This blog post talks about some of my learnings from the book “Mathematical Notation”

If one is even reasonably curious about the way our world works, one often encounters formulas, describing the universe. These formulas encapsulate a specific view of the world, and the terminology used is, model. If you are a bit more curious about any model, it is inevitable that you will encounter the assumptions behind the model, the various mathematical steps behind the formulation of the model and also possible usages of the model in practical applications; all of these are represented in mathematical notation. Most of us learn mathematical notation by reading, writing and doing math. Over a period of time, one becomes proficient at recognizing math symbols, working with math symbols and writing these symbols via $\LaTeX$ or other softwares that enable to write math notation. You might be familiar with a certain set of symbols more than other, depending on the domain you are exposed on. Do you need a formal book for learning notation ? Not necessarily. You will learn them anyway. Having said that, if you want an overview of most of the math notation you will ever come across, this little book serves as a fantastic resource. The following are some of my learnings from the book:

Named functions(such as those from trigonometry) are written as short words and written in roman; $\cos(x), \log(y), \det(A)$
Why do we see accents and decorations on symbols ? Bold alphabets are difficult to write on blackboard and hence one resorts to accents such as $\hat x$, $\tilde x$, $\overline x$
A way to write bold letters by hand is to double some portion of the shape of the letter; for example, the real numbers represented by $\mathbb R$
Why Greek Alphabets ? Somehow the 52 upper and lower case Latin letters, in various font styles(italic, bold, script) are not sufficient and hence letters from other alphabets are used. The most common choice is Greek alphabets
A variety of special notation is used to indicate disjoint union of sets; for example $\dot{\bigcup}$, $ \bigsqcup $, $ ⊕ $
Set exponentiation: If $A$ and $B$ are sets, then $B^A$ stands for the set of all functions from $A$ to $B$

$$ B^A = \{ f| f:A \to B \} $$

Difference between Scientific notation and Engineering notation
- Scientific notation: the number before the power of 10 is usually at least 1 and less than 10
- Engineering notation: the number before the power of 10 is usually at least 1 and less than 1000
- $6.022 \times 10^{23}$ is via Scientific notation. $602.2 \times 10^{21}$ is via Engineering notation
Electric engineers use $i$ to represent current, and do they use the letter $j$ to represent $\sqrt -1$. For the, complex numbers are written as $a + bj$
Modular numbers: For an integer $n\geq 2$, we write $\mathbb Z_n$ to stand for the set $\{0,1, \ldots , n-1 \}$
In the realm of real numbers, $\mathbb R$ denotes the set of extended real numbers which includes the additional values $+\infty, -\infty$.
The cardinality of integers is represented by $\aleph_0$
A function whose inputs and outputs are themselves functions is often called an operator or a transform. The application of an operator to a function might omit the parenthesis: $Lf$ denotes the evaluation of the operator $L$ on the function $f$
The notation $f(\cdot)$ is sometimes used to emphasize that $f$ is a function with the dot showing than an argument is expected.
$\lg$ stands for $\log_2 x$
A rational function denoted by $\mathbb R(x)$ is the ratio of two polynomials

$$ f(x) = { a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 \over b_m x^m + m_{m-1} x^{m-1} + \ldots + b_0 } $$

Chebyshev Polynomials
- degree-$n$ Chebyshev polynomial of the first order is denoted by $T_n(x)$; $T_n(\cos \theta) = \cos(n \theta)$
- degree-$n$ Chebyshev polynomial of the second order is denoted by $U_n(x)$;$U_n(\cos \theta)\sin \theta = \sin((n+1) \theta)$
Hermite Polynomials - $H_n(x)$
Laguerre Polynomials - $L_n(x)$
Legendre Polynomials - $P_n(x)$
The class $C^\infty$ contains function for which all orders of derivatives exist
$L^p$ denotes the functions for which the following inequality holds

$$ \int^\infty_{-\infty} |f(x)|^p dx < \infty $$

$L^p[a,b]$ denotes the functions for which the following inequality holds

$$ \int^b_{a} |f(x)|^p dx < \infty $$

$\ell^p$ denotes the functions for which the following inequality holds

$$ \sum^\infty_0 |a_k|^p < \infty $$

$C^k$ denotes the class of functions for which the $k^{\text{th}}$ derivative exists and is continuous
Multichoose notation

