Pure Math Archives - Time Travel, Quantum Entanglement and Quantum Computing

Godel’s Consistency of Axiom of Choice Paper

anuj — Tue, 21 Jan 2025 02:52:10 +0000

Gödel’s Landmark Paper

The Consistency of the Axiom of Choice and the Generalized Continuum-Hypothesis with the Axioms of Set Theory

Background

Set Theory and ZFC: The Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC) form the standard framework for modern set theory.
- The Axiom of Choice (AC) states that for any collection of non-empty sets, there exists a choice function that selects one element from each set.
- The Generalized Continuum Hypothesis (GCH) postulates that for any infinite cardinal κ, 2^κ = κ+ (the next cardinal).
Hilbert’s Program: David Hilbert asked about the consistency of mathematics, including AC and GCH within ZFC.
Motivation for Gödel’s Work: To show that AC and GCH are consistent with ZFC, assuming ZFC itself is consistent.

Gödel’s Results

Inner Models and Constructibility: Gödel introduced the constructible universe L, a class of sets built in a step-by-step, definable manner.
Axiom of Choice (AC): Gödel showed that AC holds within L.
Generalized Continuum Hypothesis (GCH): Gödel proved that 2^κ = κ+ for all infinite cardinals κ in L.
Consistency Proof: Gödel concluded that if ZFC is consistent, then ZFC + AC + GCH is also consistent.

Methodology

The Constructible Universe: Gödel defined L as a hierarchy indexed by ordinals:
- L₀ contains all hereditarily finite sets.
- L_α+1 includes subsets of L_α that are definable from parameters.
- L_λ for limit ordinals λ is the union of all L_α for α < λ.
Relative Consistency: Gödel showed that if ZFC is consistent, then so is ZFC + AC + GCH.

Impact and Subsequent Developments

Completeness vs. Independence: Paul Cohen (1963) showed that AC and GCH are independent of ZFC, meaning they can neither be proved nor disproved from ZFC.
Foundation of Modern Set Theory: Gödel’s methods laid the groundwork for later developments in set theory.
Philosophical Implications: Gödel’s work highlighted the limitations of formal systems and the existence of different “universes” of set theory.

Conclusion

Gödel’s 1940 paper demonstrated the relative consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the axioms of set theory. This foundational work had a profound impact on set theory and mathematical logic, influencing both its philosophy and technical practice.

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Construct a conformal equivalence f between the “angle” {z ∈ C | z 6= 0, 0 < arg(z) < π/3} and the unit disk D ⊂ C

anuj — Mon, 30 Dec 2024 22:42:48 +0000

Constructing a Conformal Equivalence

We aim to construct a conformal equivalence f between the “angle”
A = { z ∈ ℂ | z ≠ 0, 0 < arg(z) < π/3 }
and the unit disk
𝔻 = { w ∈ ℂ | |w| < 1 }.

1. Map the angle to a horizontal strip

Use the logarithm map:

z → w₁ = log(z) = ln|z| + i·arg(z),

where z ∈ A. Under this map:

The boundary ray arg(z) = 0 maps to the real axis (Im(w₁) = 0).
The boundary ray arg(z) = π/3 maps to the line Im(w₁) = π/3.
The region A maps to the horizontal strip
S = { w₁ ∈ ℂ | 0 < Im(w₁) < π/3 }.

2. Map the strip to the upper half-plane

The exponential stretching map:

w₁ → w₂ = exp(πi·w₁ / 3)

transforms the strip S to the upper half-plane
H = { w₂ ∈ ℂ | Im(w₂) > 0 }.

3. Map the upper half-plane to the unit disk

The Möbius transformation:

w₂ → w₃ = (w₂ – i) / (w₂ + i)

is a conformal equivalence between the upper half-plane H and the unit disk 𝔻.

4. Combine the maps

The full conformal map f: A → 𝔻 is the composition:

f(z) = [(exp(πi·log(z) / 3) – i) / (exp(πi·log(z) / 3) + i)].

