Archives for Pure Math
Nowhere Differentiable Functions and Integration of such functions
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…
Analytic Within and On the Unit Circle
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…
Curves Over Finite Fields
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…
Examples of Non-Measurable Sets in Lebesgue Measure Theory
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…
Relationship Between Measure Theory, Lebesgue Integrals, and Hilbert Spaces
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…
Lebesgue Integral Solved Problems
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 using the Lebesgue measure.…
Galois group of a polynomial
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…
The Galois Group and Representation Theory
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…
Suppose cp is a C1 function on R such that cp(x)+a and cp’(x)~b as xjoo. Prove or give a counterexample: b must be zero.
Suppose cp is a C1 function on R such that cp(x)+a and cp’(x)~b as xjoo. Prove or give a counterexample: b must be zero. To address the question, we need…
what is the langlands program?
The Langlands program is a set of far-reaching and deep conjectures proposed by Robert Langlands in 1967, which aims to relate and unify various areas of mathematics, including number theory,…