonds. You dont get any interest on them, but you can get your money back any time you want, and you are put into a weekly draw ...
8
votes
5answers
1k views

Subsets of the Cantor set

Does anyone know of any interesting subsets of the Cantor set? When I first started thinking about this, I thought that since the Cantor set $C$ is the intersection of disjoint unions of closed sets, ...
2
votes
1answer
72 views

Kirby-like diagrams for $M^n$ when $n > 4$

Are there any attempts on constructing Kirby-like diagrams for representing manifolds $M^n$ with $n > 4$. What are the references on that ? I think you run out of dimension in which you can draw ...
2
votes
1answer
247 views

How can i prove that the theory of random graph has a vaughtian pair?

I'm searching for theories that have a vaughtian pair. I've been given a hint, that $T_{RG}$ has at least one. I have also found many theorems stating in which cases a theory has no vaughtian pair, ...
2
votes
3answers
166 views

Some iterate of a linear operator over $\mathbf F_q$ is a projection

If $T$ is an endomorphism of a finite-dimensional vector space $V$ over a finite field, then how can I show that there exists a positive integer $r$ such that $T^r$ is a projection operator?

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