5. Simplified expression

Expanding the steps, the final form of f(z) is:

f(z) = [(exp(πi(ln|z| + i·arg(z)) / 3) – i) / (exp(πi(ln|z| + i·arg(z)) / 3) + i)].

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The post Construct a conformal equivalence f between the “angle” {z ∈ C | z 6= 0, 0 < arg(z) < π/3} and the unit disk D ⊂ C appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Nowhere Differentiable Functions and Integration of such functions

anuj — Wed, 09 Oct 2024 02:36:20 +0000

Nowhere Differentiable Functions

Nowhere differentiable functions are functions that are continuous everywhere but do not have a well-defined derivative at any point. They exhibit erratic behavior, and although they can be integrated (as integration measures the area under the curve), they defy differentiation in the conventional sense.

1. Weierstrass Function

The Weierstrass function is one of the first discovered examples of a continuous, nowhere differentiable function. It is defined as:

    W(x) = ∑_n=0^∞ aⁿ cos(bⁿ π x)

where:

0 < a < 1
b is an odd integer such that ab > 1 + 3/2 π

This function is continuous but has no well-defined derivative at any point due to the rapid oscillations caused by the series.

Weirstrass Function

Integration: Yes, the Weierstrass function is integrable since it’s continuous and bounded. The integral of the function over an interval exists and is well-defined, but the result might not be simple to compute due to its complex structure.

2. Cantor Function (Devil’s Staircase)

The Cantor function is another famous example of a function that is continuous but nondifferentiable almost everywhere. It’s defined on the unit interval [0, 1] using the Cantor set and is constructed by removing the middle thirds repeatedly from each remaining segment.

Integration: The Cantor function is not differentiable almost everywhere, but it is integrable. In fact, the integral of the Cantor function over the interval [0, 1] is equal to 0.5.

3. Brownian Motion (Wiener Process)

A Brownian motion path, denoted B(t), is a random process that is continuous almost surely but is nowhere differentiable with probability 1. This is commonly used in fields such as physics and finance for modeling stochastic processes.

Integration: Brownian motion is integrable in a stochastic sense (stochastic integrals), and techniques such as Itô calculus are used to handle such integrals. However, this is a special kind of integration designed to handle the irregularities of stochastic processes.

Can They Be Integrated?

Yes, most nowhere differentiable functions can be integrated, especially in the Riemann or Lebesgue sense, because integration is concerned with measuring the “area under the curve,” while differentiation is focused on the local rate of change, which is what these functions lack.

For example:

The Weierstrass function is integrable over any interval due to its continuity.
The Cantor function is also integrable, though its derivative is 0 almost everywhere.
For Brownian motion, special methods (stochastic integrals) allow for meaningful integration.

In general, continuity guarantees integrability, but differentiability is not necessary for a function to be integrable.

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Analytic Within and On the Unit Circle

anuj — Tue, 08 Oct 2024 00:30:32 +0000

Analytic Within and On the Unit Circle

When discussing whether a function is analytic within or on the unit circle, we are referring to complex analysis, which deals with functions of a complex variable.

Analytic Within the Unit Circle

A function f(z) is said to be analytic within the unit circle if it is analytic (i.e., differentiable) at every point inside the unit circle. The unit circle is defined as the set of points z ∈ ℂ such that |z| = 1, and the region inside the unit circle is where |z| < 1.

Analytic Function: A function is analytic at a point if it is differentiable not only at that point but also in a neighborhood around that point. More formally, a function f(z) is analytic if it has a Taylor series expansion that converges to f(z) in some neighborhood around the point of interest.
Within the Unit Circle: The phrase “within the unit circle” means that the function is analytic at all points where |z| < 1. This includes all points strictly inside the circle but excludes the points on the boundary |z| = 1.

For example, the function f(z) = 1/(1 – z) is analytic within the unit circle because it can be expanded as a convergent power series:

f(z) = ∑_n=0∞ zⁿ   for   |z| < 1

This series converges for all z inside the unit circle, but the function becomes singular at z = 1 (on the unit circle), where it has a pole.

Analytic On the Unit Circle

A function f(z) is said to be analytic on the unit circle if it is analytic at every point on the unit circle, meaning |z| = 1. This is a stronger condition because the function must now be analytic at all points on the boundary of the disk defined by |z| = 1, not just within it.

