Version 3 of the Open Font Format was officially published as ISO standard early this month. One of the interesting new feature is that Microsoft's MATH table has been integrated into this official specification. Hopefully, this will encourage type designers to create more math fonts and OS vendors to integrate them into their systems. But are browser vendors ready to use Open Font Format for native MathML rendering? Here is a table of important Open Font Format features for math rendering and (to my knowledge) the current status in Apple, Google, Microsoft and Mozilla products.

As announced in a previous blog post, I was invited to two Mozilla Work Weeks in Toronto and Whistler during the month of June. Before these work weeks, the only assistive technology able to read MathML in Gecko-based browsers was NVDA, via the help of the third-party MathPlayer plugin developed by Design Science, as shown in the following video:

Thanks to the effort done during these work weeks plus some additional days, we have made good progress to expose MathML via accessibility APIs on other platforms: Mac OS X, Linux, Android and Firefox OS. Note that Firefox for iOS uses WebKit, so MathML should already be exposed and handled via Apple's WebKit/VoiceOver. If you are not familiar with accessibility APIs (and actually even if you are), I recommend you to read Marco Zehe's excellent blog post about why accessibility APIs matter.

Apple was the first company to rely on accessibility APIs to make MathML accessible: WebKit exposes MathML via its NSAccessibility protocol and it can then be handled by the VoiceOver assistive technology. One of the obvious consequence of working with open standards and open accessibility APIs is that it was then relatively easy for us to make MathML accessible on Mac OS X: We basically just read the WebKit source code to verify how MathML is exposed and did the same for Gecko. The following video shows VoiceOver reading a Wikipedia page with MathML mode enabled in Gecko 41:

Of course, one of the disadvantage is that VoiceOver is proprietary and so we are dependent on what Apple actually implements for MathML and we can not easily propose patches to fix bugs or add support for new languages. This is however still more convenient for users than the proprietary MathPlayer plugin used by NVDA: at least VoiceOver is installed by default on Apple's products and well-integrated into their user & accessibility interfaces. For instance, I was able to use the standard user interface to select the French language in VoiceOver and it worked immediately. For NVDA+MathPlayer, there are several configuration menus (one for the Windows system, one for NVDA and one for MathPlayer) and even after selecting French everywhere and rebooting, the math formulas were still read in English...

The next desktop platform we worked on was Linux. We continued to improve how Gecko expose the MathML via the ATK interface but the most important work was done by Joanmarie Diggs: making Orca able to handle the exposed MathML accessibility tree. Compared to the previous solutions, this one is 100% open and I was happy to be able to submit a couple of patches to Orca and to work with the Gnome Translation Team to keep the French translation up-to-date. By the way, if you are willing to contribute to the localization of Orca into your language feel free to join the Gnome Translation Project, help will be much appreciated! The following video shows how Orca reads the previous Wikipedia page in Nightly builds:

On mobile platforms (Android and Firefox OS) we use a common Javascript layer called AccessFu to handle Gecko's internal accessibility tree. So all of this is handled by Mozilla and hence is also 100% open. As I said in my previous blog post, I was not really aware of the internal details before the Work Weeks so it was good to get more explanations and help from Yura Zenevich. Although we were able to do some preliminary work to add MathML support to AccessFu in bug 1163374, this will definitely need further improvements. So I will not provide any demo for now :-)

To conclude this overview, you can check the status of accessibility on the Mozilla MathML Project page. This page also contains a table of MathML tests and how they are handled on the various platforms. At the end of September, I will travel to Toronto to participate to the Mozilla and FOSS Assistive Technology Meetup. In particular, I hope to continue improvements to MathML Accessibility in Mozilla products... Stay tuned!

In a previous blog post about MathML in Wikipedia, I mentioned that, despite ongoing efforts there was still no accessibility support for MathML in Gecko. The situation changed two months ago: Design Science and NV Access released new versions of MathPlayer and NVDA, which in particular add MathML accessibility support on Windows, as shown in the demos below. This is exciting news and I am really willing to see this support extended to other platforms...

Last December, I also met Joanmarie Diggs at the Web Engines Hackfest and we have been able to start some preliminary work for Linux (WebKit/Gecko/Orca). I had the opportunity to refresh some of the patches written by Jonathan Wei during a Mozilla internship and to get part of his work landed into trunk. I have also made basic improvements to how we expose the accessible tree for ATK in order to prepare future support in Orca. It is certainly too early to announce anything. Just as a comparison, I also provide how Orca currently (badly) reads the MathML formulas below.

MathML accessibility support is also available in the latest versions of Safari+VoiceOver. So in theory, we "only" need to make Gecko expose the same accessible tree as WebKit in order to support the Mac platform. Jonathan Wei had a work-in-progress patch for that, see bug 1001641. Since it is far from being ready, I will cheat a bit and just show how VoiceOver reads the MathML examples in Safari.

Finally, the mobile platforms (Firefox OS and Android) are also very important. So far, I have only submitted some patches to make the GeckoView accessible and to fix some other small accessibility bugs. So I am interested in hearing more from Mozilla developers about the AccessFu stuff and how we could make MathML accessible on these platforms.

Audio Demos

The table below contains some concrete examples taken from Wikipedia (in MathML mode), Mozilla Developer Network, KaTeX and MathJax. Note that at the moment, MathJax MathML formulas are not exposed to all assistive technologies. I recommend to force native MathML using an add-on for Gecko browsers or Safari ; or to use this GreaseMonkey script.

All of these developments are still in progress and there are certainly many bugs to fix and improvements to do. Next month, I expect to have several opportunities to meet people and make progress on MathML. For people interested to help, here is my schedule:

On the 8th, I'll attend the 9th European e-Accessibility Forum in Paris where Neil Soiffer (Design Science) will present "The State of Accessible Math".

From the 11th to 22th, I'll stay in the Montréal area. I hope I'll have the chance to meet Joanmarie and the accessibility team from Mozilla Toronto during that time frame (to be confirmed).

Last but not least, I'm one of the lucky Mozilla volunteer invited to Whistler's work week from the 22th to 26th!

As mentioned in a previous blog post, I've been working on the exercises of chapter 1 of Kunen's Set Theory book. I finally uploaded my solutions this morning.

In a my previous blog post, I discussed the canonical well-ordering on
$\alpha \times \alpha $ and stated theorem 0.5 below to calculate its
order-type $\gamma (\alpha )$. Subsequent corollaries provided a bound for
$\gamma (\alpha )$, its fixed points and a proof that infinite cardinals were
among these fixed points (and so that cardinal addition and multiplication
is trivial).
In this second part, I’m going to provide the proof of this theorem.

First, we note that for all $$, there is only one order-type for
$n\times n$, since
the notion of cardinal and ordinal is the same for finite sets.
So indeed

$$

and taking the limit, we get our first infinite fixed point:

$$

For all $\alpha $ the ordering on
$(\alpha +1)\times (\alpha +1)$ is as follows: first the
elements of $\alpha \times \alpha $ ordered as $\gamma (\alpha )$, followed by
the elements $(\xi ,\alpha )$ for $$, followed by
the elements $(\alpha ,\xi )$ for $0\le \xi \le \alpha $. Hence

The important point is $1+\omega =\omega $ ; in general we will use
several times the property
$\alpha +{\omega}^{\beta}={\omega}^{\beta}$ if $$.
By a simple recurrence (see proposition 0.1),
we can generalize the expression to arbitrary $n$:

$$\gamma (\omega +n)=\omega \left(2n+1\right)+n$$

and so taking the limit

$$\gamma (\omega \cdot 2)={\omega}^{2}$$

which is a limit ordinal not fixed by $\gamma $.
We note another point that will be used later: taking the limit
“eliminates” the smallest terms.
More generally, we can perform the same calculation, starting from any
arbitrary $\alpha \ge \omega $:

As above, we can take the limit and say
$$ which is consistent with
$\gamma (\omega \cdot 2)=\omega +{\omega}^{2}={\omega}^{2}$. However, if we
consider the Cantor Normal Form
$\alpha ={\omega}^{{\beta}_{1}}{n}_{1}+\mathrm{\dots}+{\omega}^{{\beta}_{k}}{n}_{k}$,
then for all $$ we have can use the fact that “${\omega}^{{\beta}_{1}}$ will
eliminate smaller terms on its left” that is
$\alpha \cdot n={\omega}^{{\beta}_{1}}({n}_{1}n)+{\omega}^{{\beta}_{2}}{n}_{2}+\mathrm{\dots}+{\omega}^{{\beta}_{k}}{n}_{k}$. Then using the
fact that “taking the limit eliminates the smallest terms” we get
$$. So actually, we have a nicer formula where
$\alpha \cdot \omega $ is put in Cantor Normal Form:

This can be generalized by the following proposition:

Proposition 0.2.

For any $\alpha \mathrm{\ge}\omega $ and
$\beta \mathrm{\ge}\mathrm{1}$ such that ${\mathrm{log}}_{\omega}\mathit{}\mathrm{(}\alpha \mathrm{)}\mathrm{+}\mathrm{1}\mathrm{\ge}\beta $ we have

We prove by induction on $\beta $ that for all such $\alpha $
the expression is true. We just verified $\beta =1$ and the
limit case is obvious by continuity of $\gamma $ and of the sum/exponentiation
in the second variable. For the successor step, if
${\mathrm{log}}_{\omega}(\alpha )+1\ge \beta +1$ then
a fortiori
$$.
We can then use the induction hypothesis to prove by induction on
$$ that
$\gamma (\alpha +{\omega}^{\beta}\cdot n)=\gamma (\alpha )+{\omega}^{{\mathrm{log}}_{\omega}(\alpha )+\beta}\cdot n$. For $n=1$, this is
just the induction hypothesis of $\beta $ (for the same $\alpha $!). For the
successor step, we need to use the induction hypothesis of $\beta $
(for $\alpha +{\omega}^{\beta}\cdot n$) which is
$\gamma (\alpha +{\omega}^{\beta}\cdot (n+1))=\gamma (\alpha +{\omega}^{\beta}\cdot n)+{\omega}^{{\mathrm{log}}_{\omega}(\alpha )+\beta}$.
Finally,
$$, as wanted.
∎

For all $\alpha \ge 1$,
${\mathrm{log}}_{\omega}({\omega}^{\alpha})+1=\alpha +1$ so the previous
paragraph also gives
$\gamma ({\omega}^{\alpha +1})=\gamma \left({\omega}^{\alpha}+{\omega}^{\alpha +1}\right)=\gamma ({\omega}^{\alpha})+{\omega}^{\alpha \cdot 2+1}$.
Then, we find

And more generally by induction on $$, we can show that

$$\gamma ({\omega}^{n+1})={\omega}^{2n+1}$$

Then we deduce another fixed point

$$

The following proposition tries to generalize the expression of
$\gamma ({\omega}^{\alpha +1})$.

Proposition 0.3.

