zivwang – Ziv's Math World

June 2025
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Measure Theory(1)

Chapter 1: Motivation for Measure Theory As the first chapter of this series, we begin by presenting the motivation behind measure theory. Motivation Measure theory was developed to address the limitations of Riemann integration. Let’s first recall Riemann integration, which approximates the integral through upper and lower bounds, taking the limit as the partition becomes […]
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Hensel’s Lemma

We start from a problem: Problem: Find the number of elements in such that it is congruent to module for some integer . Solution. The problem is the same as finding the number of solutions to Using Chinese Remainder Theorem, the problem reduces to the system of equations Our […]
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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 […]
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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 […]
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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 […]
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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 […]
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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 […]
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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 […]
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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 […]
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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 […]
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