# All Questions

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

Prove $(n -1) = (n-1)^{n}$ mod n How would one go about doing this?
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### Tropical Lifting Lemma (Puiseux Series)

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

### What does the symbol $\otimes$ mean?

I am familiar with the direct sum of sets, $\oplus$.
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### 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; } ...
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### 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 ...
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 ...
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### 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?
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### 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 ...
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 $z \in K(d_{1},d_{2},...,d_{n})$?
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 ...
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 ...
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 ...
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### 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 ...
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.
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### 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 ...
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 $Z^{(j)}$, $j=0,1,...$ ...
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### 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 ...
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. ...
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### 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 ...
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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 $\phi:X \to (-\infty, \infty]$ ...
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 ...
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### 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 ...