# All Questions

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### Improved error estimate for Conjugate Gradient Method

Let $A \in \mathbb{R}^{n \times n}$ be SPD. The error estimate for the conjugate gradient method is given by \|x_* - x_m \|_A \leq 2 \left( \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1} ...
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### Application of Vitali's covering lemma, absolutely continuous function excercise

Show that for absolutely continuous function $f:[-1,1] \to \mathbb{R}$ limit $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}=g(x)$ exists almost everywhere and $f(x) = f(-1) + \int_{-1}^{x}g(x)dx$. I've ...
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### Propound a model. Is it right my answer?

We have three products, $x:$ gasoline, $y:$ oil and $z:$ gas To produce one unit of gasoline we need one unit of oil and one of gas, to produce one unit of oil we need $\frac1{5}$ of oil and ...
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### Proving that restrictions of partial orders are partial orders

Prove: A set has a partial-order relation $R$ on it. $P$ is a subset of this set. Prove that the restriction of $R$ to $P$ is itself a partial-order relation. Assume that this relation, $T$, ...
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### $\mathcal{Ext}^i(\mathcal{O}_{L_1}, \mathcal{O}_{L_2})$ and $\text{Ext}^i(\mathcal{O}_{L_1}, \mathcal{O}_{L_2})$ for two lines in $\mathbb{P}^3$

Let $L_1$ and $L_2$ be two lines in $\mathbb{P}^3$. I want to compute $\mathcal{Ext}^i(\mathcal{O}_{L_1}, \mathcal{O}_{L_2})$ and $\text{Ext}^i(\mathcal{O}_{L_1}, \mathcal{O}_{L_2})$. Basically, there ...
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### A simple complex rational function that leads to a difficult question.

Let $$f(z)=\dfrac{z-a}{z-b},\,\,\,\,\,\,z\not=b\not=a$$ be a complex valued rational function. How can I show that, if $|a|,|b|\lt1,$ then there is a complex number $z_0$ satisfying $|z_0|=1$ and ...
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### Operation that makes the negative real numbers, $\mathbb{R}_{<0}$, a group

The real numbers with addition, $\left( \mathbb{R}, + \right)$, and the positive-real numbers with multiplication, $\left( \mathbb{R}_{>0}, \cdot \right)$, both are Abelian groups. For reasons of ...
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### Prove polynomial and verification

Suppose we have a polynomial $f(x)=a_nx^n+...a_1x+a_0$ where $a_i$ is in real number, $i$ is between $0$ and $n$. Prove that $f(z)=0$ implies $f(\overline z)=0$ Then let $f(x)=x^4+4$. Verify that ...
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### Tangents to Parametrized Curves

I need to find the equation for the line tangent to the curve: $$x = 2\cos t, y = 2\sin t, t = \frac \pi4$$ I also need to find the value of: $$\frac{d^2y}{dx^2}$$ I'm not too sure what to do.. ...
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### Methods of Evaluating $\lim_{x\rightarrow 0} \frac{\sin x}{x}=1$ Multiple Choice Question

Which of the following techniques for evaluating limits cannot be used to show $\lim_{x\rightarrow 0} \frac{\sin x}{x}=1$ $a:$ The Squeeze Theorem $b:$ L'Hôpital's Rule $c:$ Using the graph $d:$ ...
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### Guess the number of the prize & win the prize problem

There is prize in a box. The prize has a value of a positive integer between 1 and N and you are given N. To win the prize, you have to guess its value. Your goal ...
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### How do we decide a problem is in NP, but not in P or NPC?

As I understand, NPC set contains only the problems which can be polynomially converted into each other and which are hardest in NP set/ But how do we decide which problems are in NPC and which ...
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### Axiom of Double Induction?

What would the set-theoretical axiom of induction look like for double induction* when stated in the mathematical language of first- or second-order logic? *References as to What Double Induction ...
I'm attempting to express the composite 3-point Gauss formula for integrating f(x) over an interval [a,b]. $\int_{a}^b f(x) dx = \sum_{j=0}^n f(xj)*cj$ where $cj = \int_{a}^b lj(x)dx$ $lj(x)$ are ...
From ODE I learned if $g$ is Lipchitz on $\mathbb{R}^n$ there exists a unique solution $y:\mathbb{R} \Rightarrow \mathbb{R}^n$ to the IVP \begin{eqnarray} y' &=& g(y)\\ y(t_0) &=& y_0 ...