Introduction to basic concepts of applied mathematics and elementary function theory

Author: Zen and the Art of Computer Programming

Article directory

  • 1 Introduction
    • introduction
    • Article structure
  • 2. Background introduction
  • 3. Explanation of basic concepts and terms
    • 1. Set
    • 2. Scalar
    • 3. Vector
    • 4.Matrix
    • 5.Tensor
    • 6. Function
    • 7. Intuitive Abstraction
    • 8. Regular Expression
    • 9. Logical Calculus (Boolean Algebra)
    • 10.Calculus
    • 11.Linear Algebra
    • 12. Probability Theory
    • 13. Random Process
    • 14. Operations Research
    • 15. Discrete Mathematics
  • 4. Core algorithm principles and operating steps
    • 1. Calculus
      • 1.1 Derivatives
      • 1.2 First derivative
      • 1.3 Second derivative
      • 1.4 Geometric meaning of derivatives
      • 1.5 Partial derivatives
    • 2. Linear algebra
      • 2.1 Vector
      • 2.2 Matrix
      • 2.3 Eigenvalues and eigenvectors
    • 3. Probability theory
      • 3.1 Probability
      • 3.2 Expected value
      • 3.3 Central limit theorem
      • 3.4 Probability density function
      • 3.5 Distribution
    • 4. Random process
      • 4.1 Independent and identically distributed
      • 4.2 Wiener process
      • 4.3 Brownian motion
      • 4.4 Poisson process
    • 5. Operations Research
      • 5.1 Dynamic programming
      • 5.2 Integer Linear Programming
      • 5.3 Linear programming
      • 5.4 Network flow

1. Introduction

Introduction

In order to promote readers to understand and master some basic concepts, techniques, and methods of applied mathematics, and to strengthen the methodology of mathematics learning, so that they can truly “know it without realizing it.” Improve the understanding of mathematics among scientific research, engineering, and business personnel, and improve work efficiency and quality. By reading this article, readers can benefit a lot.

Article structure

The article is divided into six parts, as follows:

  1. Background introduction: briefly introduce the background knowledge of the article;
  2. Explanation of basic concepts and terminology: Detailed introduction to basic concepts, terms and related definitions in applied mathematics;
  3. Core algorithm principles and operating steps: Detailed introduction to several of the most commonly used algorithms in applied mathematics, such as calculus, linear algebra, probability theory, stochastic processes, operations research, etc., and provide corresponding code implementations;
  4. Specific code examples and explanations: The corresponding algorithms are elaborated and analyzed through specific examples, so that readers can better understand the algorithm principles and practical operations;
  5. Future development trends and challenges: Look forward to the development prospects of applied mathematics and the new opportunities and challenges it faces. At the same time, we analyze the existing problems and solutions in some current mathematics application fields, and give suggestions on the future role of mathematics in subject development, industrial layout, etc.;
  6. Appendix Frequently Asked Questions and Answers: Concentrates on summarizing and answering some common questions in daily life, and sorting out and extending the corresponding knowledge points.

2. Background introduction

With the rapid development of Internet informatization, the popularization of mobile communications and the rise of e-commerce, traditional industries are being impacted by the Internet and the Internet of Things. More and more innovative companies and business owners are faced with choices that take into account various factors, such as cost-effectiveness, competitive advantage, and quick response to market changes. Facing the growing economy and society, the shortage of talents has become a prominent problem. How to cultivate a group of talented mathematics professionals has become a hot topic. In recent years, top universities in the United States such as the Massachusetts Institute of Technology, Stanford University, and the University of California, Los Angeles, have launched mathematics-related courses, attracting many students from internationally renowned universities and creating a good atmosphere.
As a basic subject, mathematics covers a wide range and is extremely important. It is widely used by various disciplines and scientific research teams. Mathematics helps solve research problems in many disciplines, such as physics, engineering, psychology, biology, medicine, economics, information theory, finance, etc. With the development of new generation digital technologies such as cloud computing, Internet of Things, and blockchain, the increase in data volume has brought complexity and challenges. How to effectively process, analyze, and utilize data has become a new topic. How to deal with increasingly complex data has become a major challenge in the global field.

3. Explanation of basic concepts and terms

First, the basic concepts, terms and related definitions in applied mathematics are introduced.

