Mathematics

I did not post anything for more than one month. I was having some problems. Maybe I lost my aim, maybe lost myself and maybe lost in myself. However, I have made up my mind to go on. I will continue to contribute and will ultimately finish Analysis I. Of course, I will then start to cover Analysis II. However, I am here and will be here for myself and most importantly for my small amount of readers. I put my anxiety and stress aside and cleared my mind. La mort du loup and then birth from ashes like a phoenix. Yes, it is what it is like to exist. Being in the world feels strange somehow, and writing digitally or conventionally is the cure to this. Writing anything. Any ink particule. In particular, writing with a fountain pen. It restricted me to write gibberish, to erase my mistakes and allowed me to think deeply, a little advise so to say. ...

Bernoulli Differential Equations If the differential equation is in the form of $$ \begin{align} y^{\prime}+p(x)y=q(x)y^n \end{align} $$ $$ \begin{align} y^{\prime}+p(x)y=q(x)y^n \end{align} $$ If $n=0$, then the differential equation is linear differential equation: $$ \begin{aligned} y^{\prime}+p(x)y=q(x) \end{aligned} $$ $$ \begin{aligned} y^{\prime}+p(x)y=q(x) \end{aligned} $$ If $n=1$, then the differential equation is separable differential equation: $$ \begin{aligned} y^{\prime}+p(x)y &=q(x)y\\ \dfrac{dy}{y} &= (q(x)-p(x))dx \end{aligned} $$ $$ \begin{aligned} y^{\prime}+p(x)y &=q(x)y\\ \dfrac{dy}{y} &= (q(x)-p(x))dx \end{aligned} $$ If $n \neq 1, 0$, then it is said to be Bernoulli’s Differential Equation in $y$. The differential equation is neither separable, homogenous nor linear differential equation for $n \neq 0, 1$. ...

The motivation of this post is to enlarge the steps of theorems and the definition of relativeness, compactness and more. Since the famous and brilliant book Principles of Mathematical Analysis by Walter Rudin is so terse so to say. I love “Baby Rudin”, but unfortunately it is hugely incomprehensible if you do not know what you are doing. Rudin, of course, had deliberately skipped over the details. He probably guessed that the reader was quite gifted in mathematics or the reader is taught by a brilliant teacher. ...

Consider the initial value problem (IVP) $$ \begin{aligned} x=x(t), x_0 = x(t_0) \text{ for } t \geq t_0 \end{aligned} $$ $$ \begin{aligned} x=x(t), x_0 = x(t_0) \text{ for } t \geq t_0 \end{aligned} $$ where $$ \begin{aligned} (x, t) : \vert t - t_0\vert < a, \vert x - x_0\vert < b \end{aligned} $$ $$ \begin{aligned} (x, t) : \vert t - t_0\vert < a, \vert x - x_0\vert < b \end{aligned} $$ this means that the pairs $(x, t)$ satisfies the inequalities $\vert t - t_0\vert < a, \vert x - x_0\vert < b$ for $a,b > 0$. For the sake of understanding we can say $t_0$ is initial time and $x_0$ is initial position. Therefore $(x_0, t_0)$ or $(x(t_0), t_0)$ is initial condition. In this case $t_0$ is independent variable and $x$ is dependent variable (depends on $t$). ...

Prerequisites: metric spaces $\textbf{\small Definition 1.1.:}$ (Neighbourhood) Let $X$ be a metric space. A neighbourhood of a point $p$ is a set $N_r(p)$ consisting of all points $q$ such that $d(p,q) < r$. The number $r$ is called the radius of $N_r (p)$. $$ \begin{aligned} N_r(p) = \{q : d(p,q) < r \text{ where } p,q \in X\} \end{aligned} $$ $$ \begin{aligned} N_r(p) = \{q : d(p,q) < r \text{ where } p,q \in X\} \end{aligned} $$ If we are in an Euclidean metric $(\mathbb{R}^k)$ then, ...

I was freshman when I was first introduced supremum and infimum. I could not understand a thing, yet everybody seemed to understand. I felt like stupid, inadequate and also insufficient person to learn mathematics. However, reality is not like that. In truth, I was the one to blame; I hadn’t dived deep into these topics deeply enough. I do not know if I am alone or not, but I think supremum and infimum are the topics which cannot by bypassed without learning and embracing. Today I want to show an interesting fact; if a set has least-upper-bound property and bounded below, then it has infimum. But first let us define supremum and infimum. ...

In this post we will continue to investigate more theorems of convergence tests. Tests D’Alembert Ratio Test $\textbf{\small Theorem 1.17.:} $ Let $\sum_{n=1}^{\infty} a_n$ be positive series and let $l = \lim_{n\rightarrow \infty}\frac{a_{n+1}}{a_n}$. Then, series converges if $l < 1$, series diverges if $l > 1$, and gives no information if $l = 1$ or limit fails to exist. $\textbf{\small Proof:} $ Let $l < 1$. We want to prove that $\sum_{n=1}^{\infty} a_n$ converges. Let us write what we know from assumption. Choose $\varepsilon > 0$ and $\forall n > N$ and $0 < (l + \varepsilon) < 1$: ...

Telescoping Series In this post, we will learn more about infinite series. We shall start with so-called telescoping series, that has a form of $$ \begin{aligned} \sum_{k=1}^{\infty} (a_n - a_{n+1}) \end{aligned} $$ $$ \begin{aligned} \sum_{n=1}^{\infty} (a_n - a_{n+1}) \end{aligned} $$ Let us investigate this sum step by step. Observe that the partial sum $s_n$ can be written as $$ \begin{aligned} s_n = a_1 - a_2 + a_2 - a_3 + \dots + a_n - a_{n+1} \end{aligned} $$ $$ \begin{aligned} s_n = a_1 - a_2 + a_2 - a_3 + \dots + a_n - a_{n+1} \end{aligned} $$ So we can just simplify this ...

By the intuition from our birth, we love summing things, categorising the similar things and so on. Consider a sum $A=1+2+3+4+\dots$ what we have here is infinite sum, as “$\dots$” imply “this sum goes to infinity with the order you see”. So in a some sense, it behaves like sequences, because we write the things in order like $1$, $2$, $3$, $4$, $\dots$ and then we sum them. Mathematicians tought that so, and as a result they came up with a new topic called “series”, especially “infinite series”. Spoiler alert, an infinite series can converge, if not we say series is divergent. ...

First Initiation When we first started to learn mathematics, our concern is only to solve problems, understand what equation means, comprehend what teacher says during class and etc. At high school we learn more about sets and functions. However, if we are not curious enough, then the philosophical meaning of the word “mathematical space” is not emphasized. When $f: \mathbb{R} \rightarrow \mathbb{Q}$ is written, everybody who has learned high school math can say “the function takes values from $\mathbb{R}$ and maps it to $\mathbb{Q}$. But have people reading this ever thought about the background? ...