Papers which solve the summary of my paper involving expectations, hausdorff measure, prevelant/shy sets, partitions, samples, pathways & entropy.
Question: Is there a published research paper which already solves the problems in the summary of my paper?
Since the writing in the paper is difficult, here is the summary:
Let $n\in\mathbb{N}$ and suppose $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ is a function, where $A$ and $f$ are Borel. We want a unique, satisfying average of highly discontinuous $f$, taking finite values only. For instance, consider an everywhere surjective $f$, where its graph has zero Hausdorff measure in its dimension and a nowhere continuous $f$ defined on the rationals. The problem is that the expected value of these examples of $f$, w.r.t. the Hausdorff measure in its dimension, is undefined. Thus, take any chosen family of bounded functions converging to $f$ with the same satisfying and finite expected value, where the term "satisfying" is explained in the third paragraph.
The importance of this solution is that it solves the following problem: the set of all $f\in\mathbb{R}^{A}$ with a finite expected value, forms a shy "measure zero" subset of $\mathbb{R}^{A}$. This issue is solved since the set of all $f\in\mathbb{R}^{A}$, where there exists a family of bounded functions converging to $f$ with a finite expected value, forms a prevalent "full measure" subset of $\mathbb{R}^{A}$. Despite this, the set of all $f\in\mathbb{R}^{A}$—where two or more families of bounded functions converging to $f$ have different expected values—forms a prevalent subset of $\mathbb{R}^{A}$. Hence, we need a choice function which chooses a subset of all families of bounded functions converging to $f$ with the same satisfying and finite expected value.
Notice, "satisfying" is explained in a leading question which uses rigorous versions of phrases in the former paragraph and the "measure" of the chosen families of each bounded function's graph involving partitioning each graph into equal measure sets and taking the following—a sample point from each partition, pathways of line segments between sample points, lengths of line segments in each pathway, removed lengths which are outliers, remaining lengths which are converted into a probability distribution, and the entropy of the distribution. In addition, we define a fixed rate of expansion versus the actual rate of expansion of a family of each bounded function's graph.
(Keywords: Discontinuity, Hausdorff measure, Expected Value, Function Space, Prevalent and Shy Sets, Partitions, Samples, Euclidean Distance, Entropy, Choice Function)
I used Reserachgate to find similar paper based on the keywords, but since I am an undergraduate I do not fully understand the papers. (I cannot give an accurate summary.)
Here are the most similar papers I could find.
- Ergodic Optimization For Open Expanding Multi-Valued Topological Dynamical Systems
- Typical Uniqueness in Ergodic Optimization
- Prediction of dynamical systems from time-delayed measurements with self-intersections
- A Hausdorff-measure boundary element method for acoustic scattering by fractal screens
Question 2: Do any of these papers solve the problems in the summary?
1 answer
I am not an expert on what you are doing, so what I can offer is some ways of maybe finding relevant literature.
- Snowball search means look at your references, and looking at what they refer to, and, in turn, who refers to them (you can find this in Google scholar and other places), and then iterating this. Maybe also do this to the similar articles. You'll learn fairly quickly which articles are dissimilar and which you should use further in your snowball.
- You should use your university's library service, Semantic scholar, Google scholar etc. and try searching for relevant literature. Read enough of what you find to also find better search terms or new keywords, once you see what people in your field actually use.
- You might try an artificial intelligence program like Keenious, or just one of the big language models, and ask them to provide references and links to articles similar to yours, or that might respond to the same questions.
- Ask a faculty member who is interested in this stuff at your university; preferably someone you have a good relation to.
Note that the mathematical community has its share of amateurs who are eager but do not know what they are doing. If they are insistent, they are called "cranks". You should be careful in how you present yourself to not be taken as a crank.
For example, articles in mathematics often have a fairly laconic and short abstract. I start reading yours and I start immediately wondering whether I should know what a "satisfying average" is. Is it a precise term or strange language? Then in the third paragraph, if I even come so far, I find out that "satisfying" is explained in a "leading question", which makes me wonder what you mean by a "leading question"; probably not a question that is formed to get a particular answer, which would be my understanding.
Please do not respond to explain me what you mean by the terms, but rather take this as feedback on how a particular mathematician reacts to your text. You should hope that I instead got a firm understanding of what you are out to do.
Maybe you present a notion of averaging that applies even to highly irregular functions? Then write that. And then explain how that is a good notion of averaging. Presumably it generalizes other notions of averaging and has some uniqueness properties or some type of monotonicity or some other property one associates with an average, and presumably you prove these as theorems in your article. Maybe you have an application in mind. (Maybe the second paragraph? I am left uncertain.)
In any case, the ideas like above are what you should write out, as clearly and explicitly as possible, in your abstract. Or maybe you do, but I lack the background to read it.
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