For any $\alpha \mathrm{,}\beta \mathrm{\ge}\mathrm{1}$ such that
${\mathrm{log}}_{\omega}\mathit{}\mathrm{(}\alpha \mathrm{)}\mathrm{>}{\mathrm{log}}_{\omega}\mathit{}\mathrm{(}\beta \mathrm{)}$ we have

This is done
by induction on $$ for a fixed $\alpha $. We already verified
the case $\beta =1$ in the previous paragraph and the limit case is obvious
by continuity of $\gamma $ and of the sum/exponentiation in the second variable.
For the successor step, we have
$\gamma ({\omega}^{\alpha +\beta +1})=\gamma ({\omega}^{\alpha +\beta})+{\omega}^{(\alpha +\beta )\cdot 2+1}$ and by induction
hypothesis, $\gamma ({\omega}^{\alpha +\beta})=\gamma ({\omega}^{\alpha})+{\omega}^{\alpha \cdot 2+\beta}$.
Since ${\mathrm{log}}_{\omega}(\alpha )>{\mathrm{log}}_{\omega}(\beta +1)={\mathrm{log}}_{\omega}(\beta )$ we have
$(\alpha +\beta )\cdot 2+1=\alpha +\beta +\alpha +\beta +1=\alpha \cdot 2+\beta +1>\alpha \cdot 2+\beta $ and so
${\omega}^{\alpha \cdot 2+\beta}+{\omega}^{\alpha \cdot 2+\beta +1}={\omega}^{\alpha \cdot 2+\beta +1}$.
Finally, $\gamma ({\omega}^{\alpha +\beta +1})=\gamma ({\omega}^{\alpha +\beta})+{\omega}^{\alpha \cdot 2+\beta +1}$ as wanted.
∎

For any $$
and $\alpha \ge 1$, if $$
then $$. Hence
proposition 0.3 gives
$\gamma ({\omega}^{{\omega}^{\alpha}\cdot n+\beta})=\gamma ({\omega}^{{\omega}^{\alpha}\cdot n})+{\omega}^{{\omega}^{\alpha}\cdot 2n+\beta}$
Then by continuity of $\gamma $ and of the sum/exponentiation in the second
variable, we can consider the limit $\beta \to {\omega}^{\alpha}$ to get
$\gamma ({\omega}^{{\omega}^{\alpha}\cdot (n+1)})=\gamma ({\omega}^{{\omega}^{\alpha}\cdot n})+{\omega}^{{\omega}^{\alpha}\cdot (2n+1)}$.
So continuing our calculation we have

From the relation $\gamma ({\omega}^{{\omega}^{\alpha}\cdot (n+1)})=\gamma ({\omega}^{{\omega}^{\alpha}\cdot n})+{\omega}^{{\omega}^{\alpha}\cdot (2n+1)}$,
we deduce by induction on $n$ that

We can then show by induction that all the
${\omega}^{{\omega}^{\alpha}}$ are actually fixed points, using the previous
relation at successor step, the continuity of $\gamma $ at limit step
and the fact that $\gamma (\omega )=\omega $. This means

Equipped with these four propositions, we have a way to recursively calculate
$\gamma $. We are ready to prove the main theorem:

Theorem 0.5.

For all ordinal $\alpha $, we denote $\gamma \mathit{}\mathrm{(}\alpha \mathrm{)}$ the order-type of
the canonical ordering of $\alpha \mathrm{\times}\alpha $. Then $\gamma $ can be
calculated as follows:

(1)

Finite Ordinals:
For any $$ we have

$$\gamma (n)={n}^{2}$$

(2)

Limit Ordinals:
For any limit ordinal $\alpha $,

(a)

If ${\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}$ does not
divide ${\mathrm{log}}_{\omega}(\alpha )$ then

(like the first case but we “decrement $n$ in the second factor”)

(c)

Otherwise, $\alpha ={\omega}^{{\mathrm{log}}_{\omega}(\alpha )}+\rho $ and
we write ${\mathrm{log}}_{\omega}(\alpha )={\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}m$
for some $m\ge 1$. We have

where $\gamma (\alpha )$ is determined as in the previous point.

Proof.

The “Finite Ordinals” has been discussed at the beginning and the
“Infinite Successor Ordinals” is proposition 0.1.
Now let’s consider the Cantor Normal Form
${\omega}^{{\beta}_{1}}n+\mathrm{\dots}+{\omega}^{{\beta}_{k}}{n}_{k}$ of a limit ordinal
$\alpha \ge \omega $ (so ${\beta}_{k}\ge 1$ and
${\beta}_{1}={\mathrm{log}}_{\omega}(\alpha )$). First,
from proposition 0.2 we can make successively
extract the ${n}_{k}$ terms ${\omega}^{{\beta}_{k}}$ (by left-multiplying them
by ${\omega}^{{\beta}_{1}}$), then the ${n}_{k-1}$ terms ${\omega}^{{\beta}_{k-1}}$, …
then the ${n}_{2}$ terms ${\omega}^{{\beta}_{2}}$ and finally
$n-1$ terms ${\omega}^{{\beta}_{1}}$. We obtain:

We now write ${\beta}_{1}={\omega}^{\delta}m+\sigma $ where
$\delta ={\mathrm{log}}_{\omega}{\beta}_{1}$ and $m,\sigma $ are the quotient and remainder
of the Euclidean division of ${\beta}_{1}$ by ${\omega}^{\delta}$. We can then use
proposition 0.3 to extract $\sigma $:

$\sigma \ne 0$ means that
${\omega}^{\delta}={\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}$
does not divide
${\beta}_{1}={\mathrm{log}}_{\omega}(\alpha )$. In that case,
${\omega}^{{\omega}^{\delta}\cdot (2m)+\sigma}>{\omega}^{{\omega}^{\delta}\cdot (2m-1)}$
and so
$\gamma ({\omega}^{{\beta}_{1}})={\omega}^{{\omega}^{\delta}\cdot (2m)+\sigma}$.
We note that ${\beta}_{1}\cdot 2={\omega}^{\delta}m+\sigma +{\omega}^{\delta}m+\sigma ={\omega}^{\delta}(2m)+\sigma $ since the remainder $\sigma $ is less than
${\omega}^{\delta}$. So actually $\gamma ({\omega}^{{\beta}_{1}})={\omega}^{{\beta}_{1}}{\omega}^{{\beta}_{1}}$. Coming back to the expression
of $\gamma (\alpha )$, this term can be grouped with
${\omega}^{{\beta}_{1}}(n-1)$ to
recover the Cantor Normal Form of $\alpha $ and
we finally get $\gamma (\alpha )={\omega}^{{\beta}_{1}}\alpha $.

Otherwise, $\sigma =0$, ${\beta}_{1}={\omega}^{\delta}m$ and

But then ${\beta}_{1}={\omega}^{\delta}m>{\omega}^{\delta}(m-1)$ and so
${\omega}^{{\beta}_{1}}{\omega}^{{\beta}_{1}}>{\omega}^{{\beta}_{1}}{\omega}^{{\omega}^{\delta}\cdot (m-1)}$. Hence if
$n\ge 2$, the term
$\gamma ({\omega}^{{\beta}_{1}})$ is eliminated in the expression of $\gamma (\alpha )$
and it remains

where $\rho ={\omega}^{{\beta}_{2}}{n}_{2}+\mathrm{\dots}+{\omega}^{{\beta}_{k}}{n}_{k}$. If instead $n=1$, then the term
${\omega}^{{\beta}_{1}}(n-1)$ is zero and it remains

It is well-known that the cartesian product of two infinite sets of cardinality ${\mathrm{\aleph}}_{\alpha}$ is also of cardinality ${\mathrm{\aleph}}_{\alpha}$. Equivalently, the set ${\omega}_{\alpha}\times {\omega}_{\alpha}$ can be well-ordered in order-type ${\omega}_{\alpha}$. However, the standard ordering on ${\omega}_{\alpha}\times {\omega}_{\alpha}$ does not work, since it is always of order type ${\omega}_{\alpha}^{2}$. Instead, we introduce for each ordinal $\alpha $ the canonical well-ordering of $\alpha \times \alpha $ as follows: $({\xi}_{1},{\xi}_{2})\u22b2({\eta}_{1},{\eta}_{2})$ if either $$ or $max({\xi}_{1},{\xi}_{2})=max({\eta}_{1},{\eta}_{2})$ but $({\xi}_{1},{\xi}_{2})$ precedes $({\eta}_{1},{\eta}_{2})$ lexicographically. If we note $\gamma (\alpha )$ the ordinal isomorphic to that well-ordering ordering, then we can prove that $\gamma ({\omega}_{\alpha})={\omega}_{\alpha}$ as wanted (see Corollary 0.4).

Jech’s Set Theory book contains several properties of $\gamma $. For example, $\gamma $ is increasing : for any ordinal $\alpha $, $\alpha \times \alpha $ is a (proper) initial segment of $(\alpha +1)\times (\alpha +1)$ and of $\lambda \times \lambda $ for any limit ordinal $\lambda >\alpha $. As a consequence, $\forall \alpha ,\alpha \le \gamma (\alpha )$ which can also trivially be seen by the increasing function $\xi \mapsto (\xi ,0)$ from $\alpha $ to $\alpha \times \alpha $. Also, $$ implies $$. This can not be a strict equality, or otherwise the element $(\xi ,\eta )\in \lambda \times \lambda $ corresponding to $$ via the isomorphism between $\lambda \times \lambda $ and $\gamma (\lambda )$ would be an element larger than all $(\alpha ,\alpha )$ for $$. Hence $\gamma $ is continuous and by exercise 2.7 of the same book, $\gamma $ has arbitrary large fixed points. Exercise 3.5 shows that $\gamma (\alpha )\le {\omega}^{\alpha}$ and so $\gamma $ does not increase cardinality (see Corollary 0.2 for a much better upper bound). Hence starting at any infinite cardinal $\kappa $ the construction of exercise 2.7 provides infinitely many fixed points of $\gamma $ having cardinality $\kappa $. Hence the infinite fixed points of $\gamma $ are not just cardinals and we can wonder what they are exactly…

I recently tried to solve Exercise I.11.7 of Kunen’s Set Theory book which suggests a nice characterization of infinite fixed points of $\gamma $: they are the ordinals of the form ${\omega}^{{\omega}^{\alpha}}$. However, I could not find a simple proof of this statement so instead I tried to determine the explicit expression of $\gamma (\alpha )$, from which the result becomes obvious (see Corollary 0.3). My final calculation is summarized in Theorem 0.1, which provides a relatively nice expression of $\gamma (\alpha )$. Recall that any $\alpha \ge 1$ can be written uniquely as $\alpha ={\omega}^{\beta}q+\rho $ where $\beta $ is (following Kunen’s terminology) the “logarithm in base $\omega $ of $\alpha $” (that we will denote ${\mathrm{log}}_{\omega}\alpha $ for $\alpha \ge \omega $) and $q,\rho $ are the quotient and remainder of the Euclidean division of $\alpha $ by ${\omega}^{\beta}$. Alternatively, this can be seen from Cantor’s Normal Form: $\beta $ and $q$ are the exponent and coefficient of the largest term while $\rho $ is the sum of terms of smaller exponents.

Theorem 0.1.

For all ordinal $\alpha $, we denote $\gamma \mathit{}\mathrm{(}\alpha \mathrm{)}$ the order-type of the canonical ordering of $\alpha \mathrm{\times}\alpha $. Then $\gamma $ can be calculated as follows:

(1)

Finite Ordinals: For any $$ we have

$$\gamma (n)={n}^{2}$$

(2)

Limit Ordinals: For any limit ordinal $\alpha $,

(a)

If ${\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}$ does not divide ${\mathrm{log}}_{\omega}(\alpha )$ then

(like the first case but we “decrement $n$ in the second factor”)

(c)

Otherwise, $\alpha ={\omega}^{{\mathrm{log}}_{\omega}(\alpha )}+\rho $ and we write ${\mathrm{log}}_{\omega}(\alpha )={\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}m$ for some $m\ge 1$. We have

where $r\mathrm{=}\alpha \mathrm{mod}\omega $ is the remainder in the Euclidean division of $\alpha $ by $\omega $ (i.e. the constant term in the Cantor Normal Form).

Proof.

The “Limit Ordinals” case is $r=0$ i.e. $\alpha \le \gamma (\alpha )\le {\omega}^{{\mathrm{log}}_{\omega}(\alpha )}\cdot \alpha $, which is readily seen by the previous theorem. Then we deduce for the “Infinite Successor Ordinals” case: $\gamma (\alpha +n)=\gamma (\alpha )+\alpha \cdot 2n+n\le {\omega}^{{\mathrm{log}}_{\omega}(\alpha )}\cdot \alpha +\left(\alpha +n\right)\cdot 2n$ where ${\mathrm{log}}_{\omega}(\alpha +n)={\mathrm{log}}_{\omega}(\alpha )$ and $n=\left(\alpha +nmod\omega \right)$. Since ${\omega}^{{\mathrm{log}}_{\omega}(\alpha )}\le \alpha $ we can write ${\omega}^{{\mathrm{log}}_{\omega}(\alpha )}\cdot \left(\alpha -r\right)+\alpha \cdot 2r\le \alpha \cdot \left(\alpha -r+2r\right)=\alpha \cdot (\alpha +r)$. ∎

Hence the order type of the canonical ordering $\u22b2$ of $\alpha \times \alpha $ (i.e. $\gamma (\alpha )\le \alpha (\alpha +\omega )$) is never significantly bigger than the one of the standard lexical ordering (i.e. ${\alpha}^{2}$), and even never larger for limit ordinals. Moreover, for many ordinals $\alpha $ the canonical ordering is of order type $\alpha $:

Corollary 0.3.

The fixed points of $\gamma $ are $\mathrm{0}\mathrm{,}\mathrm{1}$ and ${\omega}^{{\omega}^{\alpha}}$ for all ordinals $\alpha $.

