Archives for Pure Math - Page 2
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,…
Let R > 1 and let f be analytic on IzI < R except at z = 1, where f has a simple pole. If M n=O is the Maclaurin series for f, show that hn,+,M alL exists.
To show that the limit limn→∞nan\lim_{n \to \infty} n a_nlimn→∞nan exists for the Maclaurin series f(z)=∑n=0∞anznf(z) = \sum_{n=0}^{\infty} a_n z^nf(z)=∑n=0∞anzn of the function fff, which is analytic in ∣z∣<R|z| <…
An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.
To prove that there is exactly one point on the map of California that lies directly over the point it represents, we can use a combination of the Brouwer Fixed-Point…
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,) 4 0, and x, f(-xc,) 4 0 as n -+ o.
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,)…