$$ {\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{n}{k}\right)\kern-.3em\right)} $$

tiny arrow on top of the vector is called harpoon
In $\mathbb R^n$, the standard basis vectors are denoted by $e_1, e_2, \ldots, e_n$
$\delta(i,j)$ denoted Kronecker’s delta and takes a value of $1$ if $i=j$, else $0$
$A ˆ B $ - Hadamard product
Matrix of all ones $J_{m\times n}$
$A^H$ - Conjugate transpose of a matrix. Also denoted by $A^\ast$
$A = A^H$ hermitian matrix
$A \otimes B$ - Tensor product
$A \oplus B$ - Kronecker’s sum / Direct sum
Matrix norm represented with triple vertical line

$$ \lvert \hskip -1pt \lvert \hskip -1pt \lvert A \rvert \hskip -1pt \rvert \hskip -1pt \rvert $$

Conditional number $\kappa(A)$
Spectral radius $\rho(A)$
Operator norm : Every norm on $\mathbb C^n$ induces a matrix norm on $\mathbb C^{n \times m}$. In particular, for the $p$ norms, we have

$$ \lvert \hskip -1pt \lvert \hskip -1pt \lvert A \rvert \hskip -1pt \rvert \hskip -1pt \rvert_p = \max_{x:||x||_p=1} ||Ax||_p $$

Newton’s notation for derivative $f^\prime$
Leibniz’s notation for derivative ${df \over dx}$
Laplacian operator is denoted by $\Delta$
Curl of a function - $\nabla \times f$
Divergence $\nabla \cdot f$
When the subscript on the integral is a curve, then the integral denotes the line integral of $f$ along the curve $\gamma$.

$$ \int_\gamma f,ds $$

If the curve is closed, then the integral along the curve is denoted as

$$ \oint_\gamma f,ds $$

In some disciplines, $\langle X \rangle$ is used to denote the expected value of $X$
The symbol $\sim$ means asymptotic and asserts that the limit of the ratio of the two expressions approaches 1
One also tends to see $\propto$ used to indicate approximate sense, in which the lower order terms are neglected $ {n \choose 3} ∝ n^3$
The symbol $\asymp$ expressed a notation of approximate equality. The ratio between the two expressions need not tend to a limit; all that is required is that the ratio be bounded away from 0 and $\infty$. More precisely, the expression $f \asymp g$ means that there are positive numbers $t,a and b$ such that for all $x\geq t, a \leq f(x)/g(x) \leq b$
Big O $O(g(x))$ notation has two main areas of application:
- In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion
- In computer science, it is useful in the analysis of algorithms
Big O - The meaning of the notation $f(x) = O(g(x))$ means that that the ratio of two functions $|f(x)/g(x)| \leq M$,i.e. there is a upper bound
Big Omega - The meaning of the notation $f(x) = \Omega(g(x))$ means that that the ratio of two functions $|f(x)/g(x)| \geq m$,i.e. there is a lower bound
Theta - The meaning of the notation $f(x) = \Theta(g(x))$ means that that the ratio of two functions $m \leq |f(x)/g(x)| \leq M$. This implies that $f(x) = O(g(x))$ and $f(x) = \Theta(g(x))$
Little o notation - The notation $f(x) = o(g(x))$ indicates that $f(x)$ is very much smaller than $g(x)$ in the sense that the ratio tends to zero. The precise meaning on whether it is small $x$ or large $x$ depends on the context
Little omega notation - The notation $f(x) = \omega(g(x))$ means that $f(x)$ is much larger than $g(x)$. This means that, if $f(x) = \omega(g(x))$

$$ lim_{x \to 0} | {f(x) \over g(x)} | =\infty $$ OR $$ lim_{x \to \infty} | {f(x) \over g(x)} | =\infty $$

Takeaway

If you start learning English grammar, you will have a better sense of appreciation of any written English text. In the same way, if you are familiar with symbols and notation, you will develop a sort of cognitive friendship with these symbols. That will only help you develop a richer understanding of the subject. By spending a few hours on this book, a reader will walk away with a good understanding of a broad set of math symbols and notation that one might come across in books, papers, articles. Superbly written book.