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January 31st, 2024 (4th ed)1 When designing a system, what should you optimize? If it is a user-interface or process, you should be minimizing clicks, or process steps. But for hardware-software systems, the answer is not obvious, and a common mistake is to fail to consider the end-to-end problem. This article explores what is involved in optimizing end-to-end in hardware-software systems. The goal here is to minimize the overall complexity of the system, i.e. of the triple hardware-software-user combination. The following remarks set the stage for our discussion:
 Between complexity and simplicity, progress, and new layers of abstraction. Continue reading this article…
Revised & Expanded May 2023. First published November 1998. This article provides a selection of quotes, written mostly by mathematicians, that convey especially clearly essential aspects of mathematics and its culture. Comments are collected in the endnotes. Contents
Part 1 in Ancient Mathematics series. (Part 2: The Mathematics of Uruk and Susa 3500-3000 BCE, Part 3: Exploring Cuneiform Culture 8500-2500 BCE) Abstract  The Anthropology and Archaeology of Conceptual Thought leading to the Birth of Mathematics 4th ed. Jan 2024; 3rd ed. May 2023; 2nd ed. Dec 2009; 1st ed. Sep 2004
‘What is mathematics?’ Much ink has been spilled over this question, as can be seen from the selection of ten respected responses provided in the footnote1, with seven book-length answers, and three written in the current millenium. One might well ask, is there anything new that can be said, that should be said? We’ll start by clarifying what a good answer should look like, and then explore the answer proposed. The rest of the paper follows the structure below:
1. Criteria for a Good Definition of Mathematics
2. Definition 1: covering mathematics up to the end of the 18th century (1790s)
3. Two Perspectives
Mathematics as Dialectic (Lakatos)
Mathematics shaped by its Anthropology (Hoyrup)
4. Definition 2: covering all mathematics, including contemporary mathematics
5. The emergence of contemporary mathematical practice from 1800s onward
6. Three Facets of Mathematics
1. Mathematics as an Empirical Science
2. Mathematics as a Modeling Art
3. Mathematics as an Axiomatic Arrangement of Knowledge
7. Mathematics "from the inside": Mathematicians writing about Mathematics
8. Continue Reading
9. References
Continue reading this article…
2nd ed. June 2023; 1st ed. April 2010 The term “mathematical maturity” is sometimes used as short-hand to refer to a blend of elements that distinguish students likely to be successful in mathematics. It is a mixture of mathematical interest, curiousity, creativity, persistence, adventurousness, intuition, confidence, and useful knowledge.[1],[2],[3] With advances in machine learning, computer science, robotics, nano-materials, and many other quantitative, fascinating subjects, students today have increasingly more choice in technical studies besides mathematics. To attract and retain mathematics students, it is important that mathematics instruction be experienced as both intellectually and culturally rewarding in addition to being technically empowering. Losing students from mathematics who are otherwise capable, engaged and hard-working is tragic when it could have been avoided. In this article, building on observations gained over the years teaching and coaching students in mathematics, we consider how enriched mathematics instruction (inquiry-based/discovery learning, historiography, great ideas/survey approaches, and philosophical/humanist) can help (1) develop mathematical maturity in students from at-risk backgrounds and prevent their untimely departure from quantitative studies, (2) strengthen the understanding of those that are already mathematically inclined, (3) expand mathematical and scientific literacy in the wider population.
Mathematical Finance is an area of applied mathematics that has developed rapidly during the late 80s and 90s after the deregulation of U.S. financial markets, and accelerated further in the 2000s concurrently with the rise of data science/’big data’ and computational platforms able to run complex models in close to real-time. For its financial models for risk and pricing, Mathematical Finance draws upon the partial differential equations of mathematical physics, stochastic calculus, probabilistic modeling, mathematical optimization, statistics, and numerical methods. The implementation of these often complex numerical mathematical models requires efficient algorithms and exploiting the state-of-the-art in software engineering (real-time and embedded development, low latency network programming) and computing hardware (FPGAs, GPUs, and parallel and distributed processing). Taken together, the technical aspects of mathematical finance and the software/hardware aspect of financial engineering lie at the intersection of business, economics, mathematics, computer science, physics, and electrical engineering. For the technologically inclined, there are ample opportunities to contribute. But the relevance goes beyond mathematics. There is a kernel of core financial ideas that are at the heart of the global free market capitalist system that is in place across most of the world today. These ideas affect not only economics but also politics and society. Ideally, every citizen in a democracy should understand the essential mechanics of the modern financial world and how it has arisen, regardless of whether we agree with its principles or with the impact of the financial system on social structures. This article presents a simplified account of the rise of the modern financial marketplace including some history, and contemporary financial context. Update (2012): A highly recommended graphic novel Economix, by Michael Goodwin has just been published that presents a panoramic yet highly accessible narrative.) This is a collection of short articles and reflections on topics of current interest. For older short posts, see here: #1-199 (Feb 2014-Oct 2019) 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. |
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© 2009-2024 Assad Ebrahim To contact me on email, use: assadebrahim2000 at gmail dot com (do the obvious) |
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