# All Questions

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### Prove that if $A$ is independent of $B$, $A$ is independent of $C$, then $A$ is independent of $B\cup C$.

Prove that if $A$ is independent of $B$, $A$ is independent of $C$, then $A$ is independent of $B\cup C$. $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(AB)$ $\mathbb{P}(A)\mathbb{P}(C)=\mathbb{P}(AC)$ So ...
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### Is $f(x)=x+\sin x$ bounded variation

Does $f(x)=x+\sin x$ has a bounded variation on $\Bbb R$? I dont know the concept of bounded variation.
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### What is the min integer factor which makes $4x+2y$ integer?

Sorry for the english translation. $x$ and $y \in \Bbb R$. $15$ is the minumum integer which makes $x$ integer when you multiply. $18$ is the minumum integer which makes $y$ an integer when you ...
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### What is the Cumulative Distribution Function of $a/x^b$? [on hold]

I was just wondering what the CDF of $$\frac{a}{x^b}$$ would be? $a$ and $b$ are positive constants and $b \gt 1$ ($1.22$ to be exact). $x \in [0, \infty)$ theoretically but in practice once $x$ has ...
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### Probability that in bridge game the Players N,E,S,W have a,b,c,d spades respectively.

There are 52 cards in bridge and 13 cards of each suit. The formula for numerator is: $${13\choose a}{39 \choose 13-a}{13-a\choose b}{26+a\choose 13-b}{13-a-b\choose c}{13+a+b\choose 13-c}$$ But i ...
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### Finite Modules Isomorphism

Two vector spaces are isomorphic if and only if they have the same dimension. In particular, two vector spaces over a finite field are isomorphic if and only if they have the same cardinality. For a ...
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### Minimal polynomial of a diagonal matrix

How can I show that the minimal polynomial of a diagonal matrix is the product of the distinct linear factors $(A-\lambda_{j}I)$? In particular, if we have a repeated eigenvalue, why is it that we ...
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### How low density in LDPC codes leads to best code

I'm want to use LDPC code as a channel coding mechanism. In LDPC we are using sparse matrix for parity check matrix (H). Why we are going for low-density. What are the advantages?.why it is works for ...
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### What did Lagrange, Euler, Gauss etc. learn in order to know what they knew?

What did the great mathematician, like Lagrange, Euler and Gauss, learn in order to know what they knew? It seems that they were extremely good in the most basic rules/structures/issues of math: ...
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### Intermediate digits of 34!

Problem: Given that $34!=295232799cd96041408476186096435ab000000$. Find $a, b, c, d$. $a, b, c, d$ are single digits. I am able to find $a$ and $b$ but cant find $c, d$. I did the prime factorisation ...
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### Sample variance converge almost surely

Suppose $X_{1},X_{2},\ldots$ be i.i.d. random variables such that $E\left[X_{i}\right]=\mu$ and $Var(X_{i})=\sigma^{2}<\infty$. Let $\bar{X}=\left(X_{1}+\cdots+X_{n}\right)/n$. Show that ...
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### Prove $\log_7 n$ is either an integer or irrational

I have been trying to prove a certain claim and have hit a wall. Here is the claim... Claim: If $n$ is a positive integer then $\log_{7}n$ is an integer or it is irrational Proof ...
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### graded Hopf algebra and its dual

I am learning Hopf algebras, and there are two questions as follows: Is the tensor product of two Hopf algebras still a Hopf algebra? Let $A$ be an infinite dimensional algebra. Is the dual ...
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### What is the difference between a simple “fraction” and a “common fraction”?

I have read about a common fraction in this statement (written in a text book): Ratio is the simplest form of a common fraction, in which the numerator denotes the antecedent and the denominator ...
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### Generating function for tuples of objects based on their maximal size

This is a question which arose while working through Flajolet-Sedgewick's Analytic Combinatorics. In their terminology, the cartesian product of two combinatorial classes $\mathcal{A},\mathcal{B}$ ...