Proof.

For the “Finite Ordinals” case, we have $\gamma (n)={n}^{2}=n$ iff $n=0$ or $n=1$. For the “Infinite Successor Ordinals” case, we have $\alpha \cdot 2n>\alpha $ if $n\ge 1$ so $\gamma (\alpha +n)>\alpha +n$. Now we look at the three subcases of the “Limit Ordinals” case. For the first one, we have ${\mathrm{log}}_{\omega}(\alpha )\ge 1$ and so ${\omega}^{{\mathrm{log}}_{\omega}(\alpha )}\alpha \ge \omega \alpha >\alpha $. For the second one, we note that ${\mathrm{log}}_{\omega}(\alpha )2>{\mathrm{log}}_{\omega}(\alpha )$ and so ${\omega}^{{\mathrm{log}}_{\omega}(\alpha )2}>\alpha $ (compare the Cantor Normal Form). Since $n-1\ge 1$ by assumption, we have $\gamma (\alpha )>\alpha $. Similarly in the third case, if we expand the parenthesis the first term is $\omega $ raised to the power ${\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}\cdot \left(2m-1\right)$ which is stricly greater than ${\mathrm{log}}_{\omega}(\alpha )={\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}m$ if $m\ge 2$. Now if $m=1$, we obtain $\gamma (\alpha )={\omega}^{{\mathrm{log}}_{\omega}(\alpha )}+{\omega}^{{\mathrm{log}}_{\omega}(\alpha )}\rho $ and $\alpha ={\omega}^{{\mathrm{log}}_{\omega}(\alpha )}+\rho $. Hence $\gamma (\alpha )>\alpha $ if $\rho >0$ and $\gamma (\alpha )=\alpha $ if $\rho =0$. Finally for $\alpha \ge \omega $, $\gamma (\alpha )=\alpha $ iff $\alpha ={\omega}^{{\omega}^{{\mathrm{log}}_{\omega}\left({\mathrm{log}}_{\omega}\left(\alpha \right)\right)}}$ iff $\alpha ={\omega}^{{\omega}^{\beta}}$ for some ordinal $\beta $. ∎

Finally, now that we know that infinite fixed points of $\gamma $ are of the form $\alpha ={\omega}^{{\omega}^{\beta}}$ we only need to verify that infinite cardinals $\kappa $ are of this form to prove that $|\kappa \times \kappa |=\kappa $. This provides an alternative (less straightforward) proof of Theorem 3.5 from Jech’s Set Theory book.

Corollary 0.4.

Any infinite cardinal $\kappa $ is a fixed point of $\gamma $. Hence for any infinite cardinal ${\kappa}_{\mathrm{1}}\mathrm{,}{\kappa}_{\mathrm{2}}$ we have

For any cardinal infinite cardinal $\mu $ that is a fixed point of $\gamma $, the canonical well-ordering provides a bijection between $\mu $ and $\mu \times \mu $ i.e. ${\mu}^{2}=\mu $. Hence if ${\mu}_{1}\le {\mu}_{2}$ are two fixed points, we have

Suppose that there is a cardinal $\kappa $ which is not a fixed point of $\gamma $ and consider the smallest one. Then any infinite cardinal below $\kappa $ is a fixed point of $\gamma $ and the previous equality is still true below $\kappa $.

Suppose $\kappa >{\omega}^{{\mathrm{log}}_{\omega}\kappa}$ and write $\kappa ={\omega}^{{\mathrm{log}}_{\omega}\kappa}+\rho $ where $$ corresponds to the remaining terms in Cantor Normal Form. Then $$, $$ and ${\mu}_{1}+{\mu}_{2}=\kappa $. But the two first relations imply $$ which contradicts the third one. Hence $\kappa ={\omega}^{{\mathrm{log}}_{\omega}\kappa}$.

Now suppose ${\mathrm{log}}_{\omega}\kappa >{\omega}^{{\mathrm{log}}_{\omega}{\mathrm{log}}_{\omega}\kappa}$ and write ${\mathrm{log}}_{\omega}\kappa ={\omega}^{{\mathrm{log}}_{\omega}\kappa}+\rho $ where $$ corresponds to the remaining terms in Cantor Normal Form. Then we have

where $$ and $$ since $\alpha \mapsto {\omega}^{\alpha}$ is strictly increasing. Then $$, $$ and ${\mu}_{1}{\mu}_{2}=\kappa $. But again, the two first relations imply $$ which contradicts the third one.

Finally, $\kappa ={\omega}^{{\omega}^{{\mathrm{log}}_{\omega}{\mathrm{log}}_{\omega}\kappa}}$ and so is a fixed point of $\gamma $, a contradiction. Hence all the infinite cardinals are fixed point of $\gamma $ and so the property stated at the beginning is true for arbitrary infinite cardinals ${\mu}_{1},{\mu}_{2}$. ∎

As mentioned some time ago and as recently announced on the MathML and MediaWiki mailing lists, a MathML mode with SVG/PNG fallback is now available on Wikipedia. In order to test it, you need to log in with a Wikipedia account and
select the mode in the "Math" section of your preferences.

Some quick notes for Mozillians:

Although Mozilla intern Jonathan Wei has done some work on MathML accessibility and that there are reports about work in progress to make Firefox work with NVDA / Orca / VoiceOver, we unfortunately still don't have something ready for Gecko browsers. You can instead try the existing solutions for Safari or Internet Explorer (ChromeVox and JAWS 16 beta are supposed to be MathML-aware but fail to read the MathML on Wikipedia at the moment).

By default, the following MATH fonts are tried: Cambria Math, Latin Modern Math, STIX Math, Latin Modern Math (Web font). In my opinion, our support for Cambria Math (installed by default on Windows) is still not very good, so I'd recommend to use Latin Modern Math instead, which has the same "Computer Modern" style as the current PNG mode. To do that, go to the "Skin" section of your preferences and just add the rule math { font-family: Latin Modern Math; } to your "Custom CSS". Latin Modern Math is installed with most LaTeX distributions, available from the GUST website and provided by the MathML font add-on.

You can actually install various fonts and try to make the size and style of the math font consistent with the surrounding text. Here are some examples:

/* Asana Math (Palatino style) */
.mw-body, mtext {
font-family: Palatino Linotype, URW Palladio L, Asana Math;
}
math {
font-family: Asana Math;
}
/* Cambria (Microsoft Office style) */
.mw-body, mtext {
font-family: Cambria;
}
math {
font-family: Cambria Math;
}
/* Latin Modern (Computer Modern style) */
.mw-body, mtext {
font-family: Latin Modern Roman;
}
math {
font-family: Latin Modern Math;
}
/* STIX/XITS (Times New Roman style) */
.mw-body, mtext {
font-family: XITS, STIX;
}
math {
font-family: XITS Math, STIX Math;
}
/* TeX Gyre Bonum (Bookman style) */
.mw-body, mtext {
font-family: TeX Gyre Bonum;
}
math {
font-family: TeX Gyre Bonum Math;
}
/* TeX Gyre Pagella (Palatino style) */
.mw-body, mtext {
font-family: TeX Gyre Pagella;
}
math {
font-family: TeX Gyre Pagella Math;
}
/* TeX Gyre Schola (Century Schoolbook style) */
.mw-body, mtext {
font-family: TeX Gyre Schola;
}
math {
font-family: TeX Gyre Schola Math;
}
/* TeX Gyre Termes (Times New Roman style) */
.mw-body, mtext {
font-family: TeX Gyre Termes;
}
math {
font-family: TeX Gyre Termes Math;
}

We still have bugs with missing fonts and font inflation on mobile devices. If you are affected by these bugs, you can force the SVG fallback instead:

Four years ago I started to write some MathML add-ons using
Jetpack 0.8, now called
Add-on SDK.
I've recently made progress on this project, so that all the initial features
are now available as Firefox add-ons (my initial hope was that the Add-on SDK
would eventually be compatible with all Gecko browsers but unfortunately
that still does not seem to be the case at the moment).
The Mathzilla collection is available on AMO
but some of the add-ons are still undergoing review. Here is an overview:

The conversion of content MathML using David Carlisle's XSLT stylesheet is
now
in its own MathML-ctop add-on. There is another similar add-on to add MathML3 features missing in Gecko called MathML-mml3ff. Note that these add-ons do not rely on the Add-on SDK and
will work in any Gecko browsers. However, they should
probably be improved.

Another add-on that does not rely on the Add-on SDK
is the one adding mathematical fonts called
MathML-fonts.
I uploaded version 2.0 to use the new OpenType MATH fonts supported in Gecko 31,
but I hope that it will no longer be necessary in the future (more on this
later).

The conversion of PNG images into MathML is now provided by the
Image to MathML add-on. At the moment, it is still experimental, see
the details on mozilla.dev.tech.mathml if you want to help. It only
works for some Web sites using LaTeX in alt text
but I wish I can
find a solution for Wolfram Websites.

Since many Web sites are using MathJax and because in the meantime
MathJax moved to its slow HTML-CSS output by default
I had to write an add-on to force MathJax to use native MathML, which is
available
here.
Actually, it's even better since it disables the mml2jax preprocessor to
avoid useless work by
MathJax for Web sites that already use MathML in the source code. It also
prevents the MathJax menu to override the browser user interface (note that the
three add-ons below provide some UI features similar to what one can find in
MathJax).

The feature to copy a MathML formula is now provided by the
MathML Copy
add-on. Note that it actually copies two flavors (text and html).
It is also possible
to copy the original TeX source when it is provided (e.g. on MDN).

A new MathML Zoom
add-on provides a zooming feature similar to what MathJax does.

A new MathML Font Settings add-on allows to
configure font-family and font-size of mathematics similar to
what MathJax provides. Note however, that the list of font-family
choices in the context menu is based on the OpenType MATH fonts that will only
be supported in Gecko 31.

I believe splitting the original Mathzilla add-on into many add-ons
gives more flexibility to let people choose the desired features.
As usual,
help to localize the add-ons is very welcome.

Today the Mozilla MathML team released a new version of TeXZilla. You can download a release package or install it with npm. We fixed a few bugs, but there are known issues due to errors in the unicode.xml file of
XML Entity Definitions for Characters
or inherited from the itex2MML grammar that does not make it ready for version
1.0. The main improvements in this new release are enhancements to the
public API and to the command line interface.

Stream filter

TeXZilla can now be used as a stream filter. Each TeX expressions delimited
by the classical $ ... $, $$ ... $$, \[ ... \] and \( ... \) will be
converted into inline or display MathML.
Outside these delimiters, you can use \$ and
\\ as escaped characters. We offer three ways to apply that stream filter:

echo "This is a **Markdown** document with a *math formula*: $ z = \\sqrt{x^2 + y^2} $" | markdown | nodejs TeXZilla.js streamfilter | sed '1s/^/\n<!-- HTML5 document -->\n/'

Using the TeXZilla.filterString(aString) function, for example
TeXZilla.filterString("blah $x^2$ blah") will return the filtered string.

Using the TeXZilla.filterElement(aElement) function. This one will browse
recursively the descendants of the DOM element aElement and the
stream filter will be applied to the text leaves.

By introducting these TeXZilla.filter* function, it becomes
tempting to use TeXZilla the same way as MathJax, that is to process all the
text nodes in your Web pages and to filter the TeX strings. This is not the
intended goal of TeXZilla and it is strongly
discouraged: not only the MathML content won't appear in crawlers
(e.g. search engines or feed readers) but also
browsing all the DOM elements and appending new ones can be very slow for large
documents. Instead, it is recommend to filter your static Web page with
commonJS TeXZilla.js streamfilter before publishing it or to use a
server-side conversion for example using the Web server mode. There
are situations where you do not have other choice, though. In that
case try to reduce as much as possible the number of elements being processed
(see the example in the next section).
Of course, if you do not care about performance and MathML availibility
outside your web site, you can
just use MathJax.

New Safe and Itex-Identifier parsing modes

The most notable difference between TeXZilla and itex2MML is the handling of
some expressions like $xy$ or $Func$. By default,
TeXZilla interprets this as individual
MathML identifiers <mi>x</mi><mi>y</mi> (so that as in LaTeX, they will
render in italic) while itex2MML interprets this
as a single indentifier <mi>Func</mi>. It is now possible to configure
TeXZilla to align with itex2MML's behavior. To do that, use
TeXZilla.setItexIdentifierMode
or pass the appropriate boolean to the command
line. Consecutive non-basic letters (like Greek or Arabic) are still treated as
individual tokens. With that change, we hope that TeXZilla could be used to
parse all the commands supported by itex2MML into an equivalent output.
Together with the command line stream filter, this should allow to recover
all the nice itex2MML features.

