# All Questions

1,031,708 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### 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)...
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### 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 ...
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### 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 \varphi_n(...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### Integral which involves an exponential function and a square root

How can this integral be solved? $$\int \frac{e^{ax}}{\sqrt{b^2-x^2}}dx$$
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### 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 ...
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### 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: ...
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### 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/...
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### $X_1$,$X_2$…$X_n$ 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.
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### 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 ...
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### 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 ...
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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 ‎$... 0answers 11 views ### Regular open sets in topology generated by regular open sets Let$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$... 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? 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. 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 ... 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 ... 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 $$\... 0answers 12 views ### Relation between pullback and fiber product. Consider the following cartesian diagram of schemes:$$\begin{array} AX^{'} & \stackrel{v}{\longrightarrow} & X \\ \downarrow{u} & & \downarrow{f} \\ \mathrm{Spec}A & \stackrel{g}{\... 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 ...