# All Questions

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### How to show that the $\phi and \varphi$ satisfy the Cauchy Riemann equation

when u=(u,v)=($\frac{\partial\varphi}{\partial y},-\frac{\partial\varphi}{\partial x}$) and u(x)=grad$\phi$(x)=$\nabla\phi$(x) how can you use the equations above to prove that the $\phi$ and the ...
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### Why is P or not P is unsatisifiable by construction?

A proof of predicate logic inability to express graph reachibility (page 63) involves a formula which can be interpreted as (there is no path, no matter what is the length) or (there is some path). ...
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### If $a$ is a root for $f(x):=5x^3-7x^2+3x+6$, then find $a_{0}\in \mathbb{Z}$ s.t. $|a-a_{0}|_{7}\leq 7^{-4}$

The problem is: By Hansel's lemma I found that f has root $a\in \mathbb{Z}_{7}$ s.t. $|a-1|_{7}<1/7$. So next I am asked to find $a_{0}\in \mathbb{Z}$ s.t. $|a-a_{0}|_{7}\leq 7^{-4}$. One approach ...
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### soft (standardness of definition): “Regular sequence” in non-ommutative rings

Is it non-standard to call a sequence $x_1,..,x_n$ of commutating elements in a possible non-commutative ring "regular" if multiplication by $x_{k+1}: R/(x_1,..,x_k) \mapsto R/(x_1,..,x_k)$ is a ...
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### Topological spaces X and Y and a continuous bijection $f : X → Y$ while $f^{-1} : Y → X$ is not continuous

Give an example of topological spaces X and Y and a continuous bijection $f : X → Y$ such that $f^{-1} : Y → X$ is not continuous.
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### An MLE Exercise.

Consider $$f(x;\theta)=\frac{1}{\pi} \frac{e^{\theta x}cos(\theta \pi/2)}{cosh(x)}, x\in{\mathbb{R}}$$ be a family of densities and which is clearly exponential family. Then what is the Maximum ...
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### Reflexive. What does it mean?

I would like to know the definition for reflexive. I have not found anything on the internet or in my book.
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### set functions and relation multiplication

$\ M$ is the set of all relations on $\ A = \{1,2,3\}$ $\ K$ is the the following relation on A $\ K=\{(1,1),(2,1),(3,1)\}$ let there be $\ f :M\rightarrow M$ $\ f(R) = RK$ is f injective? ...
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### a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
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### Calculus and infinitesimals

In the definition of reimann integral, why do we put a 'dx' inside the integral sign when practically it serves no purpose except maybe telling what variable you are talking about. Then in some ...
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### How to show uniqueness of $\nabla\phi$ using Green's theorem when the value of Neumann problem exists.

It says getting function $\phi(x,y,z)$ that satisfies the following conditions $\frac{\partial\phi(x)}{\partial n}$=h(x), x$\in$S is called the Neumann problem. The problem is to show the uniqueness ...
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### Regular expression translation.

Given a set {1,2,...9} how can I construct a regular expression starts with a 3 has no 8's and has even number of 6's? Here's what I tried:  Define a new set no8 = {1,2,3,4,5,6,7,9} ...
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### A multiple choice question on the system of linear equations

Consider the following system of linear equations. $x + y + z + w = b_1$; $x - y + 2z + 3w = b_2$; $x - 3y + 3z + 5w = b_3$; $x + 3y - w = b_4$. For which of the following choices of $b_1$, $b_2$, ...
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### ($\cos^4x$)($\sin^2x$) in terms of first power of cosine

I believe that I have his correct but if someone could check it and see that'd be great. Here's a pic! [IMG]http://i58.tinypic.com/2dgm5ic.jpg[/IMG]
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### The interval $(0,\infty)$ is an open set.

I want to prove this using interior points, $\epsilon$-neighborhoods and interior sets. The interior of a set A is denoted $A^o$. To show that $(0,\infty)$ is an open set, we must show that ...
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### Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k$ on some suitable disk of ...
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### Does $f^{(n)} = 0$ imply that complex $f$ is a polynomial?