Similarly, a safe mode is now available and can be enabled
with TeXZilla.setSafeMode or by passing the appropriate boolean to the command line. This mode will forbid commands that could
be used for XSS injections like \href. With that mode and the
new TeXZilla.filterElement function, I'm now able to remove MathJax's use from
my blog (users of browsers without good MathML support can still enable
it or choose the lighter mathml.css stylesheet). MathJax was a bit overkill
for my blog since I'm only parsing visitor comments. To illustrate how the
setSafeMode and filterString functions can be used, I now just
have to do

// Process TeX fragments in blog comments and comment preview.
window.addEventListener("DOMContentLoaded", function() {
TeXZilla.setSafeMode(true);
var toProcess =
document.querySelectorAll("#comments > dl > dd, #comment-form dd.comment-preview");
for (var i = 0; i < toProcess.length; i++) {
TeXZilla.filterElement(toProcess[i]);
}
});

Inserting equations in a 2D/WebGL canvas

The new function TeXZilla.toImage has been introduced to convert a TeX fragment into
a math HTML image with a base64-encoded src attribute.
Contrary to other functions of the API, this one needs to do some
work to determine the image size and perform the conversion, so it is unlikely
to work as expected in a non-browser context. The goal is really only to have a
convenient function to generate image of mathematical formulas and insert them
into a canvas context to draw 2D or 3D scientific schemas.
At the moment, this works well only in Gecko. For
instance,

will insert a mathematical formula in the middle of a 2D canvas. Similarly,
you can insert a mathematical formula as a texture in a WebGL canvas. It
is recommended to pass aRoundToPowerOfTwo=true to TeXZilla.toImage,
so that the image will
have dimensions that are power of two. Note that
the mathematical formula will be automatically centered in the middle of the
generated image.
See this example for how to setup the formulas with three.js and make
them always oriented in the direction of the camera.

Integration in Mozilla products

The CKeditor editor plugin is now
integrated in MDN, so you can click on the square root logo in the editor toolbar to insert mathematical formulas.
By the way, the mathml.css
is now used for browsers without MathML support. See for example
the pages for acosh, atanh
or CSS transform.

The editor/ in comm-central now integrates a small input box to insert mathematical formulas, accessible from the Insert menu. This will be available in Thunderbird 31 and Seamonkey 2.28, so that you can write mathematics in your emails and in the WYSIWYG editors.

Various FirefoxOS Web math apps have been written and use TeXZilla. Raniere is also working on a math keyboard for FirefoxOS as a GSoC project, which will allow to type mathematics faster on mobile devices.

update 2014/03/11: TeXZilla is now available as an npm module.

Introduction

For the past two months, the Mozilla MathML team has been working on
TeXZilla, yet another
LaTeX-to-MathML converter. The idea was to rely on
itex2MML (which dates back from the beginning of the Mozilla MathML project) to
create a LaTeX parser such that:

It is compatible with the
itex2MML syntax and is similarly generated from a LALR(1) grammar
(the goal is only to support a restricted set of core LaTeX commands for
mathematics, for a more complete converter of LaTeX documents see
LaTeXML).

It is available as a standalone Javascript module
usable in all the Mozilla Web applications and add-ons (of course,
it will work in non-Mozilla products too).

It accepts any Unicode characters and supports right-to-left
mathematical notation
(these are important for the world-wide aspect of the Mozilla
community).

The parser is generated with the help of
Jison and relies on a
grammar based on the one of itex2MML and on the
unicode.xml file of the
XML Entity Definitions
for Characters specification. As suggested by the version number,
this is still
in development. However, we have made enough progress to
present interesting features here and get more
users and developers involved.

A live demo is
available to let you test the LaTeX-to-MathML converter with various
options and examples. For people willing to use the converter on their
mobiles a FirefoxOS Web app is also available.

Using TeXZilla in a CommonJS program or Web page

TeXZilla is made of a single TeXZilla.js file with
a public
API to convert LaTeX to MathML or extract the TeX source from a
MathML element. The converter accepts some options like
inline/display mode or RTL/LTR direction of mathematics.

You can load it the standard way in any Javascript
program and obtain a TeXZilla object that exposes the public
API.
For example in a commonJS program, to convert a TeX source into a MathML
source:

var TeXZilla = require("./TeXZilla");
console.log(TeXZilla.toMathMLString("\\sqrt{\\frac{x}{2}+y}"));

or in a Web Page, to convert a TeX source into a MathML DOM element:

<script type="text/javascript" src="TeXZilla.js"></script>
...
var MathMLElement = TeXZilla.toMathML("\\sqrt{\\frac{x}{2}+y}");

Using TeXZilla in Mozilla Add-ons

One of the goal of TeXZilla is to be integrated in Mozilla add-ons,
allowing people to write cool math applications (in particular, we would
like to have an add-on for Thunderbird).
A simple Firefox add-on has been written and passed the AMO review, which means
that you can safely include the TeXZilla.js script in your
own add-ons.

TeXZilla can be used as an
addon-sdk module. However, if you intend to use features
requiring a DOMParser instance (for example toMathML),
you need to initialize the DOM explicitly:

var {Cc, Ci} = require("chrome");
TeXZilla.setDOMParser(Cc["@mozilla.org/xmlextras/domparser;1"].
createInstance(Ci.nsIDOMParser));

More generally, for traditional Mozilla add-ons, you can do

TeXZilla has a basic command line interface. However, since
CommonJS is still being
standardized, this may work inconsistently between commonjs interpreters.
We have tested it on slimerjs (which
uses Gecko),
phantomjs and
nodejs. For example you can do

or launch a Web service (see next section). We plan to implement a stream
filter too so that it can behave the same as itex2MML: looking the LaTeX
fragments from a text document and converting them into MathML.

Using TeXZilla as a Web Server

TeXZilla can be used as a Web Server that receives POST and GET HTTP requests
with the LaTeX input and sends JSON replies with the MathML output. The typical
use case is for people willing to perform some server-side LaTeX-to-MathML conversion.

For instance, to start the TeXZilla Webserver on port 7777:

$ nodejs TeXZilla.js webserver 7777
Web server started on http://localhost:7777

that will be converted into MathML by TeXZilla and displayed in your
browser:
$\u0633\; =\; \backslash frac\{-\u0628\backslash pm\backslash sqrt\{\u0628^\u0662-\u0664\u0627\u062c\}\}\{\u0662\u0627\}$. You can
set the display/dir attributes on that <x-tex> element
and they will be applied to the <math> element. Instances of
<x-tex> elements also have a source property that
you can use to retrieve or set the LaTeX source. Of course, the MathML output
will automatically be updated when dynamic changes occur. You can
try this online demo.

While the patches for MathML integration in MediaWiki are
progressively being reviewed and merged and
while we are working on the
support for Open Type fonts with a MATH table in Gecko, I finally
found time to check the progress in Mozilla's
add-on
SDK. In particular, since the last time I tried (some years ago)
they have introduced a cleaner interface for content scripts as well
as the possibility to use XPCOM for missing features.
Hence I have been able to update some of my experimental MathML add-ons.
I have submitted two new add-ons to Mozilla's AMO that I hope
could be useful to some people:

MathJax Native MathML, an add-on to force MathJax to
switch to Gecko's MathML support
without having to use the MathJax menu to change the output mode
and works even on Websites where that menu is disabled.
This also removes MathJax's automatic rescaling and inline-block
span that are currently causing random rendering bugs
with Gecko's native MathML (and will confuse possible future
line-breaking support anyway).

MathML Copy (at the moment only partially reviewed by the AMO team), an
add-on to copy MathML and TeX into the clipboard.
For MathML, two flavors are copied: the source as plain text
(to paste in your favorite text editor)
and the MathML as HTML
(to paste in Thunderbird, MDN, any Gecko-based
HTML editor etc). Copying TeX
is only possible when it is provided via the
standard MathML
annotation method, which is the case in e.g.
LaTeXML and
Instiki
documents as well as in Wikipedia in the future.

As mentioned during the Mozilla Summit and recent MathMLmeetings, progress has recently be made to the way mathematical equations are handled on Wikipedia. This work has mainly be done by
the volunteer contributor Moritz Schubotz (alias Physikerwelt),
Wikimedia Foundation's developer Gabriel Wicke as well as members of
MathJax.
Moritz has been particularly involved in that project and he even
travelled from Germany to San Francisco in order to meet MediaWiki developers and spend one month to do volunteer work on this project.
Although the solution is essentially ready for a couple of months, the
review of the patches is progressing slowly.
If you wish to speed up the integration of what is probably the most
important improvements to MediaWiki Math to happen, please read
how you can help
below.

Current Status

The approach that has been used on Wikipedia so far is the
following:

Equations are written in LaTeX
or more precisely, using a specific
set of LaTeX commands accepted by
texvc. One issue
for the MediaWiki developers is that this program is written in
OCaml and no longer maintained, so they would like to switch to a more
modern setup.

texvc calls the LaTeX program to convert the LaTeX source into PNG images and this is the default mode.
Unfortunately, using images for representing mathematical equations
on the Web
leads to classical problems (for example alignment or rendering quality
just to mention a few of them)
that can not be addressed without changing the
approach.

For a long time, registered users have been able to switch to the MathJax mode thanks to the help of nageh, a member of the MathJax community.
This mode solves many of the issues with PNG images but
unfortunately it adds its own problems,
some of them being just unacceptable for MediaWiki developers. Again, these issues are intrinsic to the use
of a Javascript polyfill and thus yet another approach is necessary for
a long-term perspective.

Finally, registered users can also switch to the LaTeX source mode, that is only display the text source of equations.

Short Term Plan

Native MathML is the appropriate way to fix all the issues regarding the display of mathematical formulas in browsers. However, the language is still not perfectly implemented in Web rendering engines, so some fallback is necessary. The new approach will thus be:

The TeX equation will still be edited by hand but it will be
possible to use a visual editor.

texvc will be used as a filter to validate the TeX source.
This
will ensure that only the texvc LaTeX syntax is accepted and will avoid
other potential security issues.
The LaTeX-to-PNG conversion as well as OCaml language will be kept in
the short term, but the plan is to drop the former and to replace the
latter with a a PHP equivalent.

A LaTeX-to-MathML conversion followed
by a MathML-to-SVG conversion will be
performed server-side using MathJax.

By default all the users will receive the same output (MathML+SVG+PNG) but only one will be made visible, according your browser capabilities. As a
first step, native MathML will only be used in Gecko
and other rendering engines will see the SVG/PNG fallback ; but the goal is to progressively drop the old PNG output and to move
to native MathML.

Registered users will still be able to switch to the LaTeX source mode.

Registered users will still be able to use MathJax client-side, especially if they want to use the HTML-CSS output. However, this is
will no longer be a separate mode but an option to enable. That is, the MathML/SVG/PNG/Source is displayed normally and progressively replaced with MathJax's output.

Most of the features above have already been approved and integrated in the development branch or are undergoing review process.

How can you help?

The main point is that
everybody can review the patches on Gerrit.
If you know about Javascript and/or PHP, if you are interested in math
typesetting and wish to get involved in an important Open Source project
such as Wikipedia then it is definitely the right time to help
the MediaWiki Math project. The article
How to become a MediaWiki hacker is a very good introduction.

When getting involved in a new open source project one of the most
important step is to set up the development environment. There are
various ways to setup a local installation of MediaWiki but
using
MediaWiki-Vagrant might be the simplest one: just follow the
Quick Start Guide and use
vagrant enable-role math to
enable the Math Extension.

If you need more information, you can ask
Moritz
or try to reach people on the
#mediawiki (freenode) or #mathml (mozilla) channels. Thanks in advance for your help!

As I mentioned three months ago, I wanted to start a crowdfunding campaign so that I can have more time to devote to MathML developments in browsers and (at least for Mozilla) continue to mentor volunteer contributors. Rather than doing several crowdfunding campaigns for small features, I finally decided to do a single crowdfunding campaign with Ulule so that I only have to worry only once about the funding. This also sounded more convenient for me to rely on some French/EU website regarding legal issues, taxes etc. Also, just like Kickstarter it's possible with Ulule to offer some "rewards" to backers according to the level of contributions, so that gives a better way to motivate them.

