1
vote
1answer
28 views

If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(x)$ is 2, prove that the image of $f$ is an open set.

I don't see how to solve the following problem, any suggestions? Let $f:\mathbb{R}^{3}\to \mathbb{R}^{2}$ such that $f\in C^{1}$. If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(...
0
votes
0answers
31 views

Show that $\pi$ is a primitive recursive number [on hold]

Can anyone provide a proof that $\pi$ is a primitive recursive number, or suggest how I might prove it? Thanks
0
votes
1answer
33 views

How to reduce $f(k, n)$ to $\operatorname{fibonacci}(n)$?

Let's define $f(k, n)$ $f(0, n) = f(0, n - 1) + f(1, n - 1)$ $f(1, n) = f(0, n - 1)$ $f(k, 1) = 1$, for every $k$. $k$, $n$ $\subset \mathbb N$, for $0 \le k \le 1, n \ge 1$. I noted that $f(k, ...
6
votes
1answer
53 views

For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$

I am working through an old qualifying exam from another university. My course did not cover as much material as what is on this test (e.g. we did not cover cohomology). So I am just working through ...
2
votes
5answers
78 views

Monotonicity of the sequence $(a_n)$, where $a_n=\left ( 1+\frac{1}{n} \right )^n$

Define $a_n=\left ( 1+\frac{1}{n} \right )^n$ for $n\geq 1$. I want to show that it is increasing. First, we have $$\frac{a_{n+1}}{a_n}=\left ( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right )^n\left ( ...
1
vote
1answer
50 views

Vakil's FOAG, Exercise 9.2.K: Transcendental Complex Numbers

How does one realize a transcendental complex number as a maximal ideal of $\mathbb{Q}(t) \otimes_{\mathbb{Q}} \mathbb{C}$? This is the essence of Exercise 9.2.K in Vakil's FOAG. Here is what I've ...
1
vote
1answer
49 views

Calculate the probability that the running total is exactly n. (homework help)

I am working through Harvard's public Stat 110 (probability) course. Question: A fair die is rolled repeatedly, and a running total is kept (which is, at each time, the total of all the rolls up ...
-3
votes
0answers
55 views

planning on trading, need mathematical edge [on hold]

I have been looking in to binary options trading, How It Works Retail trader (maybe me) goes to broker to trade binary options. If I trade that I think Euro/USD currency pair will go down, then I ...
2
votes
0answers
26 views

If $AB = BA$ for $A,B \in \mathcal{L}(V,V)$, then $A$ and $B$ have these properties [duplicate]

There is a base such $A$ and $B$ are both upper triangular on these base, and if $A$ and $B$ are diagonalizable, then $A$ and $B$ are diagonalizable simultaneously. For the first I have no idea. To ...
4
votes
1answer
34 views

Example of field's normal closure that's not Abelian?

Suppose $K$ is a global field, $L/K$ is a field extension, and $M$ = normal closure of $L$ (over $K$). Is it possible that Gal($M/L$) is not Abelian? In all cases I know, $L$ is formed from $K$ by ...
0
votes
1answer
33 views

Expectation of the first passage time of $T_{a,b}$ [duplicate]

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space and let $W_t$ be standard Wiener process and $$T_{a,b}=\inf\{t:W_t=a+bt\}$$ where $a$ and $b$ are costant.I want to get expectation of $...
-4
votes
1answer
40 views

What is the difference between the absolute extreme values and local extreme values. [on hold]

Ex. Determine the absolute and local extreme values for $ y = x^3 + 2x^2 - x + 6$. I have calculated that the x values from the derived function of the above are $x=0.22$ and $x=-1.55$
0
votes
1answer
50 views

Having no derangements — any advantage?

Is there any problem-solving advantage when a sequence has no derangements? In an Erd\"os proof of Sylvester-Schur he identifies a few exceptions which I contend would not happen if his sequence had ...
8
votes
4answers
898 views

How unique is e?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
1
vote
2answers
41 views

Correlated brownian motions and Lévy's theorem

$W^{(1)}_t$ and $W^{(2)}_t$ are two independent Brownian motions. How can I use Lévy's Theorem to show that $$W_t:=\rho W^{(1)}_t+\sqrt{(1-\rho^2)} W^{(2)}_t,$$ is also a Brownian motion for a given ...
0
votes
0answers
13 views

Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
0
votes
0answers
35 views

Expectation and Variance of an Estimator

Imagene following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+y^2}{...
3
votes
1answer
37 views

$\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?} $

Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $ A^c = $ those element of the universe that are not in A. $ \Bbb R =$ ...
0
votes
0answers
34 views

Maximising sum of sine/cosine functions

I have got a problem and I would appreciate if one could help. I have to maximise following function that is the sum of sine/cosine functions: $$ f(x,y)=a_1 \cos(x) +b_1 \sin(x)+ a_2 \cos(y) +b_2 \...
1
vote
1answer
63 views

Sketch $f(x) =\frac{ 2x}{x^2-5x+4}$

I have already found the domain, intercepts and critical number along with max and min points. Now when finding the intervals of concavity I understand you must take the second derivative but I can't ...
-2
votes
3answers
123 views

For $\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$, show there is a choice of signs such that it is irrational [on hold]

Considering $$\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$$ where you can replace each $\pm$ with $+$ or $-$. Prove that there is at least one choice of signs such that the number is ...
3
votes
2answers
36 views

Material to learn some basic combinatorics?

I realize that I'm pretty weak when It comes to basic combinatorics, even with simple things like n choose k I don't feel confident. Furthermore, I've viewed some combinatorics books and the reasoning ...
1
vote
2answers
54 views

Find the exact critical numbers for $f(x) = 3x - \sqrt{x}$

I found the derivative of the function which I believe is $3-\frac{1}{2\sqrt{x}}$ but I am not sure how to find the $x$ value for the critical number.
2
votes
2answers
63 views

Simple question: Which is the Wikipedia definition of axiom of choice

I looked up Wikipedia, and on the top of the page it says: For every indexed family $(S_i)_{i \in I}$ of nonempty sets, there exists an indexed family $(x_i)_{i \in I}$ of elements such that $...
0
votes
0answers
21 views

on the proof of Theorem 2.11 of Ratliff's “On prime divisors of $I^n$, $n$ large”

Let us consider Ratliff's paper from 1975 entitled On prime divisors of $I^n$, $n$ large, in particular the last statement of the first paragraph of the proof of Theorem 2.11. We have a Noetherian ...
1
vote
0answers
17 views

creating a loop and maybe a potential loop within a loop in matlab [on hold]

I'm extremely new at Matlab so would like some help. I'm trying to create a loop and maybe a loop within a loop, I'm not too sure at the moment but here's what I'm trying to achieve. I began with ...
0
votes
2answers
30 views

finding out total digits in a large number

Is there any easy way to find out how many digits does the number $12^{400}$ have or such types of problems like how many digits the number $x^y$ have? ($x$ and $y$ are variables)
0
votes
3answers
60 views

Understanding Big O

If $f(x)=O(x^2)$ as $x \rightarrow 0$ and $f(x)$ is continuous at $x=0$, what does this tell us? Can we assume $f(0)=0$? Is $f(x)$ differentiable at $x=0$? I am having trouble understanding this stuff....
1
vote
1answer
56 views

What is the probability of two-pair poker hand?

To start with, this question has never been asked as how I am going to ask: What is the probability that a five card poker hand will have two pairs (with no additional cards)? Example of two-...
-2
votes
1answer
111 views

$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$ change a sign to be rational [on hold]

I have this problem: $$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$$ Prove that you need to change ONLY a sign (to convert a $+$ to $-$) of a single square root, for the sum to be rational. EDIT:...
1
vote
7answers
85 views

Why is this true $\frac{1-y^n}{1-y}=(1+y+y^2+…+ y^{n-1})$? [on hold]

I have a heard time seeing why is this true $\frac{1-y^n}{1-y}=(1+y+y^2+...+ y^{n-1})$ Could you show me some kind of proof, or an identity that would me to find this?
-1
votes
0answers
21 views

Trig funct graphs check (amplitude, period)

Hi for the following questions I was wondering if I was correct in my answers and if I am incorrect, please correct me. Thank You My solutions, please correct me if I am wrong. 2cosx, amplitude ...
0
votes
2answers
25 views

Simplfying complex rational expression

I'm trying to simplify $$ \frac {\dfrac {x}{y} - \dfrac {y}{x}}{y}.$$ My method of trying to solve this is try to simplify the numerator $\frac {x}{y}-\frac{y}{x}$ Then I find the GCD: $xy$, multiply, ...
1
vote
1answer
32 views

$\forall p\in\mathbb P\exists q,r\in\mathbb P':p^3=2q+r$, $\mathbb P'=$ set of non twin primes

Define $\mathbb P'=\{n\in\mathbb P|n-2,n+2\notin \mathbb P\}$. Conjecture: Given a prime $p>3$, then $\exists q,r\in\mathbb P':p^3=2q+r.$ Tested for the first 10000 primes. The solutions ...
0
votes
0answers
28 views

How do I find the induced Riemannian metric of a real smooth complete intersection?

