All Questions

0
votes
0answers
3 views

Is the following statement is True/false regarding a non-trivial homomorphism

Is the following statement is True/false For every integer $n \geq 2$, there is a unique non-trivial homomorphism $φ:S_n \rightarrow \mathbb{C}^*$. where $\mathbb{C}^*$,denotes the ...
0
votes
0answers
5 views

Space of sequences such thtat $\sum_{n=0}^{\infty}2^na_n<+\infty$

Consider the space of sequences of real numbers $\{a_n\}$ such that $\sum_{n=0}^{\infty}2^na_n<+\infty$. Then how could we better describe the space? Typically, does the space have countable linear ...
0
votes
0answers
5 views

Statistics/probability: Monty Hall question but with four doors.

Suppose you have been chosen to play the final round of a television game show. In this game, there are four doors, behind one of which is a terrific prize worth $100,000. Behind each of the other ...
0
votes
0answers
7 views

Partial Differential

If f is a function of x. Some times people write: Is this a rule? does it have a proof? need help. Thanks.
0
votes
0answers
8 views

Write in Logic: If professors are unhappy all students fail their exams

I have to write the following sentence "If professors are unhappy all students fail their exams" in logic and my answer is: ∀x [Prof(x) ∧ Unhappy(x)] ⇒ [∀y stud(y) ⇒ fail_exam(x,y)] However, the ...
0
votes
1answer
10 views

(12345) and (12) create a group

can someone telle me how I can show the following: Sonsider the symmetrical group S5 and show that (12345) and (12) creates the group. Thanks in advance!
0
votes
0answers
2 views

Clarification on How to derive Voronoi diagram from Delaunay triangulation in linear time

Definitions: Assume that a set of points $P=\{p_1,\dots,p_n\}$ in $\mathbb R^d $ is given. For each $p_i \in P$, the Voronoi region of $p_i$ is defined as: $Vor(p_i)=\{p\in\mathbb R^d:\forall p_j\in ...
0
votes
0answers
6 views

Calculate the power of a test with an unknown $\mu$

In order to prevent damages to the wind turbines, the windspeed is monitored every 5 seconds, and if it is determined that the variance of the windspeed is high the turbine switches off to a safe ...
-1
votes
0answers
10 views

I want to know how to prove ideal

I'm just learning about ideal and struggling with these problem. So I want to know how to prove these. About ℚ[x] ⊂ C[X], a ∈ C, I_a = {f(X) ∈ ℚ[x] | f(a) = 0} (1) show I_a is the ideal of ℚ[x] (2)...
0
votes
0answers
9 views

How to formulate an indifference curve

This question is stemming from an economic context, but I am only interested in the mathematical formulation of the indifference curve itself. They are simple to draw free-hand, but I have no idea how ...
3
votes
0answers
20 views

An unpleasant measure theory/functional analysis problem

I am currently taking a functional analysis course, and at the moment every student on the course is stumped by a specific question. We're looking at the bounded linear map \begin{equation} \varphi_n(...
0
votes
0answers
12 views

Meromorphic function on a simply connected open set $D$ has a complex anti-derivative on $D$

Let $f(z)$ be a meromorphic function on a simply connected open set $D$. Show that $f(z)$ has a complex anti-derivative on $D$, that is, there exists a meromorphic function $F(z)$ on $D$ such that $F'(...
1
vote
0answers
11 views

On a special kind of six dimensional vector subspace of $\mathbb C^9$ related to the primitive $9$-th root of unity

Let $\mu=e^{2\pi i/9}$ . Let $u_j:=(\mu^j,\mu^{2j}, \mu ^{3j},...,\mu^{9j})^T \in \mathbb C^9$, for $j=1,...,9$. Let $V$ be the vector subspace of $\mathbb C^9$ spanned by $\{u_2,u_3,u_4,u_5,u_6,u_7\...
3
votes
1answer
10 views

Right cone, you are at A and need to complete a revolution before reaching the bottom B. What is shortest distance AB?

You are on a mountain that is a right cone shape. You are trying to get to B and you are somewhere up the mountain A such that you lie on the line OB. The line AB must do one full revolution of the ...
2
votes
0answers
15 views

If $(u(x,y))^2+u(x,y)v(x,y)$ has a local maximum or minimum in $D$, then $f$ must be constant?

Let $f(z) = u(x,y)+iv(x,y)$ be an analytic function on a connected open set $D$ with $u(x,y)$ and $v(x,y)$ being the real and imaginary parts of $f(z)$, respectively. If $(u(x,y))^2+u(x,y)v(x,y)$ has ...
-5
votes
2answers
27 views

Compute $\lim\limits_{x \to 3} \dfrac{x^2-2x-3}{|x-3|}$ [on hold]

Does the limit $$\lim\limits_{x \to 3} \dfrac{x^2-2x-3}{|x-3|}$$ exist?
4
votes
1answer
16 views

For what $n$ is $W_n$ finite?