As everybody following MathML activities noticed, big companies/organizations do not want to significantly invest in funding MathML developments at the moment. So the rationale for a crowdfunding campaign is to rely on the support of the current community and on the help of smaller companies/organizations that have business interest in it. Each one can give a small contribution and these contributions sum up in enough money to fund the project. Of course this model is probably not viable for a long term perspective, but at least this allows to start something instead of complaining without acting ; and to show bigger actors that there is a demand for these developments. As indicated on the Ulule Website, this is a way to start some relationship and to build a community around a common project. My hope is that it could lead to a long term funding of MathML developments and better partnership between the various actors.

Because one of the main demand for MathML (besides accessibility) is in EPUB, I've included in the project goals a collection of documents that demonstrate advanced Web features with native MathML. That way I can offer more concrete rewards to people and federate them around the project. Indeed, many of the work needed to improve the MathML rendering requires some preliminary "code refactoring" which is not really exciting or immediately visible to users...

Hence I launched the crowdfunding campaign the 19th of November and we reached 1/3 of the minimal funding goal in only three days! This was mainly thanks to the support of individuals from the MathML community. In mid december we reached the minimal funding goal after a significant contribution from the KWARC Group (Jacobs University Bremen, Germany) with which I have been in communication since the launch of the campaign. Currently, we are at 125% and this means that, minus the Ulule commision and my social/fiscal obligations, I will be able to work on the project during about 3 months.

I'd like to thank again all the companies, organizations and people who have supported the project so far! The crowdfunding campaign continues until the end of January so I hope more people will get involved. If you want better MathML in Web rendering engines and ebooks then please support this project, even a symbolic contribution. If you want to do a more significant contribution as a company/organization then note that Ulule is only providing a service to organize the crowdfunding campaign but otherwise the funding is legally treated the same as required by my self-employed status; feel free to contact me for any questions on the project or funding and discuss the long term perspective.

Finally, note that I've used my savings and I plan to continue like that until the official project launch in February. Below is a summary of what have been done during the five weeks before the holiday season. This is based on my weekly updates for supporters where you can also find references to the Bugzilla entries. Thanks to the Apple & Mozilla developers who spent time to review my patches!

Collection of documents

The goal is to show how to use existing tools (LaTeXML, itex2MML, tex4ht etc) to build EPUB books for science and education using Web standards. The idea is to cover various domains (maths, physics, chemistry, education, engineering...) as well as Web features. Given that
many scientific circles are too much biased by "math on paper / PDF" and closed research practices, it may look innovative to use the Open Web but to be honest the MathML language and its integration with other Web formats is well established for a long time. Hence in theory it should "just work" once you have native MathML support, without any circonvolutions or hacks. Here are a couple of features that are tested in the sample EPUB books that I wrote:

Rendering of MathML equations (of course!). Since the screen size and resolution vary for e-readers, automatic line breaking / reflowing of the page is "naturally" tested and is an important distinction with respect to paper / PDF documents.

CSS styling of the page and equations. This includes using (Web) fonts,
which are very important for mathematical publishing.

Using SVG schemas and how they can be mixed with MathML equations.

Using non-ASCII (Arabic) characters and RTL/LTR rendering of both the text and equations.

Interactive document using Javascript and <maction>, <input>, <button> etc. For those who are curious, I've created some videos for an algebra course and a lab practical.

Using the <video> element to include short sequences of an experiment in a physics course.

Using the <canvas> element to draw graphs of functions or of physical measurements.

Using WebGL to draw interactive 3D schemas. At the moment, I've only adapted a chemistry course and used ChemDoodle to load Crystallographic Information Files (CIF) and provide 3D-representation of crystal structures. But of course, there is not any problem to put MathML equations in WebGL to create other kinds of scientific 3D schemas.

WebKit

I've finished some work started as a MathJax developer, including the maction support requested by the KWARC Group. I then tried to focus on the main goals: rendering of token elements and more specifically operators (spacing and stretching).

I improved LTR/RTL handling of equations (full RTL support is not implemented yet and not part of the project goal).

I improved the maction elements and implemented the toggle actiontype.

I refactored the code of some "mrow-like" elements to make them all behave like an <mrow> element. For example while WebKit stretched (some) operators in <mrow> elements it could not stretch them in <mstyle>, <merror> etc Similarly, this will be needed to implement correct spacing around operators in <mrow> and other "mrow-like" elements.

I analyzed more carefully the vertical stretching of operators. I see at least two serious bugs to fix: baseline alignment and stretch size. I've uploaded an experimental patch to improve that.

Preliminary work on the MathML Operator Dictionary. This dictionary contains various properties of operators like spacing and stretchiness and is fundamental for later work on operators.

I have started to refactor the code for mi, mo and mfenced elements. This is also necessary for many serious bugs like the operator dictionary and the style of mi elements.

I have written a patch to restore support for foreign objects in annotation-xml elements and to implement the same selection algorithm as Gecko.

Gecko

I've continued to clean up the MathML code and to mentor volunteer contributors. The main goal is the support for the Open Type MATH table, at least for operator stretching.

Xuan Hu's work on the <mpadded> element landed in trunk. This element is used to modify the spacing of equations, for example by some TeX-to-MathML generators.

On Linux, I fixed a bug with preferred widths of MathML token elements. Concretely, when equations are used inside table cells or similar containers there is a bug that makes equations overflow the containers. Unfortunately, this bug is still present on Mac and Windows...

James Kitchener implemented the mathvariant attribute (e.g used by some tools to write symbols like double-struck, fraktur etc). This also fixed remaining issues with preferred widths of MathML token elements. Khaled Hosny started
to update his Amiri and XITS fonts to add the glyphs for Arabic mathvariants.

I finished Quentin Headen's code refactoring of mtable. This allowed to fix some bugs like bad alignment with columnalign. This is also a preparation for future support for rowspacing and columnspacing.

After the two previous points, it was finally possible to remove the private "_moz-" attributes. These were visible in the DOM or when manipulating MathML via Javascript (e.g. in editors, tree inspector, the html5lib etc)

Khaled Hosny fixed a regression with script alignments. He started to work on improvements regarding italic correction when positioning scripts. Also, James Kitchener made some progress on script size correction via the Open Type "ssty" feature.

I've refactored the stretchy operator code and prepared some patches to read the OpenType MATH table. You can try experimental support for new math fonts with e.g. Bill Gianopoulos' builds and the MathML Torture Tests.

Blink/Trident

MathML developments in Chrome or Internet Explorer is not part of the
project goal,
even if obviously MathML improvements to WebKit could
hopefully be imported to Blink in the future. Users keep asking for MathML in IE and I hope that a solution will be found to save MathPlayer's work. In the meantime, I've sent a proposal to Google and Microsoft to implement fallback content (alttext and semantics annotation) so that authors can use it. This is just a couple of CSS rules that could be integrated in the user agent style sheet. Let's see which of the two companies is the most reactive...

Note: some parts of this blog post (especially the Javascript program) may be lost when exported to Planet or other feed aggregators. Please view it on the original page.

I recently took a look at the description of the CSS 2D / SVG transform
matrix(a, b, c, d, e, f) on MDN and I added a
concrete example showing the effect of such a
transform on an SVG line, in order to make this clearer for people who
are not familiar with affine transformations or matrices.

This also recalled me a small algorithm to decompose an arbitrary
SVG transform into a composition of basic transforms (Scale, Rotate,
Translate and Skew) that I wrote 5 years ago for the Amaya SVG
editor.
I translated it into Javascript and I make it available here. Feel free
to copy it on MDN or anywhere else. The convention used to represent
transforms as 3-by-3 matrices
is the one of the SVG specification.

Live demo

Enter the CSS 2D transform you want to
reduce and decompose or pick one example from the list
. You can also choose between LU-like or QR-like decomposition:
.

CSS

Here is the reduced CSS/SVG matrix as computed by your rendering engine ? and its matrix representation:

After simplification (and modulo rounding errors), an SVG
decomposition into simple transformations is ? and it renders
like this:

After simplification (and modulo rounding errors), a CSS decomposition
into simple transformations is ? and it renders like this:

CSS

A matrix decomposition of the original transform is:

Mathematical Description

The decomposition algorithm is based on the classical
LU and
QR
decompositions. First remember the SVG specification: the transform
matrix(a,b,c,d,e,f) is represented by the matrix

which shows the classical factorization into a composition of a linear
transformation $\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)$
and a translation $\left(\begin{array}{c}e\\ f\end{array}\right)$. Now let's focus on the matrix
$\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)$ and denote $\Delta =ad-bc$ its determinant. We first
consider the LDU decomposition. If $a\ne 0$, we can use it as a pivot and
apply one step of Gaussian's elimination:

Hence if $a\ne 0$, the transform matrix(a,b,c,d,e,f) can be
written
translate(e,f) skewY(atan(b/a)) scale(a, Δ/a) skewX(c/a).
If $a=0$ and $b\ne 0$ then we have $\Delta =-cb$ and we
can write (this is approximately "LU with full pivoting"):

and so the transform becomes
translate(e,f) rotate(90°) scale(b, Δ/b) skewX(d/b). Finally,
if $a=b=0$, then we already have an LU decomposition and we can just write

and so the transform is
translate(e,f) scale(c, d) skewX(45°) scale(0, 1).

As a consequence, we have proved that any transform
matrix(a,b,c,d,e,f)
can be decomposed into a product of simple
transforms. However, the decomposition is not always what we want, for example
scale(2) rotate(30°) will be decomposed into a product that
involves skewX and skewY instead of preserving the
nice factors.

We thus consider instead the QR decomposition.
If $\Delta \ne 0$, then by applying the Gram–Schmidt process to the columns
$\left(\begin{array}{c}a\\ b\end{array}\right),\left(\begin{array}{c}c\\ d\end{array}\right)$
we obtain

where $r=\sqrt{{a}^{2}+{b}^{2}}\ne 0$. In that case, the transform becomes
translate(e,f) rotate(sign(b) * acos(a/r)) scale(r, Δ/r)
skewX(atan((a c + b d)/r^2)). In particular, a similarity transform
preserves orthogonality and length ratio and so
$ac+bd=\left(\begin{array}{c}a\\ b\end{array}\right)\cdot \left(\begin{array}{c}c\\ d\end{array}\right)=0$
and $\Delta =\parallel \left(\begin{array}{c}a\\ b\end{array}\right)\parallel \mid \left(\begin{array}{c}c\\ d\end{array}\right)\parallel \mathrm{cos}(\pi /2)={r}^{2}$. Hence for a
similarity transform we get
translate(e,f) rotate(sign(b) * acos(a/r)) scale(r) as wanted.
We also
note that it is enough to assume the weaker hypothesis
$r\ne 0$ (that is $a\ne 0$ or $b\ne 0$)
in the expression above and so the decomposition applies in that case too.
Similarly, if we let $s=\sqrt{{c}^{2}+{d}^{2}}$ and instead assume
$c\ne 0$ or $d\ne 0$ we get

Hence in that case the transform is
translate(e,f) rotate(90° - sign(d) * acos(-c/s)) scale(Delta/s, s) skewY(atan((a c + b d)/s^2)). Finally if $a=b=c=d=0$, then the transform is
just
scale(0,0).

The decomposition algorithms are now easy to write. We note that none of
them gives the best result in all the cases (compare for example how they factor
Rotate2 and Skew1). Also, for completeness we have included the noninvertible
transforms in our study
(that is $\Delta =0$) but in practice they are not really useful (try
NonInvertible).

This morning, Deyan Ginev announced on the LaTeXML mailing list that the first alpha version of LaTeXML with LaTeX to EPUB support is now available. This is a very good news for people willing to encourage researchers to move from offline formats to more modern Web formats. Although, some people
had already been successful to combine LaTeX-to-XHTML converters
and XHTML-to-EPUB converters, this is the first tool that I'm aware of that can do the direct LaTeX to EPUB3 (XHTML+MathML) conversion. I already mentioned a couple of Gecko-based EPUB tools in my previous blog post, so let's have a look at three of them. Feel free to mention more Gecko-based EPUB tools in the comments, I'm particularly interested to hear about FirefoxOS applications that would be similar
to Apple's iBooks.

