1
vote
1answer
125 views

Prove $(n -1) = (n-1)^{n}$ mod n

Prove $(n -1) = (n-1)^{n}$ mod n How would one go about doing this?
4
votes
1answer
178 views

Tropical Lifting Lemma (Puiseux Series)

The general result in tropical geometry is $K$ algebraically closed valued field $I$ ideal of $K[x_1, \cdots, x_n]$ $V(I) = \lbrace \bar{a}\in K^n: f(\bar{a})=0 \text{ for all } f \in I \rbrace$. ...
8
votes
2answers
804 views

The modular curve X(N)

I have a question about the modular curve X(N), which classifies elliptic curves with full level N structure. (A level N structure of an elliptic curve E is an isomorphism from $Z/NZ \times Z/NZ$ to ...
0
votes
0answers
57 views

Randomized methods

Randomized methods are often used in the probability theory as a kind fo numerical methods to obtain some results which cannot be easily (or even hardly) obtained analytically. One of the most famous ...
0
votes
1answer
2k views

What does the symbol $\otimes$ mean?

I am familiar with the direct sum of sets, $\oplus$.
0
votes
1answer
130 views

Correct form of sum expression

I want to create equation that represents following piece of code (it is much more complicated I simplify it for clearance) int v = 23; sum = 0; for (int i = v; i > 0; i= i - 2) { sum = sum + i; } ...
4
votes
1answer
572 views

Kullback-Leibler divergence based kernel

I'm looking to paper "A Kullback-Leibler Divergence Based Kernel for SVM Classification in Multimedia Applications". Author suggest to use kernel function for two distributions $p$ and $q$: $k(p,q)= ...
5
votes
2answers
692 views

Euler's phi function and distinct primes

It is true that $\phi(p) = (p-1)$ only if p is a prime. I had also proven (I am not sure if this is a trivial fact or not) that $\phi(pq) = (p-1)(q-1)$ only if p and q are distinct primes. However, I ...
1
vote
2answers
227 views

If a quadratic is reflected on the y axis, why does $b=0$?

If a quadratic in the form $y=ax^2+bx+c$ is reflected on the $y$ axis, why then must $b=0$, making the equation in the form of $y=ax^2+c$? I can remember the rule, but the reasoning behind it has ...
0
votes
1answer
67 views

How do I arrange the following to solve for

This is electronics related but more algebra than electronics. I'm a bit rusty having not done anything involving mathemtics for night-on 10 years. I have the following thing to solve for $R_X$: ...
10
votes
4answers
515 views

What is undecidability

What does it mean that some problem is undecidable? For instance the halting problem. Does it mean that humans can never invent a new technique that always decides whether a turing machine will ...
3
votes
0answers
289 views

Rational independence

The specific question is the following: I am given a set of $[L/2]$ numbers $$g(n) = \sqrt{ c(n)^2 + \alpha c(n) + \beta},$$ where $c(n) = \cos(2\pi n/L)$ (so both $c(n)$ and $g(n)$ depend on $L$ ...
0
votes
2answers
454 views

Every subgroup $H$ of a free abelian group $F$ is free abelian

I am working through a proof that every subgroup $H$ of a free abelian group $F$ is free abelian (for finite rank) For the inductive step, let $\{ x_1, \ldots, x_n \}$ be a basis of $F$, let $F_n = ...
6
votes
3answers
275 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
10
votes
1answer
515 views

Constructive Proof of Kronecker-Weber?

This question is motivated by my attempt at solving Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ Consider ...
1
vote
2answers
765 views

What is a 2-regular graph?

What is a 2-regular graph? Is is the same thing as an 2-connected graph where a 2-connected graph is a graph G such that G-V ( G minus a vertex V) is still connected?
6
votes
3answers
641 views

Mastering Math - Grade school- to College-level?

I'm having difficulty with my math, fractions and up. I used to understand it all, but it's been so long since I've touched the book (I finished it a couple of months ago, picked it up to review ...
4
votes
2answers
545 views

About the Three Reflections Theorem

I recently solved this exercise from Hartshorne's Classical Geometry. Given three lines $a$, $b$, $c$ through a point $O$, show that there exists a unique fourth line $d$ such that ...
7
votes
1answer
257 views

The idea's clear; the proof isn't

I'm working through Enderton's book on set theory. In chapter 3, there are a series of exercises regarding functions. For instance Prove that if $F$ and $G$ are functions, $dom(F) = dom(G)$, and ...
0
votes
1answer
149 views

How to find the max of the value?

