point-by-point convergence (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun

Phonetic Transcription

/pɔɪnt baɪ pɔɪnt kənˈvɜrdʒəns/

Meaning and Usage

Point-by-point convergence refers to a type of convergence of sequences or functions where convergence is examined at each individual point in a specific context, such as in analysis or topology. In mathematical terms, a sequence of functions converges point-by-point to a limit function if, at each point in the domain, the sequence of function values converges to the value of the limit function.

This term is frequently used in written contexts, particularly in mathematical literature, research papers, and educational materials. While it can occasionally appear in oral speech among professionals or academics discussing convergence in mathematics or related fields, it's predominantly a term found in written expressions.

Example Sentences

1. The mathematician explained point-by-point convergence during his lecture on function sequences.
2. Математик объяснил сходство по точкам во время своей лекции о последовательностях функций.
3. The concept of point-by-point convergence is crucial for understanding uniform convergence.
4. Концепция сходства по точкам имеет решающее значение для понимания равномерной сходимости.
5. In his recent publication, he explored the implications of point-by-point convergence in functional analysis.
6. В своей недавней публикации он исследовал последствия сходства по точкам в функциональном анализе.

Idiomatic Expressions

While "point-by-point convergence" itself is not commonly found in idiomatic expressions, the concept of convergence can be applied in various mathematical and scientific idioms.

1. Convergence of ideas
2. After many discussions, there was a convergence of ideas on how to improve the project.

3. После многих обсуждений произошла конвергенция идей о том, как улучшить проект.

4. Converging paths

5. The two students found their paths converging as they both chose to study environmental science.

6. Два студента обнаружили, что их пути сходятся, когда они оба решили изучать экологическую науку.

7. At the point of convergence

8. The scientists met at the point of convergence, where their research overlapped significantly.

9. Ученые встретились в точке схождения, где их исследования пересекались значительным образом.

10. Converging opinions

11. During the debate, the panelists' opinions started converging towards a common solution.
12. Во время дебатов мнения членов панели начали сходиться к общему решению.

Etymology

The term "convergence" comes from the Latin word "convergere," which means "to incline together." The prefix "point-by-point" indicates a methodical, detailed approach, suggesting that the convergence is being analyzed at each individual point. This combination reflects the careful and precise nature of mathematical analysis.

Synonyms and Antonyms

Synonyms: - Pointwise convergence - Sequential convergence

Antonyms: - Divergence - Discrepancy