I have updated the LaTeXML samples based on Boris Zbarsky's thesis that we demonstrated at the Innovation Fairs in Santa Clara & Brussels. This shows how to generate the traditional PDF version, the Web version, the Web version with MathJax fallback and now the EPUB version! Here are some screenshots using the Firefox extension Lucifox:

Boris' Thesis in Lucifox ; page 2

Boris' Thesis in Lucifox ; page 4

I have intentionally not shown the diagram that are incorrectly converted by LaTeXML due to missing Xy-pic support (this is still in development). However,
Gecko supports mixing SVG and MathML via the foreignObject element so this would not be a problem for Gecko-based EPUB readers. Here are some screenshots of an ebook about
regular polygon that can be constructed with compass and straightedge that I have created with the help of itex2MML. They are viewed in EPUBReader which is another Firefox extension:

EPUBReader, Constructible Numbers

EPUBReader, Cyclic Galois Extension

Lucifox and EPUBReader have a big drawback: they do not support EPUB pages with the "scripted" property. This means that you can not use Javascript to create dynamic ebooks with live samples or interactive exercices... but this is one of the reason to use Web formats! Fortunately, there is a XUL application called AZARDI that supports this feature. I have created another ebook that shows an interactive
course on matrices. Click on the image to see the video on YouTube:

update 2013-10-15: since I got feedback, I have to say that my funding plan is independent of my work at MathJax ; I'm not a MathJax employee but I have an independent contractor status. Actually, I already used my business to fund an intern for Gecko MathML developments during Summer 2011 :-)

Retrospect

Since last April, I have been allowed by the MathJax Consortium to dedicate a small amount of my time to do MathML development in browsers, until possibly more serious involvement later. At the same time, we mentioned this plan to Google developers but unfortunately they just decided to drop the WebKit MathML code from Blink, making external contributions hard and unwelcome...

Hence I have focused mainly on Gecko and WebKit: You can find the MathML bugs that have been closed during that period on bugzilla.mozilla.org and bugs.webkit.org. For Gecko, this has allowed me to finish some of the work I started as a volunteer before I was involved full-time in MathJax as well as to continue to mentor MathML contributors. Regarding WebKit, I added a few new basic features like MathML lengths, <mspace> or <mmultiscripts> while I was getting familiar with the MathML code and WebKit organization/community. I also started to work
on <semantics> and <maction>.
More importantly, I worked with Martin Robinson to address the design concerns of Google developers and a patch to fix these issues finally landed early this week.

However, my progress has been slow so as I mentioned in my previous blog post, I am planning to find
a way to fund MathML developments...

Why funding MathML?

Note: I am assuming that the readers of this blog know why MathML is important and are aware of the benefits it can bring
to the Web community. If not, please check
Peter Krautzberger's Interview by Fidus Writer or the MozSummit MathML slides for a quick introduction.
Here my point is to explain why we need more than volunteer-driven
development for MathML.

First the obvious thing: Volunteer time is limited so if
we really want to see serious progress in MathML support we need to give a
boost to MathML developments. e-book publishers/readers, researchers/educators who are stuck outside the Web in a LaTeX-to-PDF world, developers/users of accessibility tools or the MathML community in general want good math support in browsers now and not to wait again for 15 more years until all layout engines catch up with
Gecko or that the old Gecko bugs are fixed.

There are classical misunderstandings from people thinking that non-native
MathML solutions and other polyfills are the future or that math on the Web could be implemented
via PNG/SVG images or Web Components.
Just open a math book and you will see
that e.g. inline equations must be correctly aligned with the text or
participate
in line wrapping. Moreover we are considering math on the Web not math on paper,
so we want it to be compatible with HTML, SVG, CSS, Javascript,
Unicode, Bidi etc and also something that is fast and responsive. Technically,
this means that a clean solution must be in the core rendering engine,
spread over several parts of the code and must have strong interaction with the
various components like the HTML5 parser, the layout tree,
the graphic and font libraries, the DOM module, the style tree and so forth.
I do not see any volunteer-driven
Blink/Gecko/WebKit feature off the top of my head that has this
characteristic and actually even SVG or any other kind of language for
graphics have less interaction with HTML than MathML has.

The consequence of this is that it is extremely difficult for volunteers
to get involved in native MathML
and to do quick progress because they have to understand
how the various components of the Blink/Gecko/WebKit code work and be sure to do
things correctly. Good mathematical rendering is already something hard by
itself, so that is even more complicated when you are not writing an isolated
rendering engine for math on which you can have full control.
Also, working at the Blink/Gecko/WebKit level requires technical skills above the average
so finding volunteers who can work with the high-minded engineers of
the big browser companies is not something easy.
For instance, among the enthusiastic people coming to me and
willing to help MathML in Gecko, many got stuck when e.g. they tried to build
the Firefox source or do something more advanced and I never heard back from
them.
In the other direction, Blink/Gecko/WebKit paid developers are generally
not familiar with
MathML and do not have time to learn more about it
and thus can not always provide a relevant review of the code, or they may
break something while trying to modify code they do not entirely understand.
Moreover,
both the volunteers and paid staff have only a small amount of time to do
MathML stuff while the other parts of the engine evolve very quickly,
so it's sometimes hard to keep everything in sync.
Finally,
the core layout engines have strong security requirements that are difficult to satisfy in a volunteer-driven situation...

Beyond volunteer-driven MathML developments

At that point, there are several options. First the lazy one: Give up
with native math rendering, only focus on features that have impact on the
widest Web audience (i.e. those that would allow browser vendors to get more market share and thus increase their profit), thank the math people for creating the Web and kindly ask them to use
whatever hacks they can imagine to display equations on the Web. Of course as a
Mozillian, I think people
must decide the Web they want and thus exclude this option.

Next there is the ingenuous option: Expect that browser companies
will understand the importance of math-on-the-Web and start investing
seriously in MathML support. However, Netscape and Microsoft
rejected the <MATH> tag from the 1995 HTML 3.0 draft and the browser
companies have kept repeating they would only rely on volunteer contributions
to move MathML forward, despite the repeated requests from MathML folks and other scientific communities. So that option is excluded too, at least in the short
to medium term.

So it remains the ambitious option: Math people and other interested parties
must get together and try to fund native MathML developments. Despite the effort
of my manager at MathJax to convince partners and raise funds, my situation has
not changed much since April and it is not clear when/if the MathJax Consortium
can take the lead in native MathML developments. Given my expertise
in Gecko, WebKit and MathML, I feel the duty to do something.
Hence I wish to reorganize
my work time: Decrease my involvement in MathJax core in order to increase
my involvement in Gecko/WebKit developments. But I need the help of the
community for that purpose. If you run a business with interest for math-on-the-Web
and are willing to fund my work, then feel free to contact me directly by
mail for further discussion. In the short term, I want
to experiment with
Crowd Funding as
discussed in the next section. If this is successful we can think
of a better organization for MathML developments in the long term.

Crowd Funding

Wikipedia defines
Crowd funding as
"the collective effort of individuals who network and pool their money, usually
via the Internet, to support efforts initiated by other people or organizations". There are several Crowd Funding platforms with similar rule/interface.
I am considering Catincan which is specialized in Open Source Crowd Funding, can be used by any backer/developer around the world, can rely on Bugzilla to track the bug status and
seems to have good process to collect the
fund from backers and to pay developers.
You can easily login to the Catincan Website if
you have a GitHub, Facebook or Google account (apparently
Persona is not supported yet...). Finally, it seems to have a communication interface between backers and
developers, so that everybody can follow the development on the funded
features.

One distinctive feature of catincan is that only well-established Open
Source projects can be funded and only developers from these projects can
propose and work on the new features ; so that backers can trust that the
features will be implemented. Of course, I have been working on Gecko, WebKit and
MathML projects
so I hope people believe I sincerely want to improve
MathML support in browsers and that I have the skills to do so ;-)

As said in my previous blog post, it is not clear at all (at least to me)
whether Crowd Funding can be a reliable method, but it is worth trying. There are
many individuals and small businesses showing interest in MathML, without
the technical knowledge or appropriate staff to improve MathML in browsers. So if each
one fund a small amount of money, perhaps we can get something.

One constraint is that each feature has 60 days to reach the
funding goal. I do not have any idea about how many people are willing
to contribute to MathML and how much money they can give.
The statistical sample of projects currently funded is too small to extract relevant
information. However, I essentially see two options:
Either propose small features
and split the big ones in small steps, so that each catincat submission
will need less work/money and improvements will be progressive with
regular feedback to backers ;
or propose larger features so they look more attractive and exciting to people
and will require less frequent submissions to catincat.
At the beginning, I plan to start with the former and if the crowd funding is
successful perhaps try the latter.

Status in Open Source Layout Engines

Note: Obviously, Open Source Crowd Funding does not apply
to Internet Explorer, which is the one main rendering engine not mentioned below. Although
Microsoft has done a great job on MathML for Microsoft Word, they did not
give any public statement about MathML in Internet Explorer and all the bug
reports for MathML have been resolved "by design" so far. If you are interested
in MathML rendering and accessibility in Internet Explorer, please check
Design Science blog for the latest updates
and tools.

Blink

Note: I am actually focusing on the history of Chromium here but of course there are other Blink-based browsers. Note that programs like QtWebEngine (formerly WebKit-based) or Opera (formerly Presto-based) lost the opportunity to get MathML support when they switched to Blink.

Alex Milowski and François Sausset's first MathML implementation did not
pass Google's security review. Dave Barton fixed many issues in that implementation and as far as I know, there were not any known security vulnerabilities when Dave submitted his last version. MathML was enabled in Chrome 24 but Chrome developers had some concerns about the design of the MathML implementation in WebKit, which indeed violated some assumptions of WebKit layout code. So MathML was disabled in Chrome 25 and as said in the introduction, the source code was entirely removed when they forked.

Currently, the Chromium Dashboard indicates that MathML is shipped in Firefox/Safari, has positive feedback from developers and is an established standard ; but the Chromium status remains "No active development".
If I understand correctly,
Google's official position is that
they do not plan to invest in MathML development but will accept external
contributions and may re-enable MathML when it is ready
(for some sense of "ready" to be defined).
Given the MathML story in
Chrome, it seems really unlikely that any volunteer will magically show up and be willing to
submit a MathML patch. Incidentally, note the
interesting work
of the ChromeVox team regarding MathML accessibility:
Their recent video
provides a good overview of what they achieve (where Volker Sorge politely regrets
that "MathML is not implemented in all browsers").

Although Google's design concerns have now been addressed in WebKit, one
most serious remark from one Google engineer is that the WebKit MathML implementation is
of too low quality to be shipped so they just prefer to have no MathML
at all. As a consequence, the best short term strategy seems to be improving
WebKit MathML support and, once it is good enough, to submit a patch to
Google. The immediate corollary is that if you wish to see MathML in Chrome
or other Blink-based browsers you should
help WebKit MathML development. See the next section for
more details.

chromatic

Actually, I tried to import MathML into Blink one day this summer. However,
there were divergences between the WebKit and Blink code bases that made that
a bit difficult. I do not plan to try again anytime soon, but if someone is
interested, I have published my script and patch on GitHub. Note there may be even more divergences now and the patch is
certainly bit-rotten. I also thought about creating/maintaining a "Chromatic"
browser (Chrome + mathematics) that would be a temporary fork to let Blink
users benefit from native MathML until it is integrated back in Blink. But
at the moment, that would probably be too much effort for one person and
I would prefer to focus on WebKit/Gecko developments for now.

WebKit

The situation in WebKit is much better. As said before, Google's concerns
are now addressed and MathML will be enabled again in all WebKit releases
soon.
Martin Robinson is interested in helping the MathML developments in
WebKit and his knowledge of fonts will be important to improve operator
stretching, which is one of the biggest issue right now.
One new volunteer contributor, Gurpreet Kaur, also started to
do some work on WebKit like support for the *scriptshifts
attributes or for the <menclose> element. Last but
not least, a couple of Apple/WebKit developers reviewed and accepted
patches and even helped to fix a few bugs, which made possible to move
development forward.

When he was still working on WebKit, Dave Barton opened bug 99623 to track the top priorities. When the bugs below and their related dependencies are fixed, I think the rendering in WebKit will be good enough to be usable for advanced math notations and WebKit will pass the MathML Acid 1 test.