To be analytic on the unit circle, the function must:

Curves Over Finite Fields

anuj — Tue, 08 Oct 2024 00:29:31 +0000

Curves Over Finite Fields

Curves over finite fields are algebraic curves defined by equations where the coefficients come from a finite field. A finite field is a field with a finite number of elements, often denoted as 𝔼_q, where q is a prime power (i.e., q = pⁿ for some prime p and positive integer n).

Basic Concept

An algebraic curve is given by a polynomial equation in two variables, say x and y, of the form f(x, y) = 0. When the coefficients of this polynomial come from a finite field, we call the curve a curve over a finite field.

Example 1: Curve Over 𝔼₅

Consider the field 𝔼₅ = {0, 1, 2, 3, 4}, which is the field of integers modulo 5. A simple curve over 𝔼₅ is given by the equation:

y² = x³ + 2x + 1 (mod 5)

This is an example of an elliptic curve over the finite field 𝔼₅.

To find the points on the curve, we substitute values of x from 𝔼₅ into the equation and check if the resulting value of y² has a solution in 𝔼₅.

For x = 0, y² = 1, so y = 1 or y = 4.
For x = 1, y² = 4, so y = 2 or y = 3.
For x = 2, y² = 4, so y = 2 or y = 3.
For x = 3, y² = 0, so y = 0.
For x = 4, y² = 4, so y = 2 or y = 3.

Thus, the points on the curve are:

(0, 1), (0, 4), (1, 2), (1, 3), (2, 2), (2, 3), (3, 0), (4, 2), (4, 3)

Example 2: Line Over 𝔼₃

Consider the field 𝔼₃ = {0, 1, 2}, and the line defined by the equation:

y = 2x + 1 (mod 3)

To find the points on this line, substitute values of x from 𝔼₃ into the equation:

For x = 0, y = 1.
For x = 1, y = 0.
For x = 2, y = 2 · 2 + 1 = 5 ≡ 2 (mod 3).

Thus, the points on this line are:

(0, 1), (1, 0), (2, 2)

Key Concepts

Finite Fields: A finite field 𝔼_q contains q elements. For example, 𝔼₂ = {0, 1}, 𝔼₃ = {0, 1, 2}, etc.
Solutions Over Finite Fields: A curve over a finite field has a finite number of solutions (points), as both the x and y values are restricted to elements of the finite field.

Applications

Curves over finite fields have deep applications in number theory, coding theory, and cryptography. For instance:

Elliptic curves over finite fields are used in Elliptic Curve Cryptography (ECC), where the security of the cryptosystem relies on the difficulty of solving the elliptic curve discrete logarithm problem.
Error-correcting codes like Reed-Solomon codes use curves over finite fields for constructing robust codes to detect and correct errors in data transmission.

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Examples of Non-Measurable Sets in Lebesgue Measure Theory

anuj — Tue, 01 Oct 2024 19:46:00 +0000

Examples of Non-Measurable Sets in Lebesgue Measure Theory

1. Vitali Set

One of the most famous examples of a non-measurable set is the Vitali set. The construction begins by considering the unit interval [0, 1].
We define an equivalence relation on ℝ such that two real numbers x and y are equivalent if x - y
is a rational number (i.e., x - y ∈ ℚ). Using the axiom of choice, we can select exactly one representative from each equivalence class.

The resulting set is called a Vitali set, and it can be shown that this set is non-measurable. This is because the translation invariance of Lebesgue measure
would lead to a contradiction if such a set were measurable.

2. Axiom of Choice-based Subsets of the Real Line

More generally, any set whose existence relies on the axiom of choice and does not have a constructive definition (such as the Vitali set) is likely to be non-measurable.
These sets are often subsets of ℝ or other uncountably infinite sets where elements are chosen based on arbitrary selection rather than a definable rule.

3. Bernstein Set

A Bernstein set is a subset of ℝ such that both the set and its complement intersect every uncountable closed subset of ℝ,
but neither contains any uncountable closed subset entirely. The Bernstein set can be shown to be non-measurable.

