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html markup of previous sample

To: w3cmatherb@w3.org

Subject: html markup of previous sample

From: Ron Whitney <RFW@math.ams.org>

Date: Tue, 02 Jul 1996 22:43:38 0400 (EDT)

From RFW@math.ams.org Tue Jul 2 22: 44:01 1996

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MessageId: <836361818.256501.RFW@MATH.AMS.ORG>
I've made an approximation to HTML markup (of the math) for the example
I posted yesterday. In the ensuing portion of this message, I've enclosed
some notes and comments, followed by alternation between TeX and HTML
markup of the passages. Questions about semantics are of course much
more difficult to answer, but I'd be interested in any comments people
have on how they think the semantics of this example should be treated
(or rather how our "presemantics" will set things up). And please
do let me know if the HTML markup needs adjustment.
Ron
************************************************************************
Notes and questions:
1. I haven't made an effort to substitute character names as they exist
in TR 9573. By and large, I've substituted SGMLlike entities for TeX
characters. Identifiers such as \div in TeX come out \div in our current
proposal, although it's my understanding that "\div\nu" changes to
"\div ν" since ν is letterlike. True?
2. The differentials "Dx", etc. have been changed to "ⅅx", etc.
Should "\partial\nu" be changed to "&ppartial;ν" since the \partial
symbol itself might be reserved for meaning "boundary" and &ppartial;
provides some mnemonic consistency? [[BTW, later in this article
we have integrals with differential "d\omega^\epsilon(y)"]]
I take it that the form "differential &over; differential" will be
interpreted as a derivative by software which renders the expression
tree semantically.
Bruce spoke of visual renderers having the freedom to display &over;
with an inline /. Does the freedom also exist in reverse?
3. I've left the alignment of the first display for a later iteration.
The <sep> and <?space> markup only indicate a need to have column
separators and some indication of "separation" or distinction between
equation and condition.
4. I added SGML markup to "<mo>(ν\cdot\nabla)</mo>" since the
expression within the parens is an operator. The parens are not
properly part of the operator, though. How should this have been
handled?
5. How is "\div ν" interpreted? It appears to be two identifiers
juxtaposed and is therefore implicitly a product of the two. I've
inserted &FunctionApplication; where no parens appeared after the
operator.
6. Have we decided that we can use the _ and ^ characters for the lower
and upper limits of sums and integrals?
7. I'm guessing that we won't be able to simply enter  uncharacterized as
left or right (when the  occurs in a matching pair), so I've used
&leftvert; and &rightvert; for the pair.
8. BTW, in using the convenient screen display form for <math> elements,
I've segregated these elements so that the text of the paragraph after
</math> starts on a new line. I take it this adds significant space in an
SGML document.
9. The \max and \lim identifiers are juxtaposed to the left of
integral signs. Is the relation interpreted as multiplication?
Should I have placed a &FunctionApplication; entity between the two?
10. None of the bracketing operators should stretch in these passages.
Should they have been marked as stretchy="false" somehow?
11. I added some <mo> markup to the ad hoc : operator of the last
display. Left and right precedences are undefined. I take it that
the same locution has to be carried through to all other occurrences
of : (with the same meaning).
** TeX *******************************************************************
The Euler equations for an inviscid incompressible 2D fluid flow are
given by
%
$$\aligned & D\nu/Dt = \nabla p, &\qquad& x\in R^2, t>0 \\
& \div\nu = 0, &\qquad& \nu(x,0) = \nu_0(x)
\endaligned$$
%
where $\nu = {}^t(\nu_1,\nu_2)$ is the fluid velocity, $p$ is the
scalar pressure, $D\nu/Dt = \partial\nu/\partial t +
(\nu\cdot\nabla)\nu$, and $\nu_0$ is an initial incompressible
velocity field, i.e.~$\div\nu_0=0$.
