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In the study and discussion of mathematics, one may encounter the expression “the derivative vanishes.” This phrase is used to succinctly describe certain properties or situations involving derivatives. While it may not be a familiar expression in Japanese mathematical education, it is quite common in English. In this article, we will delve into the meaning…

The catenary is a curve represented by the mathematical equation: \[ y=\frac{\cosh ax}{a}\ \ (-b \leq x \leq b) \] where \( \cosh \) is the hyperbolic cosine function, and \(a \) is a positive constant. The curve is defined within \( [-b,b] \), with the endpoints fixed at \( x=-b \) and \( x=b…

The concept of “almost everywhere” (a.e.) is fundamental in analysis and probability theory, especially in the study of Lebesgue integration. It provides a flexible framework to describe properties that hold “almost everywhere” in a domain, allowing exceptions over sets of measure zero. Below, we define this concept and explore its notation and applications in detail.…

The empty set refers to a set that contains no elements. For example, the set defined by the condition “\( x>5 \) and \( x<3 \)” for any real number \(x\) is an empty set. As it contains nothing, it holds a special place in mathematics as the “empty set.” This article delves into the…

This article provides a detailed explanation of the concept of “supremum (sup)” in the context of subsets \( A\subset \mathbb{R} \) of real numbers. Specifically, it describes how the supremum is defined and its properties. By understanding the intuitive meaning, mathematical formalization, and examples, we can better grasp the significance of this important concept. What…