Subsequent to the completion of this paper, a widely discussed study by Shojaee et al. at Apple (The Illusion of Thinking, 2025) presented empirical evidence that several advanced reasoning models (e.g., Claude 3.7 Sonnet, DeepSeek-R1) exhibit a...
The easy way to conquer calculus
Calculus is hard—no doubt about it—and students often need help understanding or retaining the key concepts covered in class. Calculus Workbook For Dummies serves up the concept review and practice problems with...
The Chernoff approximation method is a powerful and flexible tool of functional analysis, which allows in many cases to express exp(tL) in terms of variable coefficients of a linear differential operator L. In this paper, we prove a theorem that...