Its non-measurability stems from the fact that it cannot satisfy the conditions for Lebesgue measure. Intuitively, there’s no way to assign a consistent measure to such a set
given the intricate way it interacts with closed sets.

4. Hamel Basis of ℝ over ℚ

A Hamel basis is a basis for the real numbers ℝ considered as a vector space over the rationals ℚ.
The elements of a Hamel basis are such that every real number can be expressed uniquely as a finite linear combination of these elements with rational coefficients.

A Hamel basis is non-measurable because it can be used to translate a measurable set in ways that violate the translation invariance of the Lebesgue measure.
Essentially, shifting the basis elements by rational numbers can lead to contradictions in measure theory.

5. Example Based on the Banach-Tarski Paradox

While the Banach-Tarski paradox primarily involves 3-dimensional spaces, its construction hints at the existence of non-measurable sets.
The paradox shows that a solid ball in 3D space can be decomposed into a finite number of non-measurable pieces, which can then be reassembled into two balls identical to the original.

The sets used in this decomposition are non-measurable under Lebesgue measure.

In summary, non-measurable sets often arise from constructions that require the axiom of choice and exhibit strange properties, particularly with respect to translation and countable additivity. The Vitali set and Hamel basis are classic examples of such sets.

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Relationship Between Measure Theory, Lebesgue Integrals, and Hilbert Spaces

anuj — Tue, 01 Oct 2024 00:14:14 +0000

Understanding the Relationship Between Measure Theory, Lebesgue Integrals, and Hilbert Spaces

To understand the relationship between measure theory, Lebesgue integrals, and Hilbert spaces, we need to break down each concept and then see how they connect.

1. Measure Theory

Measure theory is the mathematical foundation for defining and understanding the “size” or “volume” of sets, especially in contexts that go beyond simple geometric areas or volumes.

A measure is a function that assigns a non-negative real number (or infinity) to subsets of a space, allowing us to generalize notions like length, area, and probability.
For example, the Lebesgue measure in the real number line is a way of defining the length of intervals.
Measure theory provides the tools to work with infinite or uncountable sets in a rigorous way, making it essential for dealing with functions that aren’t necessarily continuous everywhere (such as discontinuous or complex functions).

2. Lebesgue Integrals

The Lebesgue integral extends the classical Riemann integral by integrating functions with respect to a measure, rather than just over intervals. It is designed to handle a broader class of functions, especially those that might be irregular or have discontinuities.

In contrast to the Riemann integral, which sums “heights” of functions over intervals, the Lebesgue integral sums the “measures” of function values over sets where the function takes on specific values.
The Lebesgue integral is powerful because it allows for integration of functions that might not be Riemann integrable (for instance, highly oscillatory functions or those with many discontinuities), as long as they are “measurable.”

Relationship between Measure and Lebesgue Integrals:

A function is Lebesgue integrable if the area under its curve can be described using measure theory. This means we are essentially summing “measured” sets of points where the function reaches certain values.
Lebesgue integration helps avoid some of the limitations of Riemann integration, such as dealing with functions that have complex discontinuities.

3. Hilbert Spaces

A Hilbert space is a generalization of Euclidean space, but it can be infinite-dimensional. It is a complete inner product space where distances and angles can be measured, making it a natural setting for various kinds of mathematical analysis, particularly in functional analysis and quantum mechanics.

The notion of distance in Hilbert spaces is defined using the inner product. This generalizes the concept of the dot product in Euclidean space.
L² spaces (spaces of square-integrable functions) are a crucial example of Hilbert spaces. In this context, the functions are integrated using the Lebesgue integral, and the “distance” between two functions is the square root of the integral of the square of their difference:
```
‖f - g‖ = √(∫ |f(x) - g(x)|² dμ(x))
```
where μ is a measure, often the Lebesgue measure in real analysis.

