Tag Archives: mathematics

White Paper: Situations in Life Where Gambler’s Ruin Threatens Success—and Why

Posted on November 20, 2025 by nathanalbright

Executive Summary Gambler’s ruin is a concept from probability theory describing how a participant in a series of risky, repeated events can be inevitably bankrupted—even when the odds of each individual event seem favorable—if losses cannot be absorbed and the … Continue reading →

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Course Title: Numeracy for Adults: Practical Quantitative Literacy for Life, Work, and Citizenship

Posted on October 29, 2025 by nathanalbright

Course Overview This course is designed to help adults build the kind of mathematical literacy that directly impacts their daily lives—financial management, critical evaluation of data, informed decision-making, and civic participation. The focus is on reasoning rather than rote calculation, … Continue reading →

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White Paper: The Numeracy That Matters: Essential Mathematical Literacy for Adult Life

Posted on October 29, 2025 by nathanalbright

Executive Summary Modern societies depend upon numeracy as much as literacy, yet the type of numeracy adults actually need is not always the kind emphasized in school curricula. While advanced mathematics undergirds modern technology and finance, most adults require a … Continue reading →

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White Paper: The Implicit Logic of Numbers: The Ordinal Implication of “First” and the Emergent Logic of Numerical Systems

Posted on October 19, 2025 by nathanalbright

Abstract This white paper examines the logical structure implicit in ordinal numbering systems, focusing on the semantic and philosophical implications of the word first as presupposing the existence of a second. It situates this phenomenon within the broader logic of … Continue reading →

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White Paper: From Anecdote to Data: Determining the Threshold of Meaningful Evidence

Posted on October 3, 2025 by nathanalbright

Executive Summary In policy debates, scientific inquiry, and business decision-making, anecdotal evidence often acts as the seed of inquiry but rarely as the foundation of rigorous conclusions. Yet there is a critical question: at what point does a collection of … Continue reading →

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White Paper: A Typology of Geometries Derived from Relaxations of Euclid’s Axioms

Posted on September 30, 2025 by nathanalbright

Executive Summary Euclidean geometry, as systematized in Elements, rests on five foundational postulates (axioms) and common notions. Relaxing or altering these axioms has historically generated new branches of mathematics, each with unique properties and applications. This paper surveys the typology … Continue reading →

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White Paper: A Typology of Fruitful Areas of Conjecture and Theorem Generation in Algebra Using AI Methods

Posted on September 27, 2025 by nathanalbright

Executive Summary The integration of artificial intelligence (AI) into algebraic research opens new avenues for conjecture and theorem generation. While classical mathematics has relied on human creativity, analogy, and intuition, AI models—especially those leveraging symbolic computation, large language models, and … Continue reading →

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White Paper: Practical Applications of Algebra as a Basis for Instruction

Posted on August 30, 2025 by nathanalbright

Executive Summary Algebra is often perceived by students as abstract and detached from real-world concerns. Yet, algebra provides the foundational tools needed for problem-solving in science, engineering, economics, and daily decision-making. This white paper explores the practical applications of algebra … Continue reading →

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Theorem (square-gcd count, closed form)

Posted on August 26, 2025 by nathanalbright

For a positive integer n, let A_2(n):=\#\{\,1\le k\le n:\ \gcd(k,n)\ \text{is a perfect square}\,\}. Then A_2(n)\;=\;\sum_{d^2\mid n}\varphi\!\left(\frac{n}{d^2}\right) \;=\;\prod_{p^{\alpha}\parallel n}\frac{p^{\alpha+1}-(-1)^{\alpha+1}}{p+1}, where \varphi is Euler’s totient function and the product runs over the prime powers p^\alpha dividing n exactly. Explanation. We are … Continue reading →

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White Paper: The Relevance of the No Free Lunch Theorems to Unguided Stepwise Searches for Solutions

Posted on August 11, 2025 by nathanalbright

Abstract The No Free Lunch (NFL) theorems, originally formulated in the context of optimization and machine learning, establish that averaged over all possible objective functions, all search and optimization algorithms perform equally well. Any algorithm’s advantage in one class of … Continue reading →

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