-
A Problem on Outer Measure
There is a problem in Real Analysis by Halsey L. Royden, Patrick M. Fitzpatrick. Suppose that outer-measure is defined by covering sets by countable collections of closed, bounded intervals rather than coverings by open, bounded intervals. Show that the outer- measure remains unchanged. I give my answer here. Proof. Let be a set of real […]
-
1. Argument by Contradiction
Problem 1: Determine all functions satisfying for all positive integers and . idea and solution: We could attempt some functions. We find that constant function works, does not work. does not work. This prompt us, maybe the only possible solution is ? This is because the left and right hand side has different […]
-
How to understand Weierstrass infinite product?
One of the famous expression of is Here, we are writing as a some product of polynomials with zeros apparently shown. Remember that the right hand side has to be carefully chose to ensure its convergence. We are remained to consider: Can be factorize every entire function in the form of product of […]
-
Some thoughts on proving Fourier Inversion(Complex analysis method)
In complex analysis, we have the following well-known theorem: Theorem If , then the Fourier inversion holds, namely, given We have The proof is not easy for a beginner who just touched this theorem. As for myself, I spent a tons of time on managing the proof. So I would like […]
-
Harmonic Number(2)
Our goal is to prove the following theorem by Carlo Sanna. The proof is adapted and changed slightly from his paper. Theorem. For any prime number and any , we have Lemma 1 If is prime, , and , then and Proof. . Each of the sum is by […]
-
Harmonic Number (Starting from an AMC problem) (1)
The -th Harmonic number, denoted is defined by the sum of the reciprocals of the first natural numbers: This number has quite interesting properties. On thing we could consider when is finite is the quotient of the sum. If we write it as Then many problems raise. Here, we have a problem […]
-
An inequality problem
Let be a continuous integrable real function on , then Solution 1 By Cauchy-Schwarz inequality, Solution 2 By Jensen’s inequality, since is a convex function, we have So Solution 3 By Hölder’s Inequality, we get Choose , we get the case of solution 1. We […]
-
Abstract Algebra – Sylow Theorems and Its applications
In abstract algebra, finite group theory is in its core. Now from Lagrange theorem, we have Lemma 1 Let be a subgroup of a finite group . Then the order of is a divisor of the order of . This theorem states the necessary condition for a finite group to be a subgroup of a […]
-
An approximation problem
Problem: Let us call an irrational number well approximated by rational numbers if for any natural numbers there exists a rational number such that Solution: 1. Pick Why does it work ? Let’s say we let . Then we have and We now proceed in proving that […]
-
On IMO2017 P2
I did IMO 2017 P2 with my friends yesterday. Now I would like to share my thoughts on this problem. Problem: Let be the set of real numbers. Determine all functions such that, for any real numbers and , The initial step must be taking certain values, testing its properties and then start […]
My name is Ziv. I am a high school Chinese student who love exploring mathematics! Currently, my favorite fields include complex analysis and abstract algebra!
Last post
Posted on
by
A Problem on Outer Measure
There is a problem in Real Analysis by Halsey L. Royden, Patrick M. Fitzpatrick. Suppose that outer-measure is defined by covering sets by countable collections of closed, bounded intervals rather than coverings by open, bounded intervals. Show that the outer- measure remains unchanged. I give my answer here. Proof. Let be a set of real…