infimum

简明释义

英[ˈɪnfɪməm]美[ɪnˈfaɪməm]

n. 下确界

英英释义

单词用法

同义词

反义词

例句

1.The infimum and supremum of operators, and generalized inverse of operators are heated topics in operator theory and also have important value in both theory and application.

算子的确界性质和广义逆是近年来算子理论中比较活跃的一些研究课题，在算子理论的研究中有着重要的理论价值和应用价值。

2.The infimum and supremum of operators, and generalized inverse of operators are heated topics in operator theory and also have important value in both theory and application.

算子的确界性质和广义逆是近年来算子理论中比较活跃的一些研究课题，在算子理论的研究中有着重要的理论价值和应用价值。

3.According to the definite condition, the paper USES the theorem of supremum and infimum to study characteristic of mapping function and get its important law of the mapping.

运用上确界与下确界存在定理，在一定条件下，研究了连续函数的映射特点，得到一个连续函数所独有的映射规律。

4.In mathematics, the infimum of a set is the greatest lower bound of that set.

在数学中，一个集合的infimum是该集合的最大下界。

5.To find the infimum of a function over an interval, we analyze its critical points.

要找到一个函数在某个区间上的infimum，我们需要分析其临界点。

6.The infimum of a bounded below set exists in the real numbers.

在实数中，一个有下界的集合的infimum是存在的。

7.The infimum of the sequence converges to a limit as n approaches infinity.

当n趋向于无穷大时，该序列的infimum收敛到一个极限。

8.In optimization problems, identifying the infimum can help in finding the minimum value.

在优化问题中，识别infimum可以帮助找到最小值。

作文

In mathematics, particularly in the field of real analysis, the concept of infimum plays a crucial role in understanding the bounds of sets. The infimum of a set is defined as the greatest lower bound of that set. This means that it is the largest value that is less than or equal to every number in the set. To illustrate this concept, consider the set of all positive real numbers less than 1, denoted as S = {x ∈ ℝ | 0 < x < 1}. The infimum of this set is 0, since 0 is the largest number that is less than every element in S. However, it is important to note that 0 is not an element of the set itself. This distinction between the infimum and the minimum is vital; while the minimum is an actual member of the set, the infimum may not be.Understanding the infimum can lead to deeper insights in various mathematical contexts, such as optimization problems and limit processes. For instance, when trying to minimize a function, we often seek the infimum of the function’s values over a certain domain. In calculus, the concept of limits is closely related to the idea of infimum. As we approach a boundary, we may find that the function does not actually reach a particular value but instead approaches its infimum.Moreover, the infimum is not only limited to real numbers. It can be extended to other mathematical structures, such as sequences and functions. For example, if we take a sequence of numbers that converges to a point, the infimum of that sequence will help us understand its behavior as it approaches that limit. This makes the infimum a versatile tool in both theoretical and applied mathematics.The importance of the infimum extends beyond pure mathematics; it has practical applications in economics, engineering, and computer science. In economics, for instance, the infimum can represent the lowest possible price that a good might reach in a competitive market. In engineering, optimizing resource allocation often involves finding the infimum of a cost function. Similarly, in computer science, algorithms may rely on the infimum to efficiently solve problems related to data sorting and searching.To summarize, the concept of infimum is fundamental in various branches of mathematics and its applications. By understanding the infimum, one can gain valuable insights into the structure and behavior of sets, functions, and sequences. This knowledge not only enhances mathematical reasoning but also equips individuals with analytical tools applicable in diverse fields. As we continue to explore the depths of mathematics, the infimum remains a key concept that bridges various disciplines, emphasizing the interconnectedness of mathematical ideas and their real-world implications.

在数学中，特别是在实分析领域，infimum的概念在理解集合的界限方面起着至关重要的作用。infimum是一个集合的最大下界。这意味着它是小于或等于集合中每个数的最大值。为了说明这一概念，考虑所有小于1的正实数的集合，记作S = {x ∈ ℝ | 0 < x < 1}。这个集合的infimum是0，因为0是小于集合S中每个元素的最大数。然而，重要的是要注意，0并不是集合本身的元素。这一点对于infimum和最小值之间的区别至关重要；虽然最小值是集合的实际成员，但infimum可能不是。理解infimum可以在各种数学背景中带来更深刻的见解，例如优化问题和极限过程。例如，在试图最小化一个函数时，我们通常会寻找该函数在某一域上的infimum。在微积分中，极限的概念与infimum的思想密切相关。当我们接近一个边界时，我们可能会发现函数实际上并没有达到某个特定值，而是接近其infimum。此外，infimum不仅限于实数。它可以扩展到其他数学结构，如序列和函数。例如，如果我们取一个收敛到某个点的数列，该数列的infimum将帮助我们理解它在接近该极限时的行为。这使得infimum成为理论和应用数学中一个多功能的工具。infimum的重要性超越了纯数学；它在经济学、工程学和计算机科学中具有实际应用。例如，在经济学中，infimum可以表示在竞争市场中商品可能达到的最低价格。在工程学中，优化资源分配通常涉及找到成本函数的infimum。同样，在计算机科学中，算法可能依赖于infimum来高效解决与数据排序和搜索相关的问题。总之，infimum的概念在各种数学分支及其应用中是基础性的。通过理解infimum，人们可以获得对集合、函数和序列的结构和行为的宝贵见解。这种知识不仅增强了数学推理能力，还为个人提供了适用于多个领域的分析工具。当我们继续探索数学的深度时，infimum仍然是一个关键概念，它连接着各种学科，强调了数学思想及其现实世界影响的相互联系。