summability

简明释义

英[ˌsʌməˈbɪlɪti]美[ˌsʌməˈbɪlɪti]

n. [数] 可和性

英英释义

单词用法

同义词

反义词

例句

1.As applications, the corresponding limit results for some classical summability methods are also established.

把这些结果应用到经典的可和方式，获得了相应的结果。

2.The generalized Riesz summability operators of Laplace series are introduced.

引进了拉普拉斯级数的广义黎斯可和算子。

3.As applications, the corresponding limit results for some classical summability methods are also established.

把这些结果应用到经典的可和方式，获得了相应的结果。

4.Mathematicians often study the summability of series to understand their behavior.

数学家们常常研究级数的可加性以理解其行为。

5.The summability of functions plays a key role in the theory of Fourier series.

函数的可加性在傅里叶级数理论中起着关键作用。

6.A sequence is said to have summability if the sum of its terms approaches a limit.

如果一个序列的项之和趋近于某个极限，则该序列被称为具有可加性。

7.The concept of summability is crucial in functional analysis.

在泛函分析中，可加性的概念至关重要。

8.In probability theory, the summability of a sequence of random variables can determine convergence.

在概率论中，一系列随机变量的可加性可以决定其收敛性。

作文

In mathematics, the concept of summability plays a crucial role in understanding the behavior of sequences and series. Summability refers to the property of a sequence or series that allows it to be assigned a sum, even if the conventional sum does not converge. This idea is particularly useful in various branches of analysis, where we often deal with infinite processes. The study of summability helps mathematicians extend the notion of summation beyond traditional limits and provides tools to handle divergent series effectively.One of the primary motivations for exploring summability is found within the realm of Fourier series. In signal processing and harmonic analysis, we frequently encounter functions that can be represented as infinite sums of sinusoidal components. However, not all such representations converge in the classical sense. Here, the concept of summability becomes essential, as it allows us to assign meaningful values to these otherwise divergent sums, enabling us to analyze and manipulate signals more effectively.There are various methods of summability, each with its own criteria and applications. For instance, Cesàro summability is one of the most well-known techniques. It involves taking the average of the partial sums of a series. If this average converges to a limit, we say that the series is Cesàro summable. This technique is particularly valuable because it allows for the summation of certain divergent series that would otherwise be considered meaningless. Moreover, the concept of summability extends to other forms, such as Abel summability and Borel summability, each providing unique insights into the nature of convergence and divergence.The implications of summability extend beyond pure mathematics; they find applications in physics, engineering, and computer science. In physics, for example, the analysis of wave functions and quantum states often requires dealing with infinite sums. Understanding summability allows physicists to extract physical quantities from these mathematical constructs, leading to meaningful interpretations of physical phenomena.Furthermore, in computer science, algorithms that involve infinite processes or iterative methods benefit from the principles of summability. When designing algorithms that approximate solutions to complex problems, ensuring that the sequences generated by these algorithms are summable can lead to more reliable and efficient computations. Thus, the concept of summability is not merely an abstract mathematical idea; it has practical implications in real-world applications.In conclusion, the notion of summability is a powerful tool in mathematics that allows us to extend our understanding of series and sequences beyond traditional convergence. By employing various summability methods, mathematicians and scientists can work with divergent series in a meaningful way, opening up new avenues for research and application. Whether in the context of Fourier analysis, physics, or computer science, summability serves as a bridge between theoretical inquiry and practical implementation, demonstrating the profound interconnectedness of different fields of study. As we continue to explore the depths of mathematics, the concept of summability will undoubtedly remain a significant area of focus, revealing further insights into the nature of infinity and convergence.

在数学中，概念summability在理解序列和级数的行为方面发挥着至关重要的作用。Summability指的是序列或级数的性质，它允许其被赋予一个和，即使常规和不收敛。这个想法在分析的各个分支中尤其有用，因为我们经常处理无限过程。对summability的研究帮助数学家扩展了求和的概念，超越了传统的限制，并提供了有效处理发散级数的工具。探索summability的主要动机之一可以在傅里叶级数的领域中找到。在信号处理和谐波分析中，我们经常遇到可以表示为无限正弦成分和的函数。然而，并非所有这样的表示在经典意义上都收敛。在这里，summability的概念变得至关重要，因为它使我们能够将有意义的值分配给这些否则会发散的和，从而使我们能够更有效地分析和处理信号。有多种summability方法，每种方法都有其自身的标准和应用。例如，Cesàro求和是最著名的技术之一。它涉及对级数的部分和取平均。如果这个平均值收敛于一个极限，我们就说该级数是Cesàro可求和的。这种技术特别有价值，因为它允许对某些发散级数进行求和，这些级数在其他情况下将被认为是无意义的。此外，summability的概念扩展到其他形式，如Abel求和和Borel求和，每种形式都提供了对收敛性和发散性本质的独特见解。Summability的影响超越了纯数学；它在物理学、工程学和计算机科学中找到了应用。例如，在物理学中，波函数和量子态的分析通常需要处理无限和。理解summability使物理学家能够从这些数学构造中提取物理量，从而对物理现象进行有意义的解释。此外，在计算机科学中，涉及无限过程或迭代方法的算法受益于summability的原则。在设计近似复杂问题解决方案的算法时，确保这些算法生成的序列是summable的，可以导致更可靠和高效的计算。因此，summability的概念不仅仅是一个抽象的数学思想；它在现实世界的应用中具有实际意义。总之，summability的概念是数学中的一个强大工具，它使我们能够扩展对级数和序列的理解，超越传统的收敛。通过采用各种求和方法，数学家和科学家可以以有意义的方式处理发散级数，为研究和应用开辟新的途径。无论是在傅里叶分析、物理学还是计算机科学的背景下，summability都是理论探究与实际实施之间的桥梁，展示了不同研究领域之间的深刻相互联系。随着我们继续探索数学的深度，summability的概念无疑将仍然是一个重要的关注领域，揭示关于无限和收敛本质的进一步见解。