These are some examples of the use of MathJax to convert $\LaTeX$ into nice mathematics that can be embedded in an HTML document/file. They are here for me to reference later as I begin to use this feature more often.
Here is some text with some math in it
$$\sqrt{\alpha^2} = |\alpha|$$
and here is another equation involving vectors:
\[{\bf{v}} \,=\, v_x{\bf{i}} + v_y{\bf{j}}\]
These are inline equation examples: $y = mx + b$, and $\sin^2{\theta} + \cos^2{\theta} = 1$.
What is this? Ah, I see. A test of using brackets in an expression. They need the forward slash in front of them to render properly:
$\{ax + b\}$
How about color, does this render here? It doesn't on Quora:
$\color{red}{y = mx + b}$
What about this from math.stackexchange, can I use align environment?
$$\begin{align}y^T\left(I-\frac{2}{t^2+\|x\|^2}xx^T\right)y&=\frac{t^2\|y\|^2+\|x\|^2\|y\|^2-2\|x^Ty\|^2}{t^2+\|x\|^2}\\&\ge \frac{t^2\|y\|^2+\|x\|^2\|y\|^2-2\|x\|^2\|y\|^2}{t^2+\|x\|^2}\\&>0\end{align}$$
YES!!
************************************
A question from Alon Ray, Tel Aviv University, on ResearchGate:
I am reading a Fritz John's article on ultrahyperbolic PDE, and I am little bit puzzled as to what is a line function.
He states that since $v$ is a line function, then:
$$v(\theta \xi +(1-\theta)\eta , \eta) = v(\xi , \theta \eta+(1-\theta)\xi)=v(\xi ,\eta)$$
where $\xi = (\xi_1,\xi_2,\xi_3) ,\,\, \eta=(\eta_1,\eta_2,\eta_3)$.
Now what in this context does line function mean, so that I could infer the same identities?
Answer:
This is my first attempt to answer a question at ResearchGate, and I noticed immediately that there is apparently no capability to render LaTeX via, for example, MathJax. Has anyone ever asked the admin of RG to make that available? At any rate, Alon, I don't have access to the articles at Duke Mathematical Journal since I am at a small college, so can't see Fritz John's paper. But perhaps you have noticed that his "line function" appears to be related to the parametric equations defining the ${\textit {line segment}}$ joining any two points $\xi$ and $\eta$ in $\mathbb{R}^3$, that is,
$$L(\xi ,\eta) = \{z \in \mathbb{R}^3 \,|\, z = \theta \xi +(1-\theta)\eta, 0 < \theta < 1\}$$
see, for example, ${\textit {Mathematical Analysis}}$, T. M. Apostol, p. 117, 1957. So a line function may simply be any function $v(\xi, \eta)$ ${\textit {defined}}$ on the line segment joining the two points by:
$$v(\xi ,\eta) = v(L(\xi, \eta) , \eta) = v(\xi , L(\eta, \xi))$$
I don't know if this is helpful, but thought it might be worth mentioning.
************************
$${\operatorname{vec}}(ACX) = (C^T\otimes A){\operatorname{vec}}(X)$$
Here is some text with some math in it
$$\sqrt{\alpha^2} = |\alpha|$$
and here is another equation involving vectors:
\[{\bf{v}} \,=\, v_x{\bf{i}} + v_y{\bf{j}}\]
These are inline equation examples: $y = mx + b$, and $\sin^2{\theta} + \cos^2{\theta} = 1$.
What is this? Ah, I see. A test of using brackets in an expression. They need the forward slash in front of them to render properly:
$\{ax + b\}$
How about color, does this render here? It doesn't on Quora:
$\color{red}{y = mx + b}$
What about this from math.stackexchange, can I use align environment?
$$\begin{align}y^T\left(I-\frac{2}{t^2+\|x\|^2}xx^T\right)y&=\frac{t^2\|y\|^2+\|x\|^2\|y\|^2-2\|x^Ty\|^2}{t^2+\|x\|^2}\\&\ge \frac{t^2\|y\|^2+\|x\|^2\|y\|^2-2\|x\|^2\|y\|^2}{t^2+\|x\|^2}\\&>0\end{align}$$
YES!!
************************************
A question from Alon Ray, Tel Aviv University, on ResearchGate:
I am reading a Fritz John's article on ultrahyperbolic PDE, and I am little bit puzzled as to what is a line function.
He states that since $v$ is a line function, then:
$$v(\theta \xi +(1-\theta)\eta , \eta) = v(\xi , \theta \eta+(1-\theta)\xi)=v(\xi ,\eta)$$
where $\xi = (\xi_1,\xi_2,\xi_3) ,\,\, \eta=(\eta_1,\eta_2,\eta_3)$.
Now what in this context does line function mean, so that I could infer the same identities?
Answer:
This is my first attempt to answer a question at ResearchGate, and I noticed immediately that there is apparently no capability to render LaTeX via, for example, MathJax. Has anyone ever asked the admin of RG to make that available? At any rate, Alon, I don't have access to the articles at Duke Mathematical Journal since I am at a small college, so can't see Fritz John's paper. But perhaps you have noticed that his "line function" appears to be related to the parametric equations defining the ${\textit {line segment}}$ joining any two points $\xi$ and $\eta$ in $\mathbb{R}^3$, that is,
$$L(\xi ,\eta) = \{z \in \mathbb{R}^3 \,|\, z = \theta \xi +(1-\theta)\eta, 0 < \theta < 1\}$$
see, for example, ${\textit {Mathematical Analysis}}$, T. M. Apostol, p. 117, 1957. So a line function may simply be any function $v(\xi, \eta)$ ${\textit {defined}}$ on the line segment joining the two points by:
$$v(\xi ,\eta) = v(L(\xi, \eta) , \eta) = v(\xi , L(\eta, \xi))$$
I don't know if this is helpful, but thought it might be worth mentioning.
************************
$${\operatorname{vec}}(ACX) = (C^T\otimes A){\operatorname{vec}}(X)$$