** HTML ******************************************************************
The Euler equations for an inviscid incompressible 2D fluid flow are
given by
<math>
<alignment_structure>
<mrow> <sep> ⅅν/ⅅt = ∇p, <sep><?space><sep>
x&element; R^2, t>0 </mrow>
<mrow> <sep> \div&FunctionApplication;ν = 0, <sep><?space><sep>
ν(x,0) = ν_0(x) </mrow>
</alignment_structure>
</math>
where <math mode="inline">ν= (ν_1,ν_2)^^^t</math> is the fluid
velocity, <math mode="inline">p</math> is the scalar pressure,
<math mode="inline">ⅅν/ⅅt = &ppartial;ν/&ppartial;t +
<mo>(ν\cdot\nabla)</mo>ν</math>, and
<math mode="inline">ν_0</math> is an initial incompressible velocity
field, i.e. <math mode="inline">\div&FunctionApplication;ν_0=0</math>.
** TeX *******************************************************************
In this paper, we study the detailed limiting behavior of approximate
solution sequences for 2D Euler with vortex sheet initial data. A
sequence of smooth velocity fields $\nu^\epsilon(x,t)$ is an {\it
approximate solution sequence} for 2D Euler provided that the $\nu$
is incompressible, i.e.~$\div \nu=0$, and satisfies the following
properties:
** HTML ******************************************************************
In this paper, we study the detailed limiting behavior of approximate
solution sequences for 2D Euler with vortex sheet initial data. A
sequence of smooth velocity fields
<math mode="inline">ν^ε(x,t)</math> is an <em>approximate
solution sequence</em> for 2D Euler provided that the
<math mode="inline">ν</math> is incompressible,
i.e. <math mode="inline">\div&FunctionApplication;ν=0</math>,
and satisfies the following properties:
** TeX *******************************************************************
(1) The velocity fields $\nu^\epsilon$ have uniformly bounded local
kinetic energy, i.e.
%
$$\max_{0\leq t\leq T}\int_{x\leq R} \nu^\epsilon(x,t)^2\,dx\leq C$$
%
for any $R,T>0$.
** HTML ******************************************************************
(1) The velocity fields <math mode="inline">ν^ε</math> have
uniformly bounded local kinetic energy, i.e.
<math>\max_{0≤ t≤ T}∫_{&leftvert;x&rightvert;≤ R}
&leftvert;ν^ε(x,t)&rightvert;^2ⅆx≤ C</math>
for any <math mode="inline">R,T>0</math>.
** TeX *******************************************************************
(2) The corresponding vorticity, $\omega^\epsilon=\curl\nu^\epsilon$,
is uniformly bounded in $L^1$, i.e.
%
$$ \max_{0\leq t\leq T}\int\omega^\epsilon(x,t)\,dx\leq C$$
%
for any $T>0$.
** HTML ******************************************************************
(2) The corresponding vorticity,
<math mode="inline">ω^ε=
\curl&FunctionApplication;ν^ε</math>,
is uniformly bounded in <math mode="inline">L^1</math>, i.e.
<math> \max_{0≤ t≤ T}
∫&leftvert;ω^ε(x,t)&rightvert;ⅆx≤ C
</math>
for any <math mode="inline">T>0</math>.
** TeX *******************************************************************
(3) the vortex field $\nu^\epsilon$ is weakly consistent with 2D
Euler, i.e.~for all smooth test functions, $\phi\in
C^\infty(R^2\times(0,\infty))$ with $\div \phi=0$,
%
$$ \lim_{\epsilon\rightarrow0}\int\int\phi_t\cdot\nu^\epsilon +
\nabla\phi : \nu^\epsilon\otimes\nu^\epsilon\,dx\,dt=0.$$
%
Here $\nu\otimes\nu = (\nu_i \nu_j)$, $\nabla\phi =
(\partial\phi_i/\partial x_j)$, and $A:B$ denotes the matrix product
$\sum_{i,j}a_{ij}b_{ij}$. We remark in passing ...
** HTML ******************************************************************
(3) the vortex field <math mode="inline">ν^ε</math>
is weakly consistent with 2D Euler, i.e. for all smooth test functions,
<math mode="inline">φ&element;
C^&infinity;(R^2×(0,&infinity;))</math>
with <math mode="inline">\div&FunctionApplication;φ=0</math>,
<math> \lim_{ε→0}
∫∫φ_t·ν^ε +
∇φ<mo infix="true" leftprec=? rightprec=?>:</mo>
ν^ε&circletimes;ν^ε
ⅆx ⅆt=0.
</math>
Here <math mode="inline">ν&circletimes;ν = (\nu_i \nu_j)</math>,
<math mode="inline">∇φ = (&ppartial;φ_i/&ppartial; x_j)</math>,
and <math mode="inline">A<mo infix="true" leftprec=? rightprec=?>:</mo>B</math>
denotes the matrix product
<math mode="inline">∑_{i,j}a_{ij}b_{ij}</math>. We remark in passing ...
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