What kind of math notation is this?

[muffinman123]

what kind of math language is this?

$I\subset \reals$
$$\gamma_i:I_i\rightarrow M,\quad I_i\subset\reals,\quad i=1,2$$
$\sigma_{\alpha\beta}$ , let $$\rJ\sigma_{\alpha\beta}:V_{\alpha\beta}\rightarrow \Mat_{n,n}(\reals)$$


I have seen this notation thrown around in this forum, but I never understood what they mean.

 

[slider142]

This is index notation in order to express concisely a list of many equations or functions.
For example, the first line that you write states that there are two distinct functions γ1 and γ2 such that γ1 maps I1 into M where I1 is a subset of the real numbers and γ2 maps I2 into M where I2 is a subset of the real numbers.
The arrow notation defines the domain and codomain of the function: I1 is the domain of γ1 and M is the codomain.
The second set of notation has two indexes on each object; each index is taken to vary independently. For example, if the restrictions on α and β were explicitly given as α = 1, 2 and β = 1, 2, then the expression is a concise way of expressing the following list of expressions:
$$\text{For }\sigma_{11},\text{ let }J\sigma_{11} : V_{11}\rightarrow\text{Mat}_{n, n}(\Re)$$
$$\text{For }\sigma_{12},\text{ let }J\sigma_{12} : V_{12}\rightarrow\text{Mat}_{n, n}(\Re)$$
$$\text{For }\sigma_{21},\text{ let }J\sigma_{21} : V_{21}\rightarrow\text{Mat}_{n, n}(\Re)$$
$$\text{For }\sigma_{22},\text{ let }J\sigma_{22} : V_{22}\rightarrow\text{Mat}_{n, n}(\Re)$$
In words, the sentence defines a list of 4 functions Jσ_ab that maps each respective space V_ab into the set of nxn matrices with real components. In particular, this means that Jσ_12 takes an element of V_12 as an input and returns an nxn matrix with real components as an output. The explicit operation performed by the function on those elements of V_12 is not specified in this expression.

 

[muffinman123]

alright, let me make this question simpler, what language is this?
is it latex?

 

[slider142]

Yes. You can learn more about how it is used on this forum here.

 

[CellCoree]

Can you explain it to me?

The notation provided is a mixture of set theory, calculus, and linear algebra notation. The first line, $I\subset \reals$, indicates that the set $I$ is a subset of the set of real numbers, denoted by $\reals$. The next line introduces a function $\gamma_i$ that maps a subset $I_i$ of the real numbers to a set $M$. The notation $I_i\subset\reals$ specifies that $I_i$ is a subset of the real numbers, and the notation $i=1,2$ indicates that there are two separate functions $\gamma_1$ and $\gamma_2$ being discussed.

The next line introduces a symbol $\sigma_{\alpha\beta}$, which could represent a variety of mathematical objects such as a matrix, a function, or a set. Without more context, it is difficult to determine the exact meaning of this notation. The final line introduces a map $\rJ\sigma_{\alpha\beta}$ that takes a set $V_{\alpha\beta}$ and maps it to the set of $n\times n$ matrices with real entries, denoted by $\Mat_{n,n}(\reals)$.

Overall, this notation is a way to represent mathematical concepts and relationships using symbols and mathematical notation. It is important to have a solid understanding of mathematical notation in order to effectively communicate and understand mathematical concepts.