# All Questions

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### H is normal in G and H is not equal to <e>, then the intersection of H and C(G) is not equal to e

If $G$ is a finite p-group, $H \lhd G$ and $H \neq <e>$, then $H \cap C(G) \neq <e>$. Hint: Use the idea of the proof of Therorem 47. Let $G$ act on $H$ by conjugation, what is $H_{0}$? ...
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### Solving linear system of equations using Successive Over-Relaxation

I was solving a system of linear equations with SOR. I used different values of relaxation factor (w) for the different runs. I found that for all w > w' (1 < w' < 2), the error is the result ...
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### The Lie subgroup of the compact Lie group

$G$ is a compact connected Lie group with Lie algebra $g$ whose center is $h$. Let $h^{\bot}$ be the orthogonal complement of $h$ where the inner product is chosen to be invariant under the adjoint ...
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### Fourier Series simple question

Let $f$ be a $\mathcal{C}^r$ function such that $f(0)=f(\pi)=0$, and define $a_n := \frac{2}{\pi}\int^\pi_0 sin(nx)f(x)dx$, its easy to show that exists $C>0$ such that $|a_n|\leq \frac{C}{n^r}$. ...
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### Proof by contradiction. Which statement has to be shown to be false?

I want to prove the following statement: Show that if $B=(b_1,....,b_n)$ is a basis of a vector space V, then there is no list of vectors of length $n-1$ that spans V. I would like to prove this by ...
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### Monotonically decreasing Fourier transform

What would be the conditions on $f(x)$ such that it's Fourier transform $F(k)$ would be monotonically decreasing from $k=0$ to half range ($F(0)$ would be the maximum, and it would "fall" on both ...
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### What is a permutation reordering?

What is a permutation reordering? Example Problem Input: A sequence of $n$ numbers: $a_1,a_2,\dotsc,a_n$. Output: A permutation reordering $(a_1',a_2',\dotsc,a_n')$ of the input sequence such as ...
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### Deficienting rank of a matrix

Dear friends Let ‎$\bf{C}‎$‎ be a ‎$m \times n‎$ ‎matrix, where its elements are drawn randomly from a continious distribution, and its rank is ‎$\min (m, n)$‎ with probability one. For ‎decreasing ...
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### More contemporary resource for Greenberg

Is there a more modern textbook than Greenberg's forms in many variables, or a set of notes, that summarizes the developments about $C_i$ fields? Is there such a thing as $C_i$ field where $i$ is not ...
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### Question about the double limit of a sequence

Let $a_{mn}$ be a double sequence. Then under what conditions do we have $$\lim_m\lim_na_{mn}=\lim_n\lim_ma_{mn}$$ In particular, if the sequence is positive, and for each fixed $n$, the limit exists ...
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### How to compute probability distribution function of X+Y using the distribution functions of X and Y?

suppose X and Y are two random variables which are continuous but does not necessarily have density. How can I get the distribution function of X+Y in terms of the distributions of X and Y?
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### Orthogonal Procrustes Problem

The classical orthogonal Procrustes problem concerns finding the matrix $\Omega$ which minimizes $||A\Omega-B||_{F}$ subject to $\Omega'\Omega=I$, with A and B known matrices. Let A be the identity. I ...
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### What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G$ are in the same conjugacy class?

I know the value of this summation $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G$ are in the different conjugacy class will be zero and I know how to prove it but what about if they are in the ...
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### Method of determining dimensions from photographs of multiple angles and degrees of perspective/parallax for a math newbie

I have a project that begins with some 300+ reference photos of a scale model. The only measurements I am certain of are the overall length, and the linear length of one element of one part of the ...
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### is symmetric chi-squared distance “A” metric?

Is symmetric chi squared distance $$\int \frac{(p-q)^2}{pq}\mbox{d}\mu(x)$$ a metric? I am searching web since long time ago but I couldnt find anything. It is positive and is zero whenever $p=q$ ...
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### How many buckets options are there for putting N balls in the buckets?

N balls and how many buckets you want. You want to put the N balls into buckets. How many buckets options are there? For example, (N=)4 balls can be put in the following ways: {1,1,1,1} - 1 ball ...
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### Question on Proof of Shoelace Formula

I was looking for a way to prove the shoelace formula when I found this proof: For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ ...
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### Obstruction of such gauge choice

Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard ...
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### Power series to calculate LambertW up to infinity?

Is this an allowed operation to calculate the Lambert W function as a power series up to infinity, or is there some trouble in defining it this way? Mathematica programs: ...
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### Probability that number of successes in Bernoulli trials is divisible by some number

I have $n$ Bernoulli trials with probabilities of success and failure $p$ and $q=1-p$. How can I find a probability that number of successes is divisible by some given number. I.e. if we choose $3$ ...
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### Comparison of two error distributions to determine “goodness of fit”

I am a physicist who is a few years out of doing his last course in statistics, so I am hoping to get some advice when comparing some data I recently generated. The context is as follows. I have two ...
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### Relationship between conjugacy class and centralizer for measure preserving transformations

Let $(X, \mathcal{B}, \mu)$ be a Lebesgue probability space. Let $\Phi$ be the space of all invertible measure preserving transformations on $(X,\mathcal{B}, \mu )$, endowed with the weak topology. ...
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### General position for one-parameter family of algebraic numbers

Let $P(x,y)$ be an irreducible twovariate polynomial with rational coefficients such that $P(n,.)$ has degree $>1$ for any $n\in{\mathbb N}$. For any $n\in{\mathbb N}$, one may choose a root ...
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### Solving a System of Equations with Cosine

How do I solve a system of equations when there is a cosine. Here is the system:  \left\{ \begin{array}{c} a+b=77° \\ \cos(a)=\frac{y}{3.5} \\ \cos(a)=\frac{y+1}{3.5+x} \\ ...
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### Finite Fields: Linear Feedback Shift Register Algorithm Help

I am currently trying to generate a four digit Linear Feedback Shift Register with digits in Mod 5 using polynomials and finite fields. I am attempting to do so with the following algorithm: 1) ...
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### $C(X,Y)$ complete

I want to prove that: $C(X,Y)$ is complete in the compact-open topology, when every component of $X$ is locally compact with a countable base, and $Y$ is a complete metric space. The proof I am ...
How can I find a generator of $\mathbb{F}_{743}[x]/(x^2+1)$ ? (x-5) ??? Well it has something to do with primes that do not divide the order of 743 which is equal 2*7*53 ....? Does one of you know ...