Connection to Measure Theory and Lebesgue Integrals:

Hilbert spaces, particularly L² spaces, rely heavily on Lebesgue integrals to define inner products and norms. The inner product of two functions f and g in an L² space is defined as:
```
⟨f, g⟩ = ∫ f(x) g(x) dμ(x)
```
This requires integrating using the Lebesgue measure.
The completeness of a Hilbert space means that if a sequence of functions converges in the L² norm, then it converges to a function that is also in the space. This is crucial in many applications of analysis, including solving differential equations and in quantum mechanics.

Summary of Relationships

Measure theory provides the foundational concept of size and “measurability” necessary to define integration more generally.
Lebesgue integrals extend the classical integral to work with more complex functions by using measure theory. This is essential for defining concepts like convergence in function spaces.
Hilbert spaces, particularly L² spaces, use Lebesgue integrals to define inner products and norms, making them a natural setting for analyzing functions. The structure of Hilbert spaces allows for the rigorous study of function spaces and is critical in many areas of mathematical analysis, physics, and engineering.

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Lebesgue Integral Solved Problems

anuj — Mon, 30 Sep 2024 18:48:48 +0000

Lebesgue Integral Solved Problems

Problem 1: Simple Example of a Lebesgue Integral

Problem: Compute the Lebesgue integral of the function
f(x) = 2 over the interval [0, 3] using the Lebesgue measure.

Solution:

First, recall that in the Lebesgue integral, we’re summing over the function values multiplied by the measure of the set where the function takes those values.
Since f(x) = 2 is constant for all x ∈ [0, 3], we can simplify the calculation.
We need to compute:
```
∫₀³ 2 dμ(x)
```
where μ(x) is the Lebesgue measure, and it assigns the “length” of the interval to the set.
For a constant function, this is simply the value of the function multiplied by the length of the interval:
```
∫₀³ 2 dx = 2 × (3 - 0) = 6
```

Conclusion: The Lebesgue integral of f(x) = 2 over [0, 3] is 6.

Problem 2: Lebesgue Integral of a Highly Oscillatory Function

Problem: Compute the Lebesgue integral of the oscillatory function
f(x) = sin(1/x) over the interval (0, 1].

Solution:

The function f(x) = sin(1/x) oscillates wildly as x → 0. To manage this, we’ll need to carefully apply the Lebesgue integral by breaking the function into smaller pieces.
To compute the Lebesgue integral, we will use the absolute integrability of the function and analyze whether this integral converges.
We need to compute:
```
∫₀¹ sin(1/x) dx
```
The oscillations in sin(1/x) are extreme as x → 0, but due to the bounded nature of the sine function (i.e., |sin(⋅)| ≤ 1), we can assess whether the integral of this oscillatory function converges or not.

Strategy:

Although the function oscillates as x → 0, the crucial idea is whether the oscillations “cancel out” enough to give a convergent integral. In fact, due to the high-frequency oscillations near 0, the integral doesn’t have a limit in the classical sense.

But we can study the absolute value of the function to determine whether it’s integrable in the Lebesgue sense.

Step-by-Step:

We examine the behavior of the integral of the absolute value:
```
∫₀¹ |sin(1/x)| dx
```
Since |sin(⋅)| ≤ 1, we know that:
```
∫₀¹ |sin(1/x)| dx ≤ ∫₀¹ 1 dx = 1
```
This shows that the absolute value of the oscillatory function is bounded and integrable over (0, 1].

Conclusion:

While we cannot easily compute the exact value of this integral (as it doesn’t have a simple expression), the key insight is that Lebesgue integration allows us to conclude that the oscillatory function f(x) = sin(1/x) is Lebesgue integrable over (0, 1], thanks to the boundedness of the sine function and the convergence of the integral.

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Galois group of a polynomial

anuj — Sat, 13 Jul 2024 04:25:41 +0000

The Galois group of a polynomial is a concept in the field of algebra, specifically within Galois theory, which studies the relationship between field extensions and group theory. The Galois group of a polynomial provides deep insights into the solvability and structure of the roots of the polynomial.

Key Concepts

1. Polynomial and Its Roots

A polynomial $f (x)$ with coefficients in a field $K$ (typically $Q\mathbb{Q}$ , $R\mathbb{R}$ , or $C\mathbb{C}$ ) can be written as $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0$ .
The roots of $f (x)$ are the solutions to the equation $f (x) = 0$ .