Let $f$ be a complex function with the property that $f^{(n)} = 0$. Does this imply that $f$ is a polynomial? If so, why? Upon thinking about this problem myself, I can easily observe that every ...
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### Find the intersection of three bisection lines

Let $p_1 = (a_1, b_1), p_2 = (a_2, b_2), p_3 = (a_3, b_3)$ be three, non-colinear points in the plane. For each pair of these points, let $L_{ij}$ denote the line segment from $p_i$ to $p_j$. (a) For ...
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### A Simple yet interesting “function of a random variable” question

Given continous density functions $f_0,f_1$ on $\mathbb{R}$ and $Y$, a random variable following the density $f_0$, I am able to calculate the density function $h$, of $\ln l(Y)=\ln(f_1/f_0(Y))$ as ...
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### prove or disprove: $\lim_{x\to \infty} \frac{f(x)}{g(x)}=\lim \frac{f'(x)}{g'(x)}=0 \implies \lim \frac{\frac{f(x)}{g(x)}}{\frac{f'(x)}{g'(x)}}\ne 0$

my attempt $$\lim_{x\to \infty}\frac{\frac{f(x)}{g(x)}}{\frac{f'(x)}{g'(x)}}\text{ yields }\frac{0}{0}$$ then use l'hopital on this \lim_{x\to \infty}\frac{\frac{f(x)}{g(x)}}{\frac{f'(x)}{g'(x)}} ...
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### denumerables: prove or disprove the following

Prove or disprove: a. if $A \subseteq B$ and A is denumerable, then B is denumerable. b. $J \cup K$ isdenumerable, where J is the set of all linear functions with slope 1 and retional y intercept, ...
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### PDF of Sum of Two Random Variables [on hold]

$X$ and $Y$ are uniformly distributed on the unit disk. Thus, $f_{X,Y}(x,y) = \begin{cases} \frac{1}{\pi}, & \text{if} ~ x^2+y^2 \leq 1,\\ 0, &\text{otherwise.}\end{cases}$ If $Z=X+Y$, find ...
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### Characteristic curves of 2nd-order PDEs under invertible coordinate transformations

First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know. Given a linear, scalar, ...
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### Acceptance Probability [on hold]

10 Students are applying for postdoctoral position. The employer will only choose one candidate. Student Data 1 (PhD) 2 (MSc) 3 (PhD) 4 (BSc) 5 (BSc) 6 (MSc) 7 (MSc) 8 (BSc) ...
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### Painting a grid of squares.

Consider a $9\times9$ block of squares where each square is painted either black or white. If each square is adjacent to at most three black squares, what is the maximal number of black squares? Here ...
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### Defining a region as a data structure

Is there a way for one to define a curve or region (such as a closed, 2-d disk) as a data structure into the computer, and make an algorithm which detects if a point is a boundary point, limit points, ...
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### Show that $(1 – \cos θ – \sin θ )^2 – 2(1 – \sin θ )(1 – \cos θ ) = 0$.

Show that $(1 – \cos θ – \sin θ )^2 – 2(1 – \sin θ )(1 – \cos θ ) = 0$. What kind of formulas should I use?
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### Ambiguity definitions - accumulationpoint

The literature is a bit ambiguous in my point of view. Limit points and accumulation points seems to be the same. I can accept that; that's just two names for the same. But I've seen different ...
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### Statistics: When to use pooled vs non-pooled vs paired

I'm looking for a simple method for knowing when to use pooled vs non-pooled vs paired tests in a given statistics problem. For example, if the standard deviations are exactly the same, I'm told I can ...
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### Matlab Cropping Plots

I want to adjust the amount of whitespace that is shown when plots are generated. I found this command to do just that: ...
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### Alexander–Briggs notations for the links or knots of $N^3_m$

We can use Alexander–Briggs notations for the links or knots. For example, is three separate loops with no links. And there are many other examples of Alexander–Briggs notations for three ...
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### Expanding sum of negative power using log

Example expands the series with natural logarithm but I do not know this rules. $\sum_{a} a^{-n} = \sum_{a}1 - n \sum_{a}log(a) + \cdots$ Anybody understand this expansion? Thank you!
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### probability and applied statics 4 [on hold]

In cairo 30% of residents listen to the local fm radio . ten residents are chosen at random? a) state the distribution of the random variable b) find the smallest value of s so that P (x >or equal ...

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