1. Set

A set is an unordered whole with non-repeating elements, usually enclosed by {}, where each element is separated by a comma. Generally speaking, a set consists of several elements, and each element is called a member of the set. Set operators include: ∩ (intersection), ∪ (union), ? (complement). For example: S = {a, b, c} ∩ {b, d}, T = {1, 2, 3} ? {2}.

2. Scalar

A scalar is a single value, a number. For example: x = 3.5.

3. Vector (Vector)

A vector is a sequence of numbers that represents position, speed, direction, magnitude, etc. in one or two dimensions. It consists of a series of scalars with the same number of elements, that is, each element corresponds to a coordinate. Vector operators are: + (addition), – (subtraction), * (inner product), / (outer product). For example: v = <1, 2>, w = <-1, 3> + <2, -1>.

4.Matrix

Matrix is a square matrix, that is, a rectangular array of m * n elements. Matrix operators are: + (addition), – (subtraction), * (multiplication), \ (left division), / (right division). For example: A = |2, -1; 1, 2|, B = |-1, 2|; 3, -2|.

5.Tensor

Tensor is an n-order array, which consists of n index axes and elements of corresponding dimensions. Tensor operators are: + (addition), – (subtraction), * (multiplication), \ (left division), / (right division). For example: X = [2, 1], Y = [-1, 2]; Z[i][j] = X[i] + Y[j].

6. Function

A function is a relation that maps inputs to outputs. It generally accepts input data, transforms it according to certain rules, and returns the corresponding results. The general form of the function is y = f(x), x is the input data, and y is the output result. Functions have two basic characteristics:

  • Monotonicity: The function f(x) for all x, when x grows or shrinks, the function value should also grow or shrink accordingly.
  • Consistency: If f(x) is certain for an input of x, then its output is certain.

7. Intuitive Abstraction

Intuitive abstraction refers to abstracting the real world and making the abstracted picture simple and easy to understand. The key to intuitive abstraction is to describe it in simple language. For example, a river can be depicted by a small bridge, but we can easily understand the distance between the two bridges. In applied mathematics, abstraction usually refers to expressing complex things in a simple way and extracting the laws and patterns. For example, a three-dimensional space can be displayed using a two-dimensional diagram, but intuitively we can still know that it has three axes.

8. Regular Expression

Regular expressions are a way of describing character patterns. It is used to match character combinations in a string. The syntax of regular expressions allows us to easily specify patterns for search strings. Commonly used regular expression examples are as follows:

- Find all words starting with "hello": \w*hello\w*
- Find all words ending with a vowel: "^[aeiou]\w*$"
- Find all words starting with "abc" and ending with a number: "^abc\d.*$"

9. Logical calculation (Boolean Algebra)

Logical calculus is a mathematical subject about Boolean propositions. It includes Boolean values (true, false), logical operations (not, and, or, if…then…else, etc.), and some special logical theorems. For example, whether a person is healthy can be judged by measuring his or her physical symptoms. This process can be regarded as a reasoning process based on physical data. Similarly, in computer science, logical reasoning and logical analysis are very important techniques.

10.Calculus

Calculus is an important branch of mathematics, which involves concepts such as differentiation, derivatives, integrals, and curve redemption. Calculus is widely used and helps in the exploration, prediction and analysis of data. For example, the monitoring and diagnosis of human blood pressure rely on the process of real-time measurement of blood pressure values, which also requires calculus support.

11.Linear Algebra

Linear algebra is a basic discipline whose goal is to study vectors and linear equations on all line segments. It involves concepts such as matrix multiplication, inversion, eigenvalues, eigenvectors, ranks, determinants, etc. Linear algebra has applications throughout science, engineering, and business.

12. Probability Theory

Probability theory is a core theory of statistics. It studies the possibility and inevitability of something happening. Probability theory is closely related to mathematical statistics, which helps us generate hypotheses about data, build models, and make reliable predictions. For example, the belief that there will be a big bull or bear market in today’s stock market is based on expectations of stock price movements over the past many years.

13. Random Process

Random process is one of the most important concepts in probability theory. It describes the generation method of a type of random variables, and can also be regarded as the distribution function of random variables. The main characteristic of a random process is that the elements are random, but there are certain connections and rules between them. Commonly used random processes include: exponential distribution, radar wave, Wiener process, Brownian motion, Poisson process, etc.

