1
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0answers
30 views

Example regarding the difference quotient

Give an example of a differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f^\prime(0) = 1$, but there are no points $x,z \in \mathbb{R}$ for which $\frac{f(x)-f(z)}{x-z}=1$ and $x \neq ...
1
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0answers
23 views

Chebyshev Polynomials of $P_3$

Solve a special cubic equation $4x^3-3x=a$ by plugging $x=(u+u^{-1})/2$. Note that the equation has a form $P_3(x)=a$.
1
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0answers
83 views

Quadric equation in $\mathbb{Z}/n\mathbb{Z}$?

I would like to ask for some help about the following problem. Given is a polynomial $f(x)=ax^{2}+bx+c$ in $\mathbb{Z}/n\mathbb{Z}$, we know that this quadric equation $f(x)=0$ has exactly 8 ...
1
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0answers
15 views

Find Maximum of Function $L(p)=\prod_{i=1}^{20}\left[ (1-p)A_{i}+pB_{i}\right]$

The $A_{i}'s$ and $B_{i}'s$ are known. I seek the $p$ which maximizes $L(p)$. I thought it might be easier to maximize $\log L(p)$ instead $L(p)$, but I think it is a dead end $\log ...
1
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0answers
33 views

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 $H_{0}$? ...
1
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0answers
226 views

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 ...
1
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0answers
53 views

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 ...
1
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0answers
37 views

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 $|a_n|\leq \frac{C}{n^r}$. ...
1
vote
0answers
128 views

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 ...
1
vote
0answers
47 views

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 ...
1
vote
0answers
52 views

Vanishing criterion of pure wedges

Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
1
vote
0answers
32 views

Differentiation of trigonometry, can someone check my answer

Differentiate with respect to $x$ $f(x)=(3x^{2}-4)\sin x$ Let $u=3x^{2}-4$ $du/dx=6x$ Let $v=\sin x$ $dv/dx=\cos x$ $f'(x)=6x\sin x+(3x^{2}-4)\cos x$
1
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0answers
25 views

Probabilistic event triggered on a Markov Process transition

I would like to assess a system disponibility using a Markov Process. This system has two states : a functionning state 0 and a failure state 1, with a fault rate $\lambda$ and a mean time to repair ...
1
vote
0answers
21 views

Good notation for many random points approximating an area.

I'm trying to say that as the number of random coordinate points points you plot approaches infinity, it is equivalent to an area integral where each point is an infinitesimally small $\mathrm{d}x ...
1
vote
0answers
55 views

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 ...
1
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0answers
79 views

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 ...
1
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0answers
20 views

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 ...
1
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0answers
106 views

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 ...
1
vote
0answers
26 views

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?
1
vote
0answers
188 views

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 ...
1
vote
0answers
183 views

Cauchy repeated integration formula - different lower limits/ change of variable

Cauchy's repeated integration formula is as follows: Let $f^{(-n)}$ be a continuous function on the real line. Then the $n^{th}$ repeated integral of $f$ based at $a$, $$f^{(-n)}(x) = \int_a^x ...
1
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0answers
54 views

Gaussian quadratures: Adaptive method

In my book I was asked to find the gaussian quadrature on the form $$ \int_0^1 f(x)\mathrm{d}x \approx W_0 f(x_0) + W_1 f(x_1) $$ With weight $w(x)=1$ and where $x_0$ and $x_1$ are the zeros of ...
1
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0answers
127 views

Theorem of Lebesgue and differentiation of a parameter integral

Let $(a,b)\subset\mathbb{R}$ be an interval and let $\left\{f_t\colon\Omega\to\mathbb{R}\right\}_{t\in (a,b)}$ be a family of measurable functions on the measurable space ...
1
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0answers
49 views

is there a formula for the sum of n'th powers

is there a formula for this sum $\sum_{i=1}^{k} {i^{n}}$ i know of the formula for square and cubic sums, $\sum_{i=1}^{k} {i^{2}}$, $\sum_{i=1}^{k} {i^{3}}$ but i couldn't find a ...
1
vote
0answers
69 views

Could Someone Just Verify This Proof for Me? (Euler's Theorem)

I came up with this proof for my number theory class. Is it valid? Proposition: $u\in U_m \Rightarrow u^{\varphi(m)}=1$ (Where $U_m$ is the multiplicative group of integers modulo $m$) Attempted ...
1
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0answers
12 views

Fourier analysis of real valued function

Under what condition is it not possible to obtain the fourier transform of a real valued function?
1
vote
0answers
43 views

Family of sets. Directed acyclic graph representation.

We are given a family of sets $F=\{F_1,\ldots,F_n\}$ with each $F_i$ being a subset of a ground set $N=\{1,\ldots,n\}$. In addition, we assume for each $F_i$ that it's not the subset of another $F_j$ ...
1
vote
0answers
59 views

What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
1
vote
0answers
135 views

Question about Radon-Nikodym derivative

Question: Suppose $X$ is a finite set equipped with the algebra $\mathcal{A} $of all subsets and we are given two measures $\mu$ and $\nu$ on $X$ with the condition that $$ \mu( \{ x \} ) \neq ...
1
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0answers
49 views

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 ...
1
vote
0answers
161 views

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 ...
1
vote
0answers
236 views

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 $p=q$ ...
1
vote
0answers
106 views

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 ...
1
vote
0answers
226 views

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 $OA_{i}A_{i+1}$ ...
1
vote
0answers
30 views

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 ...
1
vote
0answers
131 views

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: ...
1
vote
0answers
33 views

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 $3$ ...
1
vote
0answers
41 views

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 ...
1
vote
0answers
41 views

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. ...
1
vote
0answers
23 views

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 ...
1
vote
0answers
112 views

Sum of all possible products when each product is truncated if too large

I have a set of sets of real numbers greater than $1$. Each set can have a different quantity of numbers. Set $A_1 = \{a_{11}, a_{12},\ldots,a_{1m_1}\}$ Set $A_2 = \{a_{21}, a_{22}, \ldots, ...
1
vote
0answers
184 views

Subgame Perfect Nash Equilibrium Problem

Suppose that there are two incumbent icecream vendors, but that there is a possibility of entry of a third vendor. Specifically, at any location on the beach a third vendor can, after observing the ...
1
vote
0answers
773 views

Laguerre polynomials recurrence relation

Laguerre polynomials has the recurrence relation $$(n+1) L_{n+1} (x)=(2n+1-x) L_n (x)-nL_{n-1} (x)$$ In proving this, I differentiate the generating function of laguerre polynomials ...
1
vote
0answers
81 views

Some problems with a proof of the Farkas Lemma

The following is a proof of the Farkas Lemma that is creating me quite some problems. [I presented the all proof simply to point out the notation used by the author.] My problem is with the last part ...
1
vote
0answers
231 views

Gamma random variables with fixed sum (different scale parameters)

Given a vector of independent random variables $\{X_i\}_{i=1..N}$, each of which is distributed according to a Gamma-distribution with pdf $Pr(X_i=x;\alpha_i,\beta_i) = \frac{1}{\Gamma ...
1
vote
0answers
48 views

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} \\ ...
1
vote
0answers
127 views

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) ...
1
vote
0answers
74 views

$C(X,Y)$ 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 ...
1
vote
0answers
20 views

Internals of a MIP Solver

I would like to learn about the internals of a Mixed Integer Programming (MIP) solver. Which concepts shall I read about? Are there a couple of standard books which can be a good start?
1
vote
0answers
94 views

Finding a generator of a finite field

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 ...

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