标题： 同质性 显示英文标题

标题： Homogeneity

摘要： 四种类型的齐次性——加法、乘法、指数和对数——被推广为描述函数$f$在其自变量缩放或平移时如何变化的变换。 这些广义的齐次函数捕捉不同的缩放行为并建立基本性质。 这些性质包括齐次性在函数运算下如何保持以及它如何确定相关构造如商函数的变换行为。 这个框架将经典的齐次概念扩展到更广泛的函数对称类，提供了一种统一的方法来分析各种数学情境中的缩放特性。 显示英文摘要

摘要： The four types of homogeneity - additive, multiplicative, exponential, and logarithmic - are generalized as transformations describing how a function $f$ changes under scaling or shifting of its arguments. These generalized homogeneity functions capture different scaling behaviors and establish fundamental properties. Such properties include how homogeneity is preserved under function operations and how it determines the transformation behavior of related constructions like quotient functions. This framework extends the classical concept of homogeneity to a wider class of functional symmetries, providing a unified approach to analyzing scaling properties in various mathematical contexts.