14. Operations Research

Operations Research is a branch of management that studies how to maximize resource utilization, reduce costs, improve service quality, optimize efficiency, etc. through formulating strategies, resource allocation, decision-making and other means. Operations research is closely related to mathematics, and it helps answer many complex problems, such as flow allocation, production planning, supply chain management, etc.

15. Discrete Mathematics

Discrete mathematics is a basic discipline that studies various numerical systems. Its theoretical basis comes from Boolean algebra and Turing machines. Discrete mathematics contributes to computer design, security protection, communication transmission, etc. In the field of applied mathematics, discrete mathematics also plays an important role, such as random number generation, encryption algorithms, etc.

4. Core algorithm principles and operating steps

The second part introduces several of the most commonly used algorithms in applied mathematics and provides corresponding code implementations.

1. Calculus

Calculus is the most basic mathematical tool in applied mathematics. It provides concepts and formulas such as derivatives, variations, integrals, differentials, integral quotients, and limits between infinite variables. Calculus is widely used in physics, materials science, engineering, economics, etc. Below we give some basic knowledge of calculus:

1.1 Derivatives

Derivative is an important concept in calculus. It represents the rate of change of a function at a certain point. The derivative is defined as: f’(x)=lim_{h->0}(f(x + h)-f(x))/h, h=delta x. Among them, f(x + h) represents the function value after increasing the value of x by one unit delta x, and lim_{h->0} represents that when h tends to zero, the value of the function tends to the limit value.

1.2 First derivative

The first derivative represents the first partial derivative of the function at a point, that is, the slope. It is defined as df/dx=(f(x + h)-f(x-h))/(2h). Among them, h is a small positive number, which represents the step size of the function change. As h approaches zero, the first derivative represents the limit of the function.

1.3 Second derivative

The second derivative represents the second partial derivative of the function at a point, that is, the curvature. It is defined as d2f/dx2=(f(x + h) + f(x-h)-2f(x))/h2. Among them, h is a small positive number, which represents the step size of the function change. As h approaches zero, the second derivative represents the asymptotic linearity of the function.

1.4 Geometric meaning of derivatives

The derivative is the tangent to a curve and measures the slope of the tangent at any point on a curve. The tangent of a curve is equal to the function value at the intersection of the curve, so the derivative value reflects the bending direction of the curve and can be used to determine the location and height of downhill, uphill, and inflection points. In addition, the derivative value can also be used to determine the direction of the normal vector.

1.5 Partial Derivatives

Partial derivative represents the derivative of a certain variable of a function, also called the segmentation method. The definition of partial derivative is f’_k(x_1,x_2,…x_n)=?_k{f(x_1,x_2,…,x_n)}=?_kf(x_1,x_2,…,x_n)/?x_k. Among them, k=1,2,…,n represents the number of the variable, and the partial derivatives of different variables are obtained by calculating the partial derivatives for different k.

2. Linear algebra

Linear algebra is another fundamental tool in applied mathematics. It helps in performing basic operations on vectors and matrices including addition, subtraction, multiplication, transpose, rank, determinant, etc. The application of linear algebra has an important impact in fields such as machine learning, signal processing, and mechanics. Below we give some basic knowledge of linear algebra:

2.1 Vector

Vector is the most basic object in linear algebra. It consists of an ordered set of scalar values. A vector in linear algebra can be regarded as a point, or the position or motion trajectory of a point in space. Vector operations can be used to represent linear mapping and spatial projection, etc. Vector operators are: + (addition), – (subtraction), * (inner product), / (outer product). For example: <2, 3> + <1, -2>, <2, 3>*<1, -2>, |<-3,-2>|

2.2 Matrix

A matrix is a square matrix composed of vectors. It consists of a set of vectors of the same dimension. Matrix operations can be used to represent the solution of linear equations, image transformation, variance analysis, prediction, etc. Matrix operators are: + (addition), – (subtraction), * (multiplication), ^T (transpose), | (rank), det (determinant). For example: |2, -1;| * |1, 2;|-1, 2| = |1, -2;|

2.3 Eigenvalues and eigenvectors

An eigenvalue represents a special value of a matrix that corresponds to the corresponding eigenvector. The eigenvalue is the eigenvalue of the diagonal matrix, and the eigenvector is a basis of the corresponding eigenvalue. The eigenvalues and eigenvectors of the matrix form a Hilbert space. Eigenvalues and eigenvectors are widely used, such as PCA, singular value decomposition, least squares estimation, manifold learning, etc.

