# All Questions

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### An application of Schwarz lemma to $af(0)+bf'(0)$ [closed]

Let $f$ be holomorphic and $|f(z)|\le 1$ for all $|z|\le 1$. If $a,b \in \mathbb{C}$, show that $$|af(0)+bf'(0)| \le (|a|^2+|b|^2)^{1/2}$$
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### Boundedness of continuous summable function

Let $f\colon\mathbb{R}\to\mathbb{C}$ be a continuous function. If we suppose that $f$ is a $L^1(\mathbb{R;C})$ function too, then can we conclude that $f$ is bounded? ADD: I asked the preceding ...
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### Derivative of a function of vector parameter. Problem with notation.

I have an error function $$err=\frac{1}{N}\left[\textbf{y}^T\ln{\textbf{x}}+(\textbf{1}-\textbf{y})^T\ln{(\textbf{1}-\textbf{x})}\right]$$ I need to find the gradient $\bigtriangledown_x{}err$, such ...
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### How to find the supremum of this?

I would like to know how to find this supremum $$\sup_{x \in [1,\infty)} \left| n\left( \sqrt{x+\frac{1}{n}}-\sqrt{x} \right) - \frac{1}{2\cdot \sqrt{x}} \right|=?$$ where $n \in \mathbb{N}$. I ...
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### Idempotents in a ring of fractions of the tensor product of Gaussian integers [duplicate]

Let $S=\{x^0,x^1,x^2,...\}\subset \mathbb{Z}$ be the multiplicatively closed subset generated by $x$. What are the nontrivial idempotents in the total quotient ring ...
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### Does this figure represent a cumulative distribution function?

Is this a c.d.f.? I have no problem for random variable $X$ at $-\infty<X<x_2$. But if p.d.f. were continuous in interval $x_2\leq X<\infty$ , then c.d.f. should have been continuous. If ...
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### A sequence of random variables $(X_n)$ such that $\mathbb E(X_n)\to -\infty$ but $X_n\to +\infty$ a.s.

Let $\xi_{1},\xi_{2},\dots$ be random variables (i.e measurable functions) such that $\mathbb{P}(\xi_{n}=-3^{n})=2^{-n}$ and $\mathbb{P}(\xi_{n}=1)=1-2^{-n}$ Let ...
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### Why is the group of units mod 8 isomorpic to the Klein 4 group?

I recently learned that $U_8\cong \mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$. I can see, through a bit of computation, that this is the case, but I was wondering if this is just a coincidence ...
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### Writing real invertible matrices as exponential of real matrices

Every invertible square matrix with complex entries can be written as the exponential of a complex matrix. I wish to ask if it is true that Every invertible real matrix with positive determinant ...
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### When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?

Let $m$ and $n$ be two given positive integers such that $m<n$. Then what are the necessary and sufficient conditions for the groups $(\mathbb Z_m,+_m)$ and $(\mathbb Z_n,+_n)$ to be homomorphic ...
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### Perpendicular at a defined distance from point on line intersects another line in coordinates?

It approximately looks like the following picture The figure may be rotated at any angle. I know the coordinates of points $A$, $B$, $C$, $D$ and the length of $BF$. $\angle ABD$ and $\angle CBD$ ...
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### Existence of solution of singular ODE

Suppose $f$ is a Lipschitz continuous function defined on $\mathbb{R}$. How can one prove that the following ODE admits at least one solution. y'' + \frac{1}{x}y' + f(y) = 0 ...
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### Computing an inverse modulo $25$

Supposed we wish to compute $11^{-1}$ mod $25$. Using the extended Euclid algorithm, we find that $15 \cdot 25 - 34 \cdot 11 =1$. Reducing both sides modulo $25$, we have $-34 \cdot 11 \equiv 1$ mod ...
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### Isolating clusters in data

This is a problem that I keep encountering in one form or another every so often. Given a large N dimension space with a sufficiently large (M) sample of points. [1] If I were to be asked to find a ...
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### How to prove: $GH\cdot DJ=GD\cdot JH$

In any $\Delta ABC$, and the incenters of the triangle $ABC$ is $I$, and let $D$, $E$, and $F$ be the points on $BC$, $AC$, $AB$, respectively, and the point $M$ is on $AD$, such that $AD$, $BM$, ...
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### Given that $gcd(a,b)=1$, prove that $gcd(a+b,a^2-ab+b^2)=1$ or $3$, also when will it equal $1$? [duplicate]

It is an exercise on the lecture that i am unable to prove. Given that $gcd(a,b)=1$, prove that $gcd(a+b,a^2-ab+b^2)=1$ or $3$, also when will it equal $1$?
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