Suppose, $W_n$ is the set of all words formed by letters '$a$' and '$b$', that do not contain $n$ same consecutive nonempty subwords (that means that for any nonempty word $u$, the word $u^n$ is not a ...
2
votes
0answers
6 views

On a special kind of $6$-dimensional vector subspace of $\mathbb C^9$

Let $V \subseteq \mathbb C^9$ be a vector subspace of dimension $6$. Suppose that there exists $A,B \in M_{3 \times 6} (\mathbb C)$ such that $V=\{(x_1,...,x_9)\in \mathbb C^9 : (x_3,x_6,x_9)^T = A (...
0
votes
0answers
9 views

Determine the number of zeros of $z^4+z^3+4z^2+\alpha z +3$ in $\{Re(z)<0\}$

Determine the number of zeros of $z^4+z^3+4z^2+\alpha z +3$ in $\{Re(z)<0\}$ It doesn't say, but I am assuming $\alpha$ is an arbitrary constant. I am preparing for an upcoming final tomorrow,...
0
votes
0answers
7 views

How do you define ordinal exponentiation without induction?

I am writing some notes on introductory set theory, starting from the basic axioms all the way to cardinal arithmetic. Right now I am up to ordinal arithmetic. I have defined ordinal addition and ...
0
votes
1answer
7 views

Basic matrices I am haveing troubles with

Hello community I am new here and I have a question which might be pretty basic. So I am trying to solve an equation. I have 3 matrices A = \begin{pmatrix}1&0&0&0&0&0&0&...
0
votes
1answer
16 views

Show closure of image equals $\mathbb{C}$ (I.e. is dense in $\mathbb{C})$

Let $f(z)$ be a nonconstant analytic function on $\mathbb{C}\backslash S$, where $S$ is a finite subset of $\mathbb{C}$. Show that $\overline{f(\mathbb{C}\backslash S)}=\mathbb{C}$ In preparation for ...
0
votes
0answers
7 views

Connection between the eigenfunctions of the compact operators $T[f](x\in H_1)=\int_{H_1}k(x,y)f(y)dy$ and $R[f](x\in H_2)=\int_{H_1}k(x,y)f(y)dy$?

Let $H_1$ and $H_2$ be Hilbert spaces. Suppose we have a compact integral operator $T:H_1 \to H_1$ given by $$ T[f](x) = \int_{H_1} k(x,y)f(y)dy, \quad \quad x \in H_1. $$ Suppose we also have a ...
0
votes
2answers
19 views

Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$.

Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$. I've tried to show this statement,but I cant ...
1
vote
0answers
10 views

Compact Hausdorf space $\implies$ not the countable union of nowhere dense sets? [duplicate]

I can sort of see this intuitively, seeing as it's a similar argument to the Baire Category theorem, but does anyone have a proof I could look at? Would it suffice to say that every locally compact ...
0
votes
1answer
21 views

Calculate the inverse of an element in $\mathbb{Q}(\sqrt 3)$

I want to find the inverse of an element in $\mathbb{Q}(\sqrt 3)$. For example $a=2-\sqrt3$. I was considering at first to find a $b= {1 \over a}$ and from the calculation emerge that $b= 2+\sqrt3$....
0
votes
1answer
27 views

Can a map be subjective but still be bijective (or simply injective or surjective)?

Sorry if this is a stupid question. This page on Math Is Fun uses the terms "general function", "surjective function", "injective function" and "bijective function". This seems to imply that a map can ...
0
votes
1answer
10 views

Let $f (z) = u+iv$ be an analytic function, then show that $ (∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2 $.

Let $f (z) = u+iv$ be an analytic function, then show that $$(∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2\,.$$ $f(z) = u + iv $ $ϕ = |f(z)|^2 = u^2 + v^2 $ $f'(z) = ∂u/∂x + i∂v/∂x $ $|f'(z)|^2 = (∂u/...
4
votes
0answers
35 views

Find all postive integer $m$ such $2^{m}+1|5^m-1$

Find all postive integer $m$ such $$2^{m}+1|5^m-1$$ it seem there no solution,I think it might be necessary to use quadratic reciprocity knowledge to solve this problem. Let $M=2^m+1$,so we have $$...
0
votes
0answers
17 views

Sum of this converging telescoping series?

I'm trying to understand which is the sum of the following telescoping series (I showed this is converging, I'm not reporting here): $$\sum\limits_{j=n}^{\infty} [\mathbb{P}(E_j) - \mathbb{P}(E_{j+1})...
-1
votes
1answer
20 views

Integral which involves an exponential function and a square root

How can this integral be solved? $$\int \frac{e^{ax}}{\sqrt{b^2-x^2}}dx$$
0
votes
2answers
12 views

Uniformly continuous extension

Given $h$ be continuous function over $\mathbb{Q}$, the set of rational numbers, show that it has uniformly continuous extension on $\mathbb{R}$. I am struggling to define the function value and ...
0
votes
0answers
9 views

Apply Euler's Criterion over Finite Fields

Let $p$ be an odd prime number. Consider $\mathbb{F}_{p^n}$ the finite field with $p^n$ elements. Suppose that $k\mid n$ and $r\in \mathbb{F}_{p^n}$ such that the order of $r$ is $p^k-1$. Question: ...
2
votes
4answers
34 views

Does $\lim\limits_{x\to\infty}$ equal to $\lim\limits_{x\to+\infty}$?