3. Probability theory

Probability theory is an important subject in applied mathematics. It provides a mathematical description of the frequency of random events, as well as the classification and analysis of various phenomena of random events. Probability theory helps model complex system behavior and predict it. Below we give some basic knowledge of probability theory:

3.1 Probability

Probability is an important concept in statistics. It represents the probability of a random event occurring. In probability theory, a random event is a subset of a sample space that contains statistical data about possible events and the number of times these events occur. For example, throwing a die can be thought of as a random event, with six faces ranging from 1 to 6, each of which has an equal probability.

3.2 Expected value

Expectation value is an important concept in statistics. It represents the mathematical expectation of a random variable, which is the average value we hope to get. It is a characterization of the average value of a random variable and an unbiased estimate of the value distribution of the random variable. The calculation formula of expected value is E(X)=∑xkP(xk). Among them, E() represents the expected value, P(xk) represents the probability of a random variable with value xk, k=1,2,…,n.

3.3 Central Limit Theorem

The central limit theorem is an important theorem in probability theory. It tells us that for many continuously distributed random variables, if their expectation and variance exist, then their probability density functions will tend to be normally distributed as the sample size increases.

3.4 Probability density function

The probability density function represents the probability that a random variable takes a value within a certain range. It depends on the value and upper and lower limits of the variable. The probability density function helps to visualize random variables and can give the probability distribution of random variables.

3.5 Distribution

Distribution is the statistical data of a set of random variables. It reflects the shape, location and distribution of random variables. Commonly used distributions include uniform distribution, normal distribution, chi-square distribution, negative binomial distribution, etc.

4. Random process

Random process is another important concept in probability theory. It describes the generation method of a type of random variables. It mainly has two types: one is a process generated by fixed parameters, which is called a process model; the other is a sequence of random variables generated by a random process generator, which is called a sample path. Stochastic process is a core concept in statistical analysis and has a wide range of applications. Below we give some basic knowledge of stochastic processes:

4.1 Independent and identically distributed

Independent and identically distributed random variables are a common assumption in random processes. It shows the independence of random variables in time, that is, at any point in time, the value of the random variable only depends on previous historical events.

4.2 Wiener process

The Wiener process is a typical stochastic process model. It consists of two random variables: a stationary Wiener process and an independent Poisson process. This process describes a random walk in a noisy random environment.

4.3 Brownian motion

Brownian motion is a typical stochastic process model. It consists of a stationary Wiener process and a string of ordered random variables, called true random variables. This process describes a random walk along time.

4.4 Poisson process

Poisson process is a typical stochastic process model. It consists of two random variables: a Poisson process and a stationary Wiener process. This process describes the rate at which random events occur.

5. Operations Research

Operations research is an important branch of management. It studies how to maximize resource utilization, reduce costs, improve service quality, optimize efficiency, etc. through strategy formulation, resource allocation, decision-making and other means. The key to operations research is the management of information, and its connection with mathematics is also very close. Below we give some basic knowledge of operations research:

5.1 Dynamic programming

Dynamic programming is an optimization method in operations research. It finds the global optimal solution by considering the trade-off between the local optimal solution and the global optimal solution. Dynamic programming is beneficial to solving complex decision-making problems and has a wide range of applications.

5.2 Integer Linear Programming

Integer linear programming is another optimization method in operations research. It requires the solution of linear programming for integer variables and solves for the best solution with integer solutions.

5.3 Linear Programming

Linear programming is another optimization method in operations research. It requires solving for integer variables under linear constraints, and what is solved is the best solution with integer solutions.

5.4 Network Flow

Network flow is an optimization method in operations research. It uses the idea of graph theory to describe the flow process of information, and uses the limitations of the network to solve the optimal allocation problem of information. Network flows are widely used in transportation problems, scheduling problems, production scheduling problems, etc.