# All Questions

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### Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
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### Conceptual Question on Cramer rao lower bound for performance measure

In system identification, parameter estimation I have found in several papers that an analytical bound is derived which is the CRB of the error variance of the estimates. For, optimal performance of ...
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### Barycentric Coordinates of the circumcenter of an arbitrary triangle

Given points $A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)$ in barycentric coordinates, and points $P(x_P, y_P, z_P), Q(x_Q, y_Q, z_Q), R(x_R, y_R, z_R)$, what would the barycentric coordinates of the ...
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### I must be misunderstanding bijection on uncountable sets. (As it relates to stereographic projection)

At least one of these things must be false: The existence of a bijection between two sets implies that those sets have an equal number of elements. A bijection exists between {all points in a ...
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### eigen values and eigen vectors of the projection

Assume that $W$ is n-dimensional subspace of an m-dimensional vector space $V$. Find all eigenvalues and all eigenvectors of the projection operators $P_W$. Here is my ideas: Since $W$ is ...
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### Show the following is a functor.

Suppose $p: Y \to X$ and $p': Y' \to X$ are covering maps, and let $\phi: Y \to Y'$ be a homeomorphism such that $p'\phi=p$. Show that the functors $p^{-1}$ and $(p')^{-1}$, from $\Pi_1(X)$ to the ...
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### Primality Test for Safe Primes

Is this proof acceptable ? Theorem Let $N$ be of the form $N=2p +1$ with $p$ prime , then $N$ is prime iff $N \mid 2^{2p}-1$ Proof In one direction , if $2p+1$ is a prime then by Fermat ...
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### Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
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### Probability from multiple trials

This questions is from a practice mid-term that I don't have a solution to. A monkey in a research lab is given 6 tiles with the letters AAABNN. On each trial the monkey randomly arranges the ...
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### How can I prove that $P \neq EXP$

It seems like $P\neq EXP$ is much easier than $P \neq NP$. How can I prove $P \neq EXP$? (Well, after all I want to know any proof technique of proving there does not exist any algorithm of certain ...
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### Laplace Transform Homework

Evaluate the following integrals (you may use the table of the Laplace transform) (a)integral of t^4 e^(−(s+2)t)dt, s > −2 (b)integral of sin3t e^(−(s^2+16)t)dt I solve the first one by using t^n ...
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### Meaning of Exponential map

I've been studying differential geometry using Do Carmo's book. There's the notion of exponential map, but I don't understand why it is called "exponential" map. How does it has something to do with ...
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### the inequality $\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge \frac{a+b+c}2$

How to show that $$\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge \frac{a+b+c}2$$ for $a,b,c>0$? I tried to prove $$\frac{a^4}{a^3+b^3}\ge \frac {5a}4+\frac{-3b}4$$ but could not ...
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### Intuitive Understanding of Second Derivative/Concavity.

In Calculus, I understand that derivatives (simply explained) are a rate of change; a slope of a function at a certain point. However, I am struggling to understand the explanations behind the second ...
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### Help me with this question

Length of AB, BC and CD are equal. length of AD=9,AE=6. Find the length of $CE^2$
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### Why the doubly non-central F distribution does not have a mean or variance if the denominator degree of freedom is less than or equal 2 ??

Normally the doubly non-central F distribution is generated by the division of two non-central chi squared Random Variables,, so what is the the problem of using any famous formula to get the mean of ...
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### How to solve differential equation $dy/dx = y^2/(1+y^2)$ by inegration

My first question is, how does one solve the following differential equation: $$y' = y^2/(1+y^2)$$ My second question is, would it be possible to solve this using ordinary integration method, ...
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### Number of preimages of a group element under a homomorphism.

I was reading the first chapter of Robert Gilmore's "Lie groups, physics and geometry" and I came across a brief statement regarding the number of preimages of an element under an homomorphism which I ...