Abstract This brief note explores the use of fuzzy classifiers, with membership functions chosen using a statistical heuristic (quantile statistics), to monitor time-series metrics. The time series can arise from environmental measurements, industrial process control data, or sensor system outputs. We demonstrate implementation using the R language on an example dataset (ozone levels in New York City). Click here to skip straight to the coded solution), or read on for the discussion.
 Fuzzy classification into 5 classes using p10 and p90 levels to achieve an 80-20 rule in the outermost classes and graded class membership in the inner three classes. Comparison with crisp classifier using the same 80-20 rule is shown in the bottom panel of the figure. Continue reading this article…
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This article looks at Propositional Logic, also called Statement Calculus, from a combinatorial and algebraic point of view (Sections 3-6), its implementation in software (Section 7), and its application to digital electronics (Section 10). Historical sections cover the shift in viewpoint from classical logic based on Aristotle’s syllogism to modern symbolic logic (Section 2) and the axiomatization of logic (Section 9). (See logic sourcebook for the original source papers (1830-1881) that drove this shift.)
In Section 7, we implement the grammar of the statement calculus in the Symbolic Logic Simulator (SLS), a program written in 28 lines of Forth code, that allows computer-aided verification of any theorem in Propositional Logic (see Appendix 1 for source code). The program makes it straight-forward to explore non-obvious logical identities, and verify any propositional logic theorem or conjecture, in particular see Appendix 2 for key identities in the statement calculus (duality, algebraic, and canonical identities).
The concept of linguistic adequacy is developed in Section 8 and the NAND Adequacy Theorem is proved showing that NAND can generate all logical operations. A corollary is that any digital logic circuit can be built up entirely using NAND gates, illustrated using the free Digital Works software.
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In this article we look at the ideas of George Peacock whose 700-page opus A Treatise on Algebra (1830) transformed classical algebra into its modern form as an abstract symbolic science, free from the physical interpretation of quantity that had previously restricted it.
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By Assad Ebrahim, on March 18th, 2020 (6,076 views) |
Topic: Maths--General Interest
Revised Nov 2022, Jan 2023
In this article we look at the evolution of logic from its earliest form in the demonstration of truths to the rapid development of mathematical logic in the 1800s at the start of the “golden century” of logic (1850-1950). We also look at the rise and surprising dashing of hopes for the formalist program.
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By Assad Ebrahim, on February 18th, 2020 (877 views) |
Topic: Maths--History
This is Part 3 in the Ancient Mathematics series. (To read earlier parts: Part 1: Prehistoric Origins of Mathematics, Part 2: The Mathematics of Uruk and Susa).
This article explores what the people of Mesopotamia wrote about, counted and produced in the last part of the fourth millenium BCE. It does so by examining the frequency of signs in the proto-cuneiform tablets from the period c.3500-3000 BCE. For those wishing to build up experiential understanding of life in the Neolithic and early Bronze Age near east, this article provides suitable references as well as practical explorations of the economic and productive activities of the people: rope-making from grass, basket-weaving from reads, baking, weaving of cloth from linen, breaking ground, sowing, reaping, making flour, etc.
The first part of the article look closely at the collection of artifacts in each period that are associated with the evolution of writing and mathematics in ancient Mesopotamia, examining noteworthy individual artifacts that showcase a key development.
The study uses the CDLI database of cuneiform documents, and software I’ve written that parse the files in the CDLI database, extracting counts, parsing signs, generating frequency distributions of signs, creating a proto-cuneiform glossary, and assisting in the quantitative analysis of artifacts and semi-automated translation (see CDLI parser software library written in Ruby).
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