All principles and methods in mathematical analysis are based on the theory of real numbers and are taken as a starting point. Therefore, it is important to have a correct knowledge for the set of real numbers in order to have a profound knowledge of the principles and methods of analysis. There is a common point between two sets of numbers, called "ordered fields," in which both operations are defined and the size relation between any two numbers holds in the algebraic sense. However, there is an essential difference between the two sets. It cannot be described simply by algebraic...

Water pollution is one of the most serious environmental problems worldwide. Accurate forecast of water quality is of great importance since it can provide useful information for the managers to minimize the impact of water pollution. However, water quality forecasting still remains a challenge due to the limited information about the pollution resources and the high uncertainties of the dynamic processes. Most of the existing water quality forecast models are based on the original data without preprocessing. Water quality data is usually of high complexity and non-stationary, and therefore,...

In recent years, Hamilton-Jacobi equations have attracted a lot of attention due to the important role that they play in various fields of applications.In this book, we introduce recent developments in adaptive numerical methods for Hamilton-Jacobi equations.We construct convergent adaptive high order schemes for time dependent Hamilton-Jacobi equations and propose a general method of constructing convergent adapive high order schemes and discuss the question of its convergence.Then, we propose a method of constructing convergent adaptive high order schemes for Hamilton-Jacobi equations on...

In analysis, series occupies a special place and is used in many fields not only in analytical science but also in various subjects of mathematics, including differential equations, computational mathematics, and in applied sciences, including signal processing, and mathematical statistics.Addition is not only the first operation introduced by mankind, but also the most used because it is very easy to understand and clear in its representation. Of course, finite additions that generalize two numbers of additions are distinct in their representation and meaning, but infinite additions, i.e.,...

Integral is a most powerful and useful tool for solving various problems in natural sciences such as mathematics, physics and so on.We proceed strictly in a logical order, with the focus on the traditional approach of analytics in developing the theory of integration. We have focused on mathematical rigour, summarized the theory of integration with this focus, and more clearly described the integrability with singular integrals.First, we introduce the concept of antiderivative, and described the definition and properties of the indefinite integral, the calculus and, in particular, the...

Real-variable functional theory is an important component of analysis, a subject that learns between analytic mathematics and functional analytics.We study measure theory, integral theory, bounded variational function theory in this book.First, we study the properties of measure spaces, measurable sets, and measurable functions.Then we study the integral of measurable functions in measure space, its properties, the relationship between the Riemann integral and the Lebesgue integral, and so on.Finally we study the addition of generalized measure concepts to bounded variational functions such...