2. Splitting Field

The splitting field $L$ of $f (x)$ over $K$ is the smallest field extension of $K$ that contains all the roots of $f (x)$ .
If $f (x)$ has degree $n$ , the splitting field $L$ will be an extension of $K$ with a degree that divides $n!$ .

3. Automorphisms

An automorphism of a field $L$ is a bijective map from $L$ to itself that preserves addition and multiplication.
The automorphisms of $L$ that fix $K$ form a group under composition.

4. Galois Group

The Galois group $Gal(L/K)\text{Gal}(L/K)$ of the polynomial $f (x)$ is the group of all automorphisms of $L$ that fix every element of $K$ .

Steps to Determine the Galois Group

Find the Roots: Determine the roots of the polynomial $f (x)$ . This can be done exactly for polynomials of degree 2, 3, and 4, but for higher degrees, it might require numerical or symbolic methods.
Construct the Splitting Field: Identify the splitting field $L$ of the polynomial over the base field $K$ .
Determine Automorphisms: Identify all automorphisms of the splitting field that fix the base field.
Form the Group: The set of these automorphisms forms the Galois group, denoted $Gal(L/K)\text{Gal}(L/K)$ .

Examples

1. Quadratic Polynomial

Consider $f(x) = x^2 – 2$ over $Q\mathbb{Q}$ :

The roots are $±2\pm \sqrt{2}$ .
The splitting field is $Q(2)\mathbb{Q}(\sqrt{2})$ .
The Galois group consists of two elements: the identity automorphism and the automorphism sending $2\sqrt{2}$ to $−2-\sqrt{2}$ .
Thus, $Gal(Q(2)/Q)≅Z/2Z\text{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}$ .

2. Cubic Polynomial

Consider $f(x) = x^3 – 2$ over $Q\mathbb{Q}$ :

The roots are $23,23ω,23ω2\sqrt[3]{2}, \sqrt[3]{2}\omega, \sqrt[3]{2}\omega^2$ , where $ω\omega$ is a primitive cube root of unity.
The splitting field is $Q(23,ω)\mathbb{Q}(\sqrt[3]{2}, \omega)$ .
The Galois group has order 6 and is isomorphic to $S_3$ , the symmetric group on 3 elements.

Importance of the Galois Group

Solvability by Radicals: A polynomial is solvable by radicals if and only if its Galois group is a solvable group.
Field Extensions: The structure of the Galois group provides information about the intermediate fields between $K$ and $L$ .
Symmetry: The Galois group reflects the symmetries of the roots of the polynomial and helps in understanding the nature of the solutions.

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The Galois Group and Representation Theory

anuj — Sat, 13 Jul 2024 03:46:37 +0000

Galois Group

Overview

The Galois group is a concept from the field of algebra, specifically in the study of field theory and polynomial equations. It is named after the French mathematician Évariste Galois.

Key Concepts

Field Extensions: A field extension $L / K$ is a pair of fields $K$ and $L$ such that $K$ is a subfield of $L$ .
Automorphisms: An automorphism of a field $L$ is a bijective map from $L$ to itself that respects the field operations (addition and multiplication).
Galois Group: Given a field extension $L / K$ , the Galois group $Gal(L/K)\text{Gal}(L/K)$ is the group of all automorphisms of $L$ that fix every element of $K$ .

Example

Consider the field extension $Q(2)/Q\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ :

The automorphisms of $Q(2)\mathbb{Q}(\sqrt{2})$ that fix $Q\mathbb{Q}$ are the identity map and the map sending $2\sqrt{2}$ to $−2-\sqrt{2}$ .
Therefore, the Galois group $Gal(Q(2)/Q)\text{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q})$ has two elements, often denoted by ${id,σ}\{ \text{id}, \sigma \}$ , where $σ(2)=−2\sigma(\sqrt{2}) = -\sqrt{2}$ .

Representation Theory

Overview

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.