I have $\frac{(a-1)}{4}+\frac{b}{2}=x$ and $0\leq a \leq 1$, $0\leq b \leq 1$. How can I find the values for a, b such that x is maximized? Thanks. EDIT: Sorry, I forgot to clarify, I am working with ...
2
votes
1answer
89 views

degree of differentiability of a manifold at a point

I am neither aware fully nor have studied differential geometry, but i'd like to learn it if i get to know the answer for this question. I am asking this question based on the very superficial ...
1
vote
1answer
122 views

Existence of elements in a extension field

Let $F/K$ be an extension field and let $D$ be a subset of $F$ and $z \in K(D)$. Why we can find a subset $\{d_{1},d_{2},...,d_{n}\} \subseteq D$ such that $z \in K(d_{1},d_{2},...,d_{n})$?
4
votes
2answers
493 views

The Strong Whitney Embedding Theorem-Any Recommended Sources?

Just about all of the standard textbooks on manifold theory give proofs of weak versions of the Whitney Embedding theorem. But other then Whitney's original 1944 paper,are there any standard sources ...
1
vote
1answer
109 views

Find connected components of a a set Y

Let Y be the union of all the circles of center (1,0) radius 1-1/n in R^2. Then, we have circles of increasing radius, finally reaching r=1 as n goes to infinity. The connected components are the ...
0
votes
1answer
171 views

On the immersion, regular value theorem

Let $g\colon N\to M$ be an immersion. Then, I think that $g^{-1}(p)$ is finite set or $0$-dimensional manifolds for all $p\in M$. Now, let $g_t\colon N\to M$ be an one-parameter family of an ...
0
votes
1answer
97 views

What is the type of the “adjoin” operator on rings?

Context: I am trying to formalize the definition of a polynomial ring in a programming language. One way to think about it, is to start by formulating the definition of a ring extension by adjoining ...
1
vote
2answers
154 views

order of a group

let $D = <x, y | x^2, y^2, (xy)^n>$. What is the order of $D$? Thank you very much.
2
votes
3answers
214 views

Generalizing an orthonormal basis for $C[a,b]$

While learning the Gram-Schmidt orthonormalization process, my text would discuss orthonormalizing the standard basis for a subspace of $C[0,1]$, which is the space of continuous differentiable ...
10
votes
2answers
257 views

A sequence of order statistics from an iid sequence

Given a sequence of iid random variables $X_i$ (without loss of generality from $U(0,1)$), an integer $k \ge 1$ and some $p \in (0,1)$, construct the sequence of random vectors $Z^{(j)}$, $j=0,1,...$ ...
0
votes
1answer
132 views

Prove that $x^2 \equiv a \pmod{2^e}$ is solvable $\forall e$ iff $a \equiv 1 \pmod{8}$

Prove that $x^2 \equiv a \pmod{2^e}$ is solvable $\forall e$ iff $a \equiv 1 \pmod{8}$. Whenever I encounter congruence proof, I'm stuck right away. How can I link $a \equiv 1 \pmod{8}$ with the ...
5
votes
4answers
444 views

Suppose that $x > 1$. Prove that $s_n \to 1$

Question: Suppose $x > 1$. Prove that $x^\frac{1}{n} \to 1$. The following is a list of what I am trying to do through out my proof. Show $s_n$ is monotone (decreasing). Show $s_n$ is bounded. ...
1
vote
1answer
195 views

Does this number exist?

Consider the finite algorithm A, and the real number $0<T<1$. The output of A on input T is all possible theorems and provable propositions in ZFC, and only that. Q1. Can such an algorithm and ...
5
votes
1answer
133 views

Proving all Solutions of a Polynomial Cannot all be Real

If, a, b, c, d and e are all real numbers how could I prove that the 5 solutions of the equation: $$f(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e == 0$$ cannot all be real valued if: $$2a^2 < 5b$$ Any ...
4
votes
1answer
174 views

Why $1$ is the only quadratic residue modulo $8$?

