# All Questions

662 views

### An additive map that is not a linear transformation over $\mathbb{R}$, when $\mathbb{R}$ is considered as a $\mathbb{Q}$-vector space [duplicate]

Possible Duplicate: On sort-of-linear functions I am looking for an example of an additive map that is not a linear transformation over $\mathbb{R}$, when $\mathbb{R}$ is considered as a ...
888 views

### Cartesian product of n sets

I'm currently writing a little application in which I would like to create all permutations of n sets. For example I have the following sets (arrays): ...
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### An epimorphism from $S_{4}$ to $S_{3}$ having the kernel isomorphic to Klein four-group

Exercise $7$, page 51 from Hungerford's book Algebra. Show that $N=\{(1),(12)(34), (13)(24),(14)(23)\}$ is a normal subgroup of $S_{4}$ contained in $A_{4}$ such that $S_{4}/N\cong S_{3}$ and ...
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### linear function

I don't get this, need some help, examples and information The linear function $f$ is given by $$f(x) = 3x - 2 ,\quad -2 \leq x \leq 4.$$ Enter the independent variable and the dependent ...
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### Matrix-Square Root

I was wondering about matrix square roots. What is the procedure to evaluate $(X^{T}X)^{-1/2}$? Is it by a spectral decomposition of $(X^{T}X)^{-1}$ as $U\lambda U^{T}$ followed by the square root $S$ ...
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### Find the convergence radius of $\sum_{n=0}^{\infty}x^n\sin (nx)$

Given this functions series : $\sum_{n=0}^{\infty}x^n\sin (nx)$, I need to find the ratio where it converges. I don't see how can I change it into a form where I'll be able to use Cauchy-Hadmard or ...
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### Möbius maps and their fixed points

Why is it true that if a Möbius map, $f(z)$ fixes distinct $z_1,z_2\in \mathbb C_\infty$ that $f(z)$ either describes a rotation or has a pair of stable and unstable fixed points, such that iterations ...
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### Proof of relative primality

How is it true that: If $a_1, a_2,\ldots,a_n$ are pairwise relatively prime positive integers, then $M_i = \dfrac{(a_1a_2\cdots a_n)}{a_i}$ is relatively prime to $a_i$ ? This is ...
328 views

### Confused about permutation cycles - Question on joint cycles of odd length

For some reason I'm finding permutation cycles to be strange and hard to deal with. Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2 = \beta^2$, then ...
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### How to show that two vector spaces $V$ and $W$ are the same

How to show that two vector spaces $V$ and $W$ are the same, if we know $\dim V = \dim W$ and $V$ is a subspace of $W$ ? Would it suffice to show there exists an isomorphism between them ? Any help ...
229 views

### General relationship between original and rotated+translated line

I have a line $L$ in the plane expressed as the points in $L = \{(x,y) \in {\mathbb{R}}^2 : x \cos \theta + y \sin \theta = r \; \wedge \; 0 > \theta > \pi/2 \}$ (note that the line cannot be ...
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### Calculate ascent from a a path with elevation data

Imagine a person takes a walk through some hilly territory, and they record their elevation at intervals as they go. How would you calculate their total ascent, given that there is some error in the ...
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### Showing $\big( \frac{a}{n\bar{x}}, \frac{b}{n\bar{x}} \big)$ is an exact confidence interval for a gamma distribution

Let $X_1,\ldots,X_n$ be exponentially distributed with parameter $\lambda$ This implies that $Y=\sum_{i=1}^nX_i$ has a gamma distribution with parameters $(\lambda,n)$ Can anyone help me show that ...
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### A complete Boolean algebra $B$ satisfies the $\kappa$-chain condition if and only if $B$ is $\kappa$-saturated

Let $B$ be a Boolean algebra. Then we say $B$ is $\kappa$-saturated if there is no partition $W$ of $B$ such that $|W| = \kappa$. We say that $B$ satisfies the $\kappa$-chain condition if there is no ...
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### Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually: Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion. Let me call it Dedekind's lemma. ...
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### Proving a subset is closed under a binary operation on a set

Suppose that $*$ is an associative and commutative binary operation on a set $S$. Let $H=\{a\in S\mid a*a=a\}$. Show that $H$ is closed under $*$. I started this problem by listing the ...
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### $T(n)=3T(n/2) + n\log n,\ T(1)=1$ [duplicate]

Possible Duplicate: $T(n) = 4T({n/2}) + \theta(n\log{n})$ using Master Theorem What is the order of this recurrence? $$T(n)=3T(n/2) + n\log n,\ \ T(1)=1$$ I found the answer where ...
230 views

### Complexity of finding solutions for a system of polynomial equations

Problem A: Given a set of polynomial equations $f_1=0,\ldots,f_m=0$, where the $f_i$'s are multivariate polynomials with $n$ variables over a field $\mathbb F$, decide whether there is a ...
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### Prove $n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2$

Prove that $$n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\frac{\log 44444444}{\log33333333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2$$ where last ...
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227 views

### Understanding characteristic functions in probability theory.

I am studying characteristic functions in probability theory and I am struggling to understand the following equality. $$\int_{-\infty}^{\infty}e^{itX}dF_X(x)=\int_{-\infty}^{\infty}e^{itX}f_X(x)dx$$ ...
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### Find irreducible but not prime element in $\mathbb{Z}[\sqrt{5}]$

I have tried various numbers of the form $a+b\sqrt{5},\ a,b \in \mathbb{Z}$, but cannot find the one needed. I would appreciate any help. Update: I have found that $q=1+\sqrt{5}$ is irreducible. Now ...
Let $X$ be a set and $A\subset X$. For a sequence $F = (f_n)_{n\geq 0}$ of elements of $X$ we say that $F$ is eventually always in $A$ if for some $N\geq0$ it holds that $f_n\in A$ for all $n\geq N$. ...