Key Concepts

Group Representations: A representation of a group $G$ is a homomorphism from $G$ to the general linear group $GL(V)\text{GL}(V)$ of a vector space $V$ . Essentially, it maps elements of $G$ to invertible matrices in such a way that group operations are preserved.
Modules: A module over a ring is a generalization of the concept of a vector space over a field. Representation theory can also be viewed as the study of modules over group algebras.
Characters: The character of a representation is a function that associates to each group element the trace of its matrix in the representation. Characters provide a powerful tool for studying and classifying representations.

Example

Consider the symmetric group $S_3$ , which is the group of all permutations of three elements:

A simple representation of $S_3$ is the permutation representation on $R3\mathbb{R}^3$ , where each permutation $σ∈S3\sigma \in S_3$ is represented by the matrix that permutes the standard basis vectors of $R3\mathbb{R}^3$ according to $σ\sigma$ .

Importance

Representation theory has applications in many areas of mathematics and science, including number theory, geometry, and physics.
It provides a way to study groups by understanding their action on vector spaces, making complex group-theoretic problems more manageable by translating them into linear algebraic terms.

By exploring the Galois groups and representation theory, mathematicians can gain deeper insights into the structure and symmetries of algebraic objects and their interrelationships.

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Pure Math Archives - Time Travel, Quantum Entanglement and Quantum Computing

Godel’s Consistency of Axiom of Choice Paper

Gödel’s Landmark Paper

The Consistency of the Axiom of Choice and the Generalized Continuum-Hypothesis with the Axioms of Set Theory

Background

Gödel’s Results

Methodology

Impact and Subsequent Developments

Conclusion

Construct a conformal equivalence f between the “angle” {z ∈ C | z 6= 0, 0 < arg(z) < π/3} and the unit disk D ⊂ C

Constructing a Conformal Equivalence

1. Map the angle to a horizontal strip

2. Map the strip to the upper half-plane

3. Map the upper half-plane to the unit disk

4. Combine the maps

5. Simplified expression

Nowhere Differentiable Functions and Integration of such functions

Nowhere Differentiable Functions

1. Weierstrass Function

2. Cantor Function (Devil’s Staircase)

3. Brownian Motion (Wiener Process)

Can They Be Integrated?

Analytic Within and On the Unit Circle

Analytic Within and On the Unit Circle

Analytic Within the Unit Circle

Analytic On the Unit Circle

Curves Over Finite Fields

Curves Over Finite Fields

Basic Concept

Example 1: Curve Over 𝔼5

Example 2: Line Over 𝔼3

Key Concepts

Applications

Examples of Non-Measurable Sets in Lebesgue Measure Theory

Examples of Non-Measurable Sets in Lebesgue Measure Theory

1. Vitali Set

2. Axiom of Choice-based Subsets of the Real Line

3. Bernstein Set

4. Hamel Basis of ℝ over ℚ

5. Example Based on the Banach-Tarski Paradox

Relationship Between Measure Theory, Lebesgue Integrals, and Hilbert Spaces

Understanding the Relationship Between Measure Theory, Lebesgue Integrals, and Hilbert Spaces

1. Measure Theory

2. Lebesgue Integrals

Relationship between Measure and Lebesgue Integrals:

3. Hilbert Spaces

Connection to Measure Theory and Lebesgue Integrals:

Summary of Relationships

Lebesgue Integral Solved Problems

Lebesgue Integral Solved Problems

Problem 1: Simple Example of a Lebesgue Integral

Solution:

Problem 2: Lebesgue Integral of a Highly Oscillatory Function

Solution:

Strategy:

Step-by-Step:

Conclusion:

Galois group of a polynomial

Key Concepts

1. Polynomial and Its Roots

2. Splitting Field

3. Automorphisms

4. Galois Group

Steps to Determine the Galois Group

Examples

1. Quadratic Polynomial

2. Cubic Polynomial

Importance of the Galois Group

The Galois Group and Representation Theory

Galois Group

Overview

Key Concepts

Example

Representation Theory

Overview

Key Concepts

Example

Importance

Example 1: Curve Over 𝔼₅

Example 2: Line Over 𝔼₃