The maximal solution to Cauchy’s problems is a solution that gives full information on the physical model. In fact, we obtain this solution by extending local solutions. In this post, we give some theorem that shows the existence of the uniqueness of the maximal solution. What is the maximal solution to Cauchy’s problems? Let $f: … Read more
In this post, we give selected algebra questions with answers. The exercises focus on group theory and arithmetic. In fact, these types of algebra exercises are classic and students need to learn them. A selection of algebra questions with answers In this section, we shall give various algebra questions with answers. An algebraic equation in … Read more
The delay equations are an important class of differential equations. In fact, these equations are studied by both mathematicians and engineers because most systems are affected by a delay. Imagine a basketball game transmission from Italy to the United States. Sometimes a lag between the voice and the television images occurs. Indeed, the voice arrives … Read more
We offer exercises on polynomials with detailed proofs. In fact, our goal is to show the student how to calculate the greatest common divisor of two polynomials, and how to use the Bezout relation between polynomials. Definition and properties of a polynomial In the sequel, $\mathbb{K}$ is a commutative ring. A polynomial $P$ is an … Read more
We show you how to calculate integrals using elementary methods. In addition, we teach you to study the properties of functions defined by an integral. It is very important to be able to calculate an integral easily because it interferes in the study of differential equations another important subject of mathematical analysis. Exercises on how … Read more
We discuss some facts about first-order differential equations for beginners. Such equations are important because many problems in our real life can be modeled as a differential equation. We assume that the reader is familiar with the concept of continuous function primitives. First-order differential equations with constant coefficients In algebra, we already studied algebraic equations … Read more
In this article, we state and prove the Cauchy-Lipschitz theorem for the existence and uniqueness of solutions to nonlinear ordinary differential equations. The key proof of this theorem is the Banach-Picard fixed point theorem. We give some applications of this theorem. Local and maximal solutions to nonlinear Cauchy problems Throughout this section, $I$ is an … Read more
We show how Binomial coefficients help simplify expressions. We notice that these coefficients are heavily used in probability calculus. On the other hand, the Binomial theorem serves in computing powers of numbers. The expression of Binomial Coefficients For natural numbers $n,k\in\mathbb{N}$ uch that $n\ge k$, we define the Binomial coefficient by \begin{align*} \binom{n}{k}=\frac{n!}{k!(n-k)!}.\end{align*} The binomial … Read more
Complex numbers are a fundamental concept in mathematics that extends the realm of real numbers. They are composed of a real part and an imaginary part, where the imaginary part is a multiple of the imaginary unit, denoted by “i”. What is a complex number? Complex numbers are represented in the form $a + bi$ … Read more
We teach you how to use mathematical induction to prove algebraic properties. This technique is very useful and simple to use. We offer examples and exercises to help you understand proofs by induction. Induction reasoning is often used to prove sequence properties. Learn about how to use mathematical induction In many mathematical situations, we need … Read more
In this article, we show a list of courses to learn advanced mathematics. Although, We offer tips and techniques that can help you overcome almost any of your math hurdles. In fact, We give courses and solved exercises on general algebra, (groups, rings, and fields; linear algebra, vector spaces, dimension, matrices, endomorphisms, eigenvalues, stables vector … Read more
Elementary mathematics refers to courses in categories of numbers, inequalities, algebraic equations, calculus-like sequences, functions, integrals, and differential equations. If the student manages to accumulate a solid foundation in this type of mathematics, then he can study advanced mathematics without any problem. This is why countries like the United States pay more attention to elementary … Read more