As per title, does $$\lim\limits_{x\to\infty}$$ mean $\lim\limits_{x\to+\infty}$ or $\lim\limits_{x\to\pm\infty}$? This link seems to tell me that it's the latter: https://qc.edu.hk/math/...
0
votes
0answers
9 views

$X_1$,$X_2$…$X_n$ are iid N($\mu,\sigma^2$). Derive a confidence interval for the parametric function $\mu+\sigma$ and $\mu/\sigma$.

I know how to find confidence interval for each of the parameters $\mu$ and $\sigma^2$ but don't know how to find in the case of parametric function.
0
votes
2answers
16 views

Is this system under determined?

My immediate thought when I see this problem is that it's under determined and therefore unsolvable, except in terms of other variables. But, maybe there's some clever physics trick that could solve ...
1
vote
0answers
10 views

Showing that a locus is a sub-manifold

I'm self-studying differential geometry using Frankel's ``The Geometry of Physics". The first problem (1.1(1)) is about determining whether or not the locus $$x^2+y^2-z^2 = c $$ is a submanifold in $\...
-3
votes
0answers
18 views

Find partial limits, upper and lower limits of the sequence.

Find partial limits, upper and lower limits of the sequence$$x_n=\left(\frac{2n}{2n+1}\right)^{2n}\cos\frac{\pi n}3,\ n\in\mathbb{N}.$$ How it is doing?
3
votes
1answer
9 views

Find the number of permutations of the word 'ESTATE' if all vowels must be adjacent

First separating the word estate into vowels and non vowels gives $EAE$ for the vowels and $S,T,T$ for the non-vowels. I interpreted this as a group of 4 where there's two T's resulting in $\frac{4!}{...
3
votes
1answer
23 views

general solution of $\frac{dy}{dx}+P(x)y=0$

Consider the homogeneous first-order linear differential equation(DE), $$\frac{dy}{dx}+P(x)y=0,$$ where $P(x)$ is continuous on an interval $I=(a,b)$. I'm trying to convince myself that its general ...
-1
votes
0answers
3 views

Question of Introduction to linear optimization by Dimitris Bertsimas

Exercise 3.23 While solving a linear programming problem by the simplex method, the following tableau is obtained at some iteration. enter image description here begin{array}{r|rrrrrr} \hline‎ ‎$ ...
0
votes
0answers
11 views

Regular open sets in topology generated by regular open sets

Let $Ro( \tau )$ denote the set of regular open sets of topology $\tau$. Is it possible for a set to be regular open in the topology generated by $Ro( \tau)$, but not in $\tau$? Obviously $\tau$ ...
-3
votes
1answer
18 views

Integrating a derivative of a function

Consider the following integral $$\int \frac{d}{dx}\left(f(x)\right)dx$$ does this integral equal to $f(x)$ or is there an additive constant?
-1
votes
0answers
15 views

Trivial frattini subalgebra

Let L be a Lie algebra with trivial frattini subalgebra. Show intersection of Z(L) and derived subalgebra is trivial.
0
votes
1answer
17 views

Condition on when two different metrics generate the same topology

I've just begun working through Lee's Introduction to Topological Manifolds and am currently kind of stuck on Example 2.4(a), which is as follows: Suppose $M$ is a set and $d,d'$ are two different ...
5
votes
0answers
31 views

How much can we rearrange a series?

There's a well-known result that if $\sum a_n$ is conditionally convergent, then for any real $c$ there exists a permutation $\pi:\mathbb{N} \to \mathbb{N}$ such that $\sum a_{\pi(n)} = c$. A ...
0
votes
1answer
9 views

Proof that $\int_0^{\Lambda}\frac{x^{d-1}}{(1+x^2)^2}dx$ remains finite for $1<d<4$

For a phase transition in Landau theory, I need to show that $$I=\int_0^{\Lambda}\frac{x^{d-1}}{(1+x^2)^2}dx$$ remains finite for $1<d<4$, as $\Lambda \rightarrow \infty$. Now I know, that $$\...
0
votes
0answers
12 views

Relation between pullback and fiber product.

Consider the following cartesian diagram of schemes: $$\begin{array} AX^{'} & \stackrel{v}{\longrightarrow} & X \\ \downarrow{u} & & \downarrow{f} \\ \mathrm{Spec}A & \stackrel{g}{\...
-1
votes
3answers
30 views

Prove by induction that $ 4^{2n}-3^{2n}-7 $ is divisible by 84 for all natural numbers.

Please, I have tried some methods of induction but I can't resolve. Sorry for my english. I cannot complete to prove. I haved factoring, dividing, adding new terms but i cannot avance for the second ...
0
votes
0answers
34 views

How to ask questions about the likelihood of “interesting” mathematical statements?

I started reading about Gödel's theorems recently and found the idea of using the tools of mathematics to understand logic and what we can and cannot do with it. While doing my problem set for a basic ...

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