Bug 44208:
For example, in expression like
$\mathrm{sin}\left(x\right)$,
the "x" should be in italic but not the "sin". This is actually slightly
more complicated: It says when the default mathvariant
value must be normal/italic.
mathvariant is more like
the
text-transform CSS property in the sense that it remaps
some characters to the corresponding mathematical characters (italic, bold, fraktur,
double-struck...) for example
$\mathfrak{A}$ (mathvariant=fraktur A)
should render exactly the same as $\U0001d504$ (U+1D504).
By the way, there is the related bug 24230 on Windows, that prevents to use plane 1 characters.
The best solution will probably be to
implement mathvariant correctly. See also Gecko's ongoing work by James Kitchener below.

Bug 99618: Implement <mmultiscripts>, that allows expressions like
${}_{6}{}^{14}\mathrm{C}$ or $R_{i}{}_{j}{}_{;}{}^{j}=\frac{1}{2}S_{;}{}_{i}$. As said in the introduction, this is fixed in WebKit Nightly.

Bug 99614: Support for stretchy operators like in
${\left(\frac{\overline{{z}_{1}+{z}_{2}}}{3}\right)}^{4}$. Currently,
WebKit can only stretch operators vertically using a few Unicode constructions
like ⎛ (U+239B) + ⎜ (U+239C) + ⎝ (U+239D) for the left parenthesis.
Essentially only similar delimiters like brackets, braces etc are supported.
For small
sizes like $(\text{}$ or for large operators like
$\sum {n}^{2}$ it is necessary to use non-unicode glyphs in various math fonts, but this
is not possible in WebKit MathML yet. All of this will require a fair amount of
work: implementing horizontal stretching, font-specific stuff,
largeop/symmetric properties etc

Bug 99620:
Implement the operator dictionary. Currently, WebKit treats all the operators the same way, so for
example it will use the same 0.2em spacing before and after parenthesis, equal sign or invisible
operators in e.g.
$f\left(x\right)={x}^{2}$. Instead it should use the information provided by the MathML operator dictionary. This dictionary also specifies whether operators are stretchy, symmetric or
largeop and thus is related to the previous point.

Bug 119038: Use the code for vertical stretchy operators to draw the radical symbols
in roots like $\sqrt{\frac{2}{3}}$. Currently,
WebKit uses graphic primitives which do not give a really good rendering.

Bug 115610: Implement <mspace> which is used by many MathML generators
to do some spacing in mathematical formulas. As said in the introduction, this is fixed in WebKit Nightly.

In order to pass the Mozilla MathML torture tests, at least displaystyle and scriptlevel must be implemented too, probably as internal CSS properties. This should also allow to pass
Joe Java's MathML test, although that one relies on the infamous <mfenced>
that duplicates the stretchy operator features and is implemented inconsistently
in rendering engines. I think passing the MathML Acid 2 test will require slightly more effort,
but I expect this goal to be achievable if I have more time to work on WebKit:

Bug 120164: Implement negative spacing for <mspace> (I have an experimental patch).

Bug 85730: Implement <mpadded>, which is also used by MathML generators to do some tweaking of formulas. I have only done some experiments, that would be a generalization of <mspace>

Bug 85733: Implement the href attribute ; well I guess the name is explicit enough to understand what it is used for! I only have some experimental patch here too. That would be mimicing what is done in SVG or HTML.

Bug 120059 and
bug 100626: Implement <maction> (at least partially) and <semantics>,
which have been asked by long-time MathML users Jacques Distler and Michael Kohlhase. I have patches ready for that and this could be fixed relatively soon, I just need to find time to finish the work.

In general passing the MathML Acid 2 test is not too hard, you merely only need to implement those few MathML elements whose exact rendering is clearly defined by the MathML specification. Passing the MathML Acid 3 test is not expected in the medium term. However, the score will
naturally increase while we improve WebKit MathML implementation. The priority
is to implement what is currently known to be important to users.
To give examples of bugs not previously mentioned: Implementing menclose or fixing various DOM issues like bugs 57695, 57696 or 107392.

More advanced features like those mentioned in the next section for Gecko
are probably worth considering later (Open type MATH, linebreaking,
mlabeledtr...). It is worth noting that Apple has already
done some work on accessibility (with MathML being readable by VoiceOver
in iOS7), authoring and EPUB (MathML is enabled in WebKit-based ebook
readers
and ibooks-author has
an integrated LaTeX-to-MathML converter).

Gecko

In general I think I have a good relationship with the Mozilla community and most people have respect for the work that has been done by volunteers for almost 15 years now. The situation has greatly improved since I joined the project, at that time some people claimed the
Mozilla MathML project was dead after Roger Sidge's departure.
One important point is that Karl Tomlinson has worked
on repairing the MathML support when Roger Sidge left the project. Hence
there is at least one Mozilla employee with good knowledge of MathML who
can review the volunteer patches. Another key ingredient is the work that has recently been made by Mozilla to increase engagement of the volunteer
community like good documentation on MDN, the #Introduction channel, Josh Matthews' mentored bugs and of course programs like GSOC. However, as said
above, it is one thing to attract enthusiastic contributors and another thing
to get long-term contributors who can work independently on more advanced features. So
let's go back to my latest Roadmap for the Mozilla MathML Project and see what has been accomplished for one year:

Font support: Dmitry Shachnev created a Debian package for the
MathJax fonts and Mike Hommey added MathJax and Asana fonts in the list
of suggested packages for Iceweasel. The STIX fonts have also been
updated in Fedora and are installed by default on
Mac OS X Lion (10.7). For Linux distributions, it would be helpful
to implement Auto Installation Support. The bug to
add mathematical fonts to Android has been assigned in June but no more progress has happened so far.
Henri Sinoven opened a bug for FirefoxOS but there has not been any progress there either.
I had some patches to restore the "missing MathML fonts" warning (using an information bar) but it was refused by Firefox reviewers. However, the code to detect missing MathML font could still be used for the similar bug 648548, which also seems inactive since January. There are still some issues on the MathJax side that prevent to integrate Web fonts for the native MathML output mode. So at the moment the solution is
still to inform visitors about MathML fonts or to add MathML Web fonts to your Web site. Khaled Hosny (font and LaTeX expert) recently updated my patches to prepare the support for Open Type fonts and he offered to help on that feature.
After James Kitchener's work on mathvariant, we realized that we will
probably need to provide Arabic mathematical fonts too.

Spacing: Xuan Hu continued to work on <mpadded> improvements and I think his patch is close to be accepted. Quentin Headen has done some progress on <mtable> before focusing on his InstantBird GSOC project. He is still far from being able to work on
mtable@rowspacing/columnspacing but a work around for that has been added
to MathJax. I fixed the negative space regression
which was missing to pass the MathML Acid 2 test and is used in MathJax. Again, Khaled Hosny is willing to help to use the spacing of the Open Type MATH, but that will still be a lot of work.

<mlabeledtr>: A work around for native MathML has been added in MathJax.

Linebreaking: No progress except that I have worked on fixing a bug with intrinsic width computation. The unrelated printing issues mentioned in the blog post have been fixed, though.

Operator Stretching: No progress. I tried to analyze the regression more carefully, but nothing is ready yet.

Tabular elements: As said above, Quentin Headen has worked a bit on cleaning up <mtable> but not much improvements on that feature so far.

Token elements: My patch for <ms> landed and I have done significant progress on the bad measurement of intrinsic width for token elements (however, the fix only seems to work on Linux right now). James Kitchener has taken over my work on improving our mathvariant support and doing related refactoring of the code. I am confident that he will be able to have something ready soon. The primes in exponents should render correctly with MathJax fonts but for other math fonts we will have to do some glyph substitutions.

Dynamic MathML: No progress here but there are not so many bugs regarding Javascript+MathML, so that should not be too serious.

Documentation: It is now possible to use MathML in code sample or
directly in the source code. The MathML project pages have been entirely migrated to MDN. Also, Florian Scholz has recently been hired by Mozilla as
a documentation writer (congrats!) and will in particular continue the work he started as a volunteer to document MathML on MDN.

I apologize to volunteers who worked on bugs that are not mentioned above or who are doing documentation or testing that do not appear here. For a complete list of activity since September 2012, Bugzilla is your friend. There are two ways to consider the progress above.
If you see the glass half full, then you see that several people have continued
the work on various MathML issues, they have made some progress and we now pass
the MathML Acid 2 test. If you see
the glass half empty, then you see that most issues have not been addressed
yet and in particular those that are blocking the native MathML to be enabled
in MathJax: bug 687807, bug 415413, the math font issues discussed in the first point, and perhaps linebreaking too. That is why I believe we should go beyond volunteer-driven MathML
developments.

Most of the bugs mentioned above are tested by the MathML Acid 3 tests and we will win a few points when they are fixed. Again, passing MathML Acid 3 test is not a goal by itself so let's consider what are the big remaining areas it contains:

Improving Tabular Elements and Operator Stretching, which are obviously important and used a lot in e.g. MathJax.

Linebreaking, which as I said is likely to become fundamental with small screens and ebooks.

Elementary Mathematics (you know addition, subtraction, multiplication, and division that kids learn), which I suspect will be important for educational tools and ebooks.

Alignment: This is the one part of MathML that I am not entirely sure is relevant to work on in the short term. I understand it is useful for advanced layout but most MathML tools currently just rely on tables to do that job and as far as I know the only important engine that implements that is MathPlayer.

Finally there are other features outside the MathML rendering engines that
I also find important but for which I have less expertise:

Transferring MathML that is implementing copy/cut/drag and paste. Currently, we can do that by treating MathML as normal HTML5 code or by using the "show MathML source" feature and copying the source code. However, it would be best to implement a standard way to communicate with other MathML applications like Microsoft Word, Mathematica, Mapple, Windows' Handwriting panel etc I wrote
some work-in-progress patches last year.

Authoring MathML: Essentially implementing things like deletion, insertion etc maybe simple MathML token creation ; in Gecko's core editor, which is used by BlueGriffon, KompoZer, SeaMonkey, Thunderbird or even MDN. Other things like integrating Javascript parsers (e.g. ASCIIMath) or equation panels with buttons like are probably better done at the higher JS/HTML/XUL level. Daniel Glazman already wrote math input panels for
BlueGriffon and
Thunderbird.

MathML Accessibility: This is one important application of MathML for which there is strong demand and where Mozilla is behind the competitors. James Teh started some experimental work on his NVDA tool before the summit.

EPUB reader for FirefoxOS (and other mobile platforms): During the
"Co-creating Action Plans" session, the Mozilla Taipei people were thinking
about missing features for FirefoxOS and this idea about EPUB reader was my modest contribution.
There are a few EPUB readers relying on Gecko and it would be good to check if they work in
FirefoxOS and if they could be integrated by default, just like
Apple has iBooks. BTW, there is a version of BlueGriffon that can edit EPUB books.

Conclusion

I hope I have convinced some of the readers about the need to fund MathMLin browsers. There is a lot of MathML work to do on Gecko and WebKit but both projects have volunteers and core engineers who are willing to help. There are also several individuals / companies relying on MathML support in rendering engines for their projects and could support the MathML developments in some way. I am willing to put more of my time on Gecko and WebKit developments, but I need financial help for that purpose. I'm proposing catincan Crowd Funding in
the short term so that anyone can contribute at the appropriate level, but other alternatives to fund the MathML development can be found like asking Peter Krautzberger about native MathML funding in MathJax,
discussing with Igalia about funding Martin Robinson to work more on WebKit
MathML or contacting me directly to establish some kind of part-time
consulting agreement.

Please leave a comment on this blog or send me a private mail, if you
agree that funding MathML in browsers is important, if you like the crowd funding idea and plan to contribute ; or if you have any opinions about alternative funding options. Also, please tell me what seem to be the priority for you and
your projects among what I have mentioned above
(layout engines, features etc) or among others that I may have forgotten. Of course,
any other constructive comment to help MathML support in browsers is welcome. I plan to submit features on catincan soon, once I have more feedback on
what people are interested in. Thank you!

I'm back from a great Mozilla Summit 2013 and I'd just like to write
a quick blog post about the MathML booths at the Innovation Fairs.
I did not have the opportunity to talk with the MathML people who
ran the booth at Santa Clara yet. However, everything went pretty
well at Brussels, modulo of course some demos failing when done in
live... If you are interested,
the slides and other resources are available on my GitHub page.

Many Mozillians did not know about MathML or that it
had been available in Gecko since the early days of the Mozilla project.
Many people who use math (or just knowing someone who does)
were curious about that feature and excited about the MathML potentials.
I appreciated to get this positive feedback from Mozillians willing
to use math on the Web and related media, instead of the scorn or hatred
I sometimes see by misinformed people. I expect to provide more
updates on LaTeXML, MediaWiki Math and MathJax when their next versions
are released. The Gecko MathML support improves slowly but there has
been interesting work by James Kitchener recently that I'd like to
mention too.

