# All Questions

9 views

### Lagrange Interpolation definition doubt

Based on some exercise which explains Lagrange Interpolation itself, I got some doubts: It introduces function $$f(x)=\frac{1}{x}$$ and given points $x_0=2$, $x_1=2.75$ and $x_2=4$ so the ...
9 views

### Regarding limits

If $f$ is positive and differentiable in $(0,\infty)$, then I want to find the following limit. $\lim\limits_{n\to \infty}(\dfrac{f(x+\dfrac{1}{n})}{f(x)})^n$. I have done as follows: ...
5 views

### Intermediate Value theorem, $nth$ root function and continuity

So this problem is.. ridiculous to be honest. I have no idea where to start or what to do. Any help is appreciated. For the record, I am using the metric spaces definition of continuity.
6 views

3 views

### Newton method and contraction map

Let $a\in \mathbb{R}, \ a>0$. Show that applying Newton's method to the function $x^2-a$ gives the formula $x_{n+1}=\frac12(x_n+\frac{a}{x_n}).$ Prove that Newton's method works for any ...
13 views

### Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$

I'm interested in the following integral: $$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$ where $K(z)$ is the complete elliptic integral of the 1ˢᵗ ...
6 views

8 views

### Annihilator of a nonzero element equals the whole ring.

Let $M$ be an $R$-module where $R$ is a commutative ring with identity. Is it true that $\text{Ann}_R\ m=R$ if and only if $m=0$ for all $m\in M$? Thanks!
15 views

### Number of Solutions to a Diophantine Equation

I am asked the following: 1) Solve the diophantine equation $y^3=x^2+2$. 2) Show that the number of integer solutions to $y^p=x^2+2$ for any odd prime $p$ is at most $p-1$. The first part is easily ...
20 views

### Let $A$ be a subset of $C$ and $B$ a proper subgroup of $C$

Let $B$ a proper subgroup of $C$, show that there exist a subset A such that $C = \langle A \rangle$ and $A \cap B = \varnothing$ I am not sure how to start with this one.
15 views

### Eigenvectors and linear operators (Mistake)

I was solving the following problem from Artin and think there might be a mistake Let $T$ be a linear operator on a finite dimensional vector space $V$ such that $T^2$ is the identity operator. Prove ...
14 views

### proof of continuity in product topology Help please

Here is our function $F$ $$F:\mathcal R^n \times I \to \mathcal R^n$$ where $(x,t)\in \mathcal R^n \times I \to (1-t)x \in \mathcal R^n$ all topologies are usual topology! Book said that this $F$ ...
17 views

### Given three points on a circle, how should they be arranged to give the maximum area triangle?

I am trying to achieve multiple solutions to this problem using technology (i.e. Geometer's sketchpad) or perhaps alternatively solving without technology. I am not sure if I am in the right ...
18 views

4 views

### Is there an unambiguous CFL whose complement is not context-free?

I'm doing a little bit of research about context-free languages. A question that's popped up is whether or not there exists an unambiguous context-free language whose complement is not a context-free ...
27 views

### a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
50 views

20 views

### Master Theorem Exercise

Could you please help me with a master theorem equation i have? for master theorem we have T(n) = aT(n/b)+f(n) where a>=1, b>1 here i have: T(n) = 5T(n/4)+12 a = 5; b = 4 and f(n) = 12; Now is c = 0 ...
76 views

### Linear algebra questions that a high-schooler could explore

Are there any deep/significant concepts in linear algebra that are not overly complicated that a high schooler could explore in depth?
9 views

### $m$-primary ideal and $M\otimes_{A} A/m \neq 0$

Let $A$ be a commutative local ring with an unique maximal ideal $m$ . Let $M$ be a (not necessarily finitely generated) $A$-module . Let $x_{1},...,x_{n}$ be a M-regular sequence such that ...
44 views

### When $E(X|Y)E(Y) = E(XY)$?

$X$ and $Y$ are random variables. In what cases is the following statement true $E(X|Y)E(Y) = E(XY)$ a.s., other than the cases when $X$ and $Y$are independent, and when $E(X|Y)=EX$ a.s. ? If ...
40 views

### how to prove $\sum_{k=0}^{m}\binom{n+k}{n}=\binom{n+m+1}{n+1}$ without induction?

$$\sum_{k=0}^{m}\binom{n+k}{n}=\binom{n+m+1}{n+1}$$ how to prove it without induction? I tried with several way but I failed anybody help me ?
56 views

### What is the dual matrix (of a sample covariance matrix)?

Let $A$ be a matrix. I am most interested in the real, symmetric case, but for full understanding let's let $A$ be complex. What does it mean for $A^D$ to be the dual matrix of $A$? Can we interpret ...
14 views

### Big O, Omega and Theta Exercises

i have a few exercises to do but i need someone to correct me if they can. I am very new to the Big O notation so please forgive me for being too basic. I need to represent everything under Θ. T(n) ...
15 views

### Union of vector subspaces, sum of dimensions of vector subspaces and direct sum of vector subspaces

I am currently reading Linear Algebra Done Right by Sheldon Axler, and I have stumbled upon some proposition that I have trouble verifying. Excerpt from the book: Proposition: Suppose V is finite ...
$A,B,C$ random variables with $p(A) = p(A\mid C)$ for the pdfs, so $A$ and $C$ are independent. When does the following hold: $p(A\mid B,C) = p(A\mid B)$? Would it additionally also require $B$ and ...
I have an elliptic curve $$x^3+17x+5 \mod 59$$ $P = (4,14)$ is given and I need to find point $8P$. to calculate $8P$, I first calculated $2P$ by using the equation sigma = 3x^2+a/2y = ...