If I have a smooth complete intersection of $f_1,\ldots,f_k \in C^\infty(\mathbb{R}^n)$, presented as the vanishing locus $$ f_1 = 0 \text{ } \cdots \text{ } f_k = 0 $$ in $\mathbb{R}^n$, how can I ...
1
vote
1answer
31 views

Algebraic number spaces

While studying about Vector spaces and subspaces I came across the following question:- $Q.$ Do $algebraic$ numbers form a subspace of the vector space $\Bbb R$? According to my knowledge of $...
0
votes
0answers
10 views

Compute Fourier coefficients of spline fit to data

Suppose you have data $$\{(x_i, 1)\}_{1\dots N}, \quad x_1=0<x_i<x_{i+1}<x_N=2\pi \ \forall i \in\{2,\dots N-1\} $$ In other words, we have a sequence of $y=1$ values at distinct random ...
1
vote
0answers
25 views

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
0
votes
1answer
40 views

Given matrix $A$ such that $\forall x : |Ax| > |x|$, and eigenvalue $\lambda$ of $A$. Show $|\lambda |\geq 1$.

Say matrix $A$ has the property that for any non-zero vector $x$, left-multiplication of $x$ by $A$ increases the magnitude. That is, $\forall x$ $$ |x| > 0 \implies |Ax| > |x| $$ Is it true ...
0
votes
1answer
33 views

In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
2
votes
3answers
46 views

Proof related to Harmonic Progression

The question is as follows: Let $m_1<m_2<m_3<\cdots<m_k$ be postive integers such that $\frac{1}{m_1}$, $\frac{1}{m_2}$, $\frac{1}{m_3}$, $\cdots$, $\frac{1}{m_k}$ are in arithmetic ...
-1
votes
2answers
20 views

How to demostrate Sub-set from flat is a open set

The set C={(x,y);2< x^2 + y^2 < 4} How can we defined this set as open using the definition.
-1
votes
1answer
32 views

help with real analysis [on hold]

Let $S \subset \mathbb{R}$ be nonempty. Show that if $u= \sup S$, then for every number $n$ belong to $\mathbb{N}$ the number $u -\frac{1}{n}$ is not an upper bound of $S$, but the number $u + \frac{1}...
1
vote
1answer
23 views

Totally real Galois extension of given degree

Let $n≥1$ be an integer. I would like to prove (or disprove) the existence of a subfield $K \subset \Bbb R$ such that $K/\Bbb Q$ is Galois and has degree $n$. It is easy to construct such a subfield ...
-1
votes
1answer
20 views

How to translate this statement to First Order Logic?

“Thus there exists a pet in this house being a cat or a dog” I am unsure of how this statements should be translated.
0
votes
0answers
21 views

Is the phrase “random number function” contradictory? [on hold]

If a Function is something that produces an output from an input(s) and is consistent, then the phrase "Random Number Function" should not be allowed, right?
2
votes
2answers
53 views

Summing a series of integrals

I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here. I have a series of integrals I would like to sum, but I don't ...
0
votes
3answers
23 views

Find elevation of a tangent in a circle

I am trying to understand how I can calculate an elevation (i.e. the distance) of a tangent line given an arc and radius. For example : Given that I know $d$ and $s$, how do I get the value for $?$...
-3
votes
1answer
15 views

Quantitative Aptitude Average [on hold]

The average weight of a group of boys and girl is $38$ kg. The average weight of boys is $42$ kg and that of girls is $33$ Kg. If the number of boys is 25, then the number of girls is?
0
votes
2answers
22 views

Finding an example of a bijection from $\Bbb N$ to $E^+$.

Give an example of a bijection $h$ from $\Bbb N$ to $E^+$ such that $h(1) = 16, h(2) =12, \text{ and } h(3) = 2. $ $\Bbb N = \text{ natural numbers }$ , $E^+= \text{ positive even integers. }$ So ...

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