This is for an Insights article: Bivariate induction proof using Calc3

In summary, the article is right where I need you to start checking, read the above boxes to, check out the picture to see examples of the kind of sequence of sets we are dealing with. I need you to read the section jusr below the first picture entitled "3.0.2 Lemma 2.1: Nesting Property of ##S_N^n##".Do you need proper containment (##\subset##) in your proposition or would you be happy with just containment (##\subseteq##)?
  • #1
benorin
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https://www.physicsforums.com/insights/a-novel-technique-of-calculating-unit-hypercube-integrals/#Images-of-the-2-d-versions-of-the-some-of-the-sequence-of-sets-S-N2-and-the-geometry-of-the-next-headings-method-of-finding-extrema
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The method of bivariate induction is laid out in the article, I just want someone to check my work, I used Lagrange Multipliers to find extrema of the sequence of sets to prove they are nested but not sure if that's enough, or do I need convexity too?
Link to my insight Article it's right where I need you to start checking, read the above boxes to, check out the picture to see examples of the kind of sequence of sets we are dealing with. I need you to read the section jusr below the first picture entitled "3.0.2 Lemma 2.1: Nesting Property of ##S_N^n##".
 
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  • #2
Do you need proper containment (##\subset##) in your proposition or would you be happy with just containment (##\subseteq##)?

If the latter, I think you can make the proof simpler by using P(n=1, N=1) as your base case rather than P(n=2, N=1). Then our base proposition is in a one-dimensional space:
$$P(1,1) \equiv S_2^1\subseteq S_1^1
\equiv \left(\left(\frac xb\right)^4 \leq 1 \Rightarrow \left(\frac xb\right)^2 \leq 1 \right)$$
which is easy to prove. In fact we can easily show that
$$\forall N\in \mathbb Z^+\ \ S_N^1=[-b,b]$$
from which it follows that P(1,N) is true for all positive integers N.

So then we just need to do induction over ##n##.
 
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  • #3
For a long time, mathematicians wrote ##A \subset B## to mean "A is a subset of B". It's easier to write than the other one. "Proper subset" arises so rarely that it can be written out in words when needed. As such, it has never needed its own symbol.

With the advent of computer typesetting the more complicated symbol ## \subseteq ## became popular. I think it has overtaken ## \subset ## in popularity. Let's say that the probability that a writer uses ## \subset ## to represent the subset relation is p. Let's say that the probability that a math formula involving some kind of subsets actually involves proper subsets is q. (Choose p and q according to your personal beliefs, or do your own research, as they say. Personally, I choose p=0.1 and q = 0.001.) A simple application of Bayes rule will tell you what ## \subset ## usually means.
 
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  • #4
In case you’re curious this is what the n=2 (ie 2-dimensional family of sets looks like
2E94685A-EF37-4D67-BA0A-9F469D454294.jpeg
 

1. What is bivariate induction proof?

Bivariate induction proof is a mathematical method used to prove statements that involve two variables. It is an extension of the more commonly known mathematical induction, which is used to prove statements involving a single variable.

2. How is Calc3 used in bivariate induction proof?

Calc3, also known as multivariable calculus, is used in bivariate induction proof to analyze and manipulate functions with two variables. It allows for the use of multiple derivatives, integrals, and other calculus techniques to prove statements involving two variables.

3. What are the steps involved in a bivariate induction proof?

The steps involved in a bivariate induction proof are similar to those of a regular mathematical induction. First, the statement is proven to be true for the base case. Then, the statement is assumed to be true for an arbitrary pair of variables. Finally, using this assumption and the principles of Calc3, the statement is proven to be true for the next pair of variables, thus completing the proof.

4. What are some common applications of bivariate induction proof?

Bivariate induction proof is commonly used in various fields of mathematics, such as number theory, combinatorics, and graph theory. It is also used in computer science, specifically in the analysis of algorithms and data structures. Bivariate induction proof can also be applied in physics and engineering to prove relationships between multiple variables.

5. Are there any limitations to bivariate induction proof?

Like any mathematical proof, bivariate induction proof is subject to certain limitations. It may not be applicable to all statements involving two variables, and it may not be the most efficient or elegant method of proof in some cases. Additionally, bivariate induction proof relies on the assumption that the statement is true for an arbitrary pair of variables, which may not always be easy to prove. It is important to carefully consider the statement and alternative methods of proof before using bivariate induction.

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