Let's do an estimation
à la Fermi:
only a few volunteers have been contributing
regularly and simultaneously to MathML in Gecko while most
Mozilla-funded Gecko projects have certainly development teams that are
3 times as large. Let's be optimistic and assume that these
volunteers have been able to dedicate a mean of 1 work day
per week, compared to 5 for full-time staff.
Given that the Mozilla MathML project will celebrate its 15 years
next May, that means that the volunteer work
transposed in terms of paid-staff time is only
$\le \frac{15}{3\cdot 5}=1$
year. To be honest, I'm disregarding here the great work made by the
Mozilla NZ team around 2007 to repair MathML after the Cairo migration.
But still, what we have achieved in quality and
completeness with such limited resources and time is really impressive.

As someone told me at the MathML booth, it's really frustating
that something that is so important for the small portion of
math-educated people
is ignored because it is useless for the vast majority of people. This
is not entirely true, since even elementary mathematics taught at school
like the one of
this blog post are not easily expressed with standard HTML
and even less in a way accessible to people with visual
disabilities. However, it summarizes well the feeling MathML folks had
when they tried to convince
Google to
accept the volunteer work on MathML, despite its low quality.

As explained at the Summit Sessions, Mozilla's mission is different and
the goal is to give people the right to control the Web they want.
The MathML project is perhaps one of the oldest and successful
volunteer-driven Mozilla project that is still active and
demonstrates concretely the idea of the Mozilla's mission with e.g. the
work of Roger Sidge who started to write the MathML
implementation when Netscape opened its source code or the one of
Florian Scholz who made MDN one of the most complete Web resource for
MathML.

Mozilla Corporation has kept saying they don't want to invest in MathML developments and the focus right now is clearly on other features like FirefoxOS. Even projects that have a larger audience than the MathML support like the mail client or the editor are not in the priorities so someone else definitely need to step in for MathML. I've tried various methods, with more or less success, to boost the MathML developments like mentoring a GSoC project, funding a summer internship or relying on mentored bugs. I'm now considering crowd funding to help the MathML developments in Gecko (and WebKit). I don't want to do another Fermi estimation now but
at first that looks like a very unreliable method. The only revenue generated by the MathML project so far are the
$2\frac{\lfloor 100\cdot \pi \rfloor}{100}=2\cdot 3.14=6.28$ dollars to the Mozilla Fundation via contributions to
my MathML-fonts add-on, so it's hard to get an idea of how much people would contribute
to the Gecko implementaton.
However, that makes sense since the only people who showed interest
in native MathML support so far are individuals or small businesses
(e.g. working on EPUB or accessibility) and I think it's worth
trying it anyway. That's definitely
something I'll consider after MathJax 2.3 is
released...

Last November, I tried to provide
some details
of the proof given in chapter 7,
regarding the fact that the continuum hypothesis implies the
existence of a Ramsey ultrafilter.
Peter Krautzberger
pointed out that the proof could
probably work assuming only Martin’s Axiom.
This was indeed proved by Booth in 1970 and the missing argument is actually
given in exercise 16.16. For completeness, I copy the details on this blog
post.

Remember that the proof involves contructing a sequence
${(X_{\alpha})}_{{\alpha<2^{{\aleph_{0}}}}}$ of infinite subsets of $\omega$.
The induction hypothesis is that at step $\alpha<2^{{\aleph_{0}}}$,
for all $\beta_{1},\beta_{2}<\alpha$ we have $\beta_{1}<\beta_{2}\implies X_{{\beta_{2}}}\setminus X_{{\beta_{1}}}$ is finite.
It is then easy to show the result for the successor step,
since the construction satisfies
$X_{{\alpha+1}}\subseteq X_{\alpha}$. However at limit step, to ensure that
$X_{\alpha}\setminus X_{\beta}$ is finite for all $\beta<\alpha$, the proof
relies on the continunum hypothesis. This is the only place where it is used.

Assume instead Martin’s Axiom and consider a limit step
$\alpha<2^{{\aleph_{0}}}$. Define the forcing notion
$P_{\alpha}=\{(s,F):s\in{[\omega]}^{{<\omega}},F\in{[\alpha]}^{{<\omega}}\}$
and $(s^{{\prime}},F^{{\prime}})\leq(s,F)$ iff
$s\subseteq s^{{\prime}}$, $F\subseteq F^{{\prime}}$ and $s^{{\prime}}\setminus s\subseteq X_{\beta}$
for all $\beta\in F$.
It is clear that the relation is reflexive and antisymmetric. The transitivity
is almost obvious, just note that if $(s_{3},F_{3})\leq(s_{2},F_{2})$ and
$(s_{2},F_{2})\leq(s_{1},F_{1})$ then
for all $\beta\in F_{1}\subseteq F_{2}$ we have
$s_{3}\setminus s_{1}\subseteq s_{3}\setminus s_{2}\cup s_{2}\setminus s_{1}%
\subseteq X_{\beta}$.

The forcing notion satisfies ccc or even property (K):
since ${[\omega]}^{{<\omega}}$ is countable,
for any uncountable subset $W$ there is $t\in{[\omega]}^{{<\omega}}$ such
that $Z=\{(s,F)\in W:s=t\}$ is uncountable. Then any
$(t,F_{1}),(t,F_{2})\in Z$ have a common refinement $(t,F_{1}\cup F_{2})$.

For all $n<\omega$, define $D_{n}=\{(s,F):|s|\geq n\}$.
Let $\beta_{1}>\beta_{2}>...>\beta_{k}$ the elements of $F$.
We show by induction on $1\leq m\leq k$ that $\bigcap_{{i=1}}^{m}X_{{\beta_{i}}}$
is infinite. This is true for $m=1$ by assumption. If it is true for
$m-1$ then

The left hand side is infinite by induction hypothesis. The second term
of the right hand side is included in $X_{{\beta_{1}}}\setminus X_{{\beta_{m}}}$ and thus
is finite. Hence the first term is infinite and the result is true for $m$.
Finally, for $m=k$, we get that
$\bigcap_{{\beta\in F}}X_{\beta}$ is infinite. Pick $x_{1},x_{2},...,x_{n}$ distinct
elements from that set and define $(s^{{\prime}},F^{{\prime}})=(s\cup\{x_{1},...x_{n}\},F)$.
We have $(s^{{\prime}},F^{{\prime}})\in D_{n}$, $s\subseteq s^{{\prime}}$, $F\subseteq F^{{\prime}}$ and
for all $\beta\in F$,
$s^{{\prime}}\setminus s=\{x_{1},...x_{n}\}\subseteq X_{\beta}$. This shows that
$D_{n}$ is dense. For each $\beta<\alpha$,
the set $E_{\beta}=\{(s,F):\beta\in F\}$ is also dense:
for any $(s,F)$ consider $(s^{{\prime}},F^{{\prime}})=(s,F\cup\{\beta\})$.

By Martin’s Axiom there is a generic filter $G$ for the family
$\{D_{n}:n<\omega\}\cup\{E_{\beta}:\beta<\alpha\}$ of
size $|\alpha|<2^{{\aleph_{0}}}$.
Let $X_{\alpha}=\{n<\omega:\exists(s,F)\in G,n\in s\}$.
For all $n<\omega$, there is $(s,F)\in G\cap D_{n}$ and so
$|X_{\alpha}|\geq|s|\geq n$. Hence $X_{\alpha}$ is infinite. Let $\beta<\alpha$
and $(s_{1},F_{1})\in G\cap E_{\beta}$.
For any $x\in X_{\alpha}$, there is $(s_{2},F_{2})\in G$ such that $x\in s_{2}$.
Hence there is $(s_{3},F_{3})\in G$ a refinement of $(s_{1},F_{2}),(s_{2},F_{2})$.
We have $x\in s_{2}\subseteq s_{3}$ and $s_{3}\subseteq s_{1}\subseteq X_{\beta}$.
Hence $X_{\alpha}\setminus X_{\beta}\subseteq s_{1}$ is finite and the induction
hypothesis is true at step $\alpha$.

The exercises from this chapter was a good opportunity to play a bit more
with the forcing method. Exercise 15.15 seemed a straightforward
generalization of Easton’s forcing but turned out to be a bit technical.
I realized that the forcing notion used in that exercise provides a
result in ZFC (a bit like Exercises 15.31 and 15.32 allow to prove some
theorems on Boolean Algebras by Forcing).

Remember that $\beth_{0}=\aleph_{0},\beth_{1}=2^{{\beth_{0}}},\beth_{2}=2^{{\beth_{1}}}...,%
\beth_{\omega}=\sup\beth_{n},...,\beth_{{\alpha+1}}=2^{{\beth_{{\alpha}}}},...$ is the normal sequence built by
application of the continuum function at successor step.
One may wonder: is $\beth_{\alpha}$ regular?

First consider the case where $\alpha$ is limit. The case $\alpha=0$ is
clear ($\beth_{0}=\aleph_{0}$ is regular) so assume $\alpha>0$.
If $\alpha$ is an inacessible
cardinal, it is easy to prove by induction that for all $\beta<\alpha$ we
have
$\beth_{\beta}<\alpha$: at step $\beta=0$ we use that $\alpha$ is uncountable,
at successor step that it is strong limit and at limit step that it is
regular. Hence $\beth_{\alpha}=\alpha$ and so is regular.
If $\alpha$ is not a cardinal then
$\operatorname{cf}(\beth_{\alpha})=\operatorname{cf}(\alpha)\leq|\alpha|<\alpha%
\leq\beth_{\alpha}$
so $\beth_{\alpha}$ is singular. If $\alpha$ is a cardinal but not strong limit
then there is $\beta<\alpha$ such that $2^{\beta}\geq\alpha$. Since
$\beta<\alpha\leq\beth_{\alpha}$ there is $\gamma<\alpha$ such that
$\beth_{\gamma}>\beta$. Then
$\beth_{\alpha}\geq\beth_{{\gamma+1}}=2^{{\beth_{\gamma}}}>2^{\beta}\geq\alpha$.
So $\operatorname{cf}(\beth_{\alpha})=\operatorname{cf}(\alpha)\leq\alpha<\beth_{\alpha}$ and
$\beth_{\alpha}$ is singular. Finally, if $\alpha$ is a singular cardinal,
then again $\operatorname{cf}(\beth_{\alpha})=\operatorname{cf}(\alpha)<\alpha\leq\beth_{\alpha}$ and
$\beth_{\alpha}$ is singular.

What about the successor case i.e. $\beth_{{\alpha+1}}$?
By Corollary 5.3 from Thomas Jech’s book any $\alpha$, we can show
that $\aleph_{{\alpha+1}}$ is a regular cardinal. The Generalized
Continuum Hypothesis says that $\forall\alpha,\aleph_{\alpha}=\beth_{\alpha}$.
Since it holds in $L$ we can not prove in ZFC that for some
$\alpha$, $\beth_{{\alpha+1}}$ is singular.

The generic extension $V[G]\supseteq V$ constructed in exercise 15.15
satisfies GCH and so it’s another way
to show that $\beth_{{\alpha+1}}$ can not be proved to be singular for some
$\alpha$. However, it provides a better result: by construction,
$V[G]\models\beth_{{\alpha+1}}^{V}={(\beth_{{\alpha}}^{V})}^{+}$ and so
$V[G]\models[\beth_{{\alpha+1}}^{V}\text{ is a regular cardinal}]$. Since
‘‘regular cardinal’’ is a $\Pi_{1}$ notion we deduce that
$\beth_{{\alpha+1}}$ is a regular cardinal in $V$.

Now the question is: is there any ‘‘elementary’’ proof of the fact that
$\beth_{{\alpha+1}}$ is regular i.e. without using the forcing method?

--update: of course, I forgot to mention that by König’s theorem,
$2^{{\beth_{{\alpha}}}}=\beth_{{\alpha+1}}\geq\operatorname{cf}(\beth_{{\alpha+%
1}})=\operatorname{cf}(2^{{\beth_{{\alpha}}}})\geq{(\beth_{{\alpha}})}^{+}$
so the singularity of $\beth_{{\alpha+1}}$ would imply the failure of the
continuum hypothesis for the cardinal $\beth_{\alpha}$ and this is not provable
in ZFC.