I'm trying to understand the Proposition 5.1.1 - Ireland and Rosen, A Classical Introduction to Modern Number Theory, p.50, however, I can't understand why this argument is true: $1$ is the only ...
3
votes
2answers
436 views

Finding best response function with probabilities (BR) given a normal-matrix representation of the game

We are given players 1, 2 and their respective strategies (U, M, D for player 1, L, C, R for player 2) and the corresponding pay-offs through the following table: $\begin{matrix} 1|2 & L & C ...
1
vote
0answers
211 views

approximate solution for bin packing problem that minimizes sum of max values of bins

I am trying to approximate the following NP-hard problem, which is similar to bin packing, but does not have a linear objective function: minimize $\Sigma_{i=1, \ldots, W}$ max{$v_s$ | s $\in$ ...
1
vote
1answer
202 views

How to find matrices with given commutator

Consider $M_2(\mathbb{Z})$. Is it possible to find two matrices A,B such that their commutator AB - BA equals a given matrix C? Is there any chance to characterize all possible occuring commutators in ...
2
votes
1answer
224 views

questions about simple groups

how to show that there is no simple group of order $1755 = 3^3 \cdot 5 \cdot 13$? Thank you very much.
1
vote
1answer
170 views

composition of multivariate power series

The composition of formal power series $g \circ f$ is well defined if f has vanishing constant term. My question is how one can generalize composition of power series to several variables? If we ...
3
votes
2answers
259 views

Help with a quintic polynomial

$$y = 0.10 + 4.060264x - 6.226862x^2 + 48.145864x^3 - 60.928632x^4 + 49.848766x^5$$ I need to be able to solve this equation for $x$. I've looked around and seem to be failing miserably and solving ...
1
vote
1answer
182 views

Evaluating Jacobi's symbol using Eisenstein's algorithm

The algorithm is described as follow, Eisenstein proposed the following algorithm for computing the Jacobi symbol $(a|n)$ where $a$ and $n$ are odd numbers. Write $n = aq + er$ where $q = ...
2
votes
1answer
1k views

Find equation of cubic function with gradient?

I have to find a cubic function, when I know the gradient of that line at its start and finish. The gradient at the start is 0.2, and at the top -0.4. Using the cubic function: ...
3
votes
0answers
83 views

Maybe there is an interpretation of something like $ S_{-3}$?

I'm doing some exploratory afternoon reading, and I'm baffled by a minor detail in this paper. Here is the background: A sequence $S = S_n$ is almost convergent to L if for any $\epsilon > 0$ we ...
2
votes
1answer
62 views

Decreasing function in the context of gradient flows

I'm studying the lecture notes by Philippe Clément about Gradient Flows in Metric Spaces. Now the following problem arises ($X$ is a Hilbert space): Definition: Let $\phi:X \to (-\infty, \infty]$ ...
1
vote
1answer
308 views

Closed form for the exponential of a Lie algebra 3x3 matrix?

Matrices of the form: $$\begin{pmatrix} iz+iy&-iz+w&-iy-w\\ -iz-w&iz+ix&-ix+w\\ -iy+w&-ix-w&iy+ix\end{pmatrix}$$ where $x,y,z,w$ may be assumed to be real, form a Lie ...
4
votes
1answer
76 views

Enhancing the monoid structure over a finite alphabet to prove Arden's rule

Suppose you have a finite-state, deterministic automaton, that you wish to convert to a regular expression. A common method, perhaps easier to apply by hand that Yamada's algorithm, is to reduce the ...
3
votes
2answers
151 views

probability -Diverging expectation

As I keep reading probability books, there are always some issues that no one considers. For example, for $\omega \in \Omega$ and $X$, $Y$ independent random variable we define $Z(\omega ...
3
votes
2answers
426 views

Birth and Death Process Question (Queuing)

A small shop has two people who can each serve one customer at a time. There is also space for two customers to wait. Anyone who arrives and sees that the shop is full will go to another store. ...
7
votes
4answers
438 views

Please help me to show, that $(\ln x)'=\frac1 x$

In school, we recently started with derivations. I looked into a list of simple derivations and tried to prove them, in order to practice. Now, I tried to find the derivative of $\ln x$, but I got ...
4
votes
1answer
292 views

Center of a polygon inside the polygon

What is the name of the point(s) in a polygon, calculated by "shrinking" the polygon until there's no surface left? Example (the light areas): Also, of possible, it would be cool to have an ...

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