<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Let me answer by forward my reply to William’s message a few days ago.<div class=""><br class=""></div><div class="">((NLFs are not treated ideally, but that seems to be a common bug; so there are some spurious extra line breaks below)):<div class=""><br class=""></div><div class=""><br class=""><blockquote type="cite" class=""><div class="">Vidarebefordrat brev:</div><br class="Apple-interchange-newline"><div style="margin: 0px;" class=""><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class=""><b class="">Från: </b></span><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class="">Kent Karlsson <<a href="mailto:kent.b.karlsson@bahnhof.se" class="">kent.b.karlsson@bahnhof.se</a>><br class=""></span></div><div style="margin: 0px;" class=""><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class=""><b class="">Ämne: </b></span><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class=""><b class="">Re: My suggestions for Unicode based math expression format(s)</b><br class=""></span></div><div style="margin: 0px;" class=""><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class=""><b class="">Datum: </b></span><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class="">13 december 2022 16:20:42 CET<br class=""></span></div><div style="margin: 0px;" class=""><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class=""><b class="">Till: </b></span><span style="font-family: -webkit-system-font, "Helvetica Neue", Helvetica, sans-serif;" class="">William_J_G Overington <<a href="mailto:wjgo_10009@btinternet.com" class="">wjgo_10009@btinternet.com</a>><br class=""></span></div><br class=""><div class=""><br class=""><br class=""><div dir="ltr" class="">Skickat från min iPhone</div><div dir="ltr" class=""><br class=""><blockquote type="cite" class="">13 dec. 2022 kl. 13:33 skrev William_J_G Overington <<a href="mailto:wjgo_10009@btinternet.com" class="">wjgo_10009@btinternet.com</a>>:<br class=""><br class=""></blockquote></div><blockquote type="cite" class=""><div dir="ltr" class=""> <div class="auto-created-dir-div" dir="auto" style="unicode-bidi: embed;">Hi<div class=""><p class=""><br class=""></p><p class="">I have never used the various existing packages that you mention.</p><p class=""><br class=""></p><p class="">May I make three observations please?</p><p class=""><br class=""></p><p class="">1. I consider that using control codes to specify layout is a problem. A way to express things without using control codes is needed.</p></div></div></div></blockquote><span class="" style="float: none; display: inline !important;">There is. Two of them.</span><br class=""><blockquote type="cite" class=""><div dir="ltr" class=""><div class="auto-created-dir-div" dir="auto" style="unicode-bidi: embed;"><div class=""><p class=""><br class=""></p><p class="">2. Would a test be that what one wants to typeset can be typeset in Microsoft WordPad? One might need to copy and paste characters from a WordPad file that has one of each character in it, as if it were a typecase. For the avoidance of doubt I am not suggesting that all typesetting should be done in WordPad, not at all, but I am saying that if it cannot be typeset in WordPad then a format may be too complicated or too expensive or too inaccessible for widespread use.</p></div></div></div></blockquote><span class="" style="float: none; display: inline !important;">That is a question for someone else than me…</span><br class=""><blockquote type="cite" class=""><div dir="ltr" class=""><div class="auto-created-dir-div" dir="auto" style="unicode-bidi: embed;"><div class=""><p class=""><br class=""></p><p class="">3. Back in the early 1990s I was involved in a discussion of how to express mathematical equations using just 7-bit ASCII characters in a monospaced display typical of mainframe visual display units terminals at the time.</p><p class=""><br class=""></p><p class="">Some of my suggestions might be relevant here.</p><p class=""><br class=""></p><p class="">I suggested using :pom: to express a plus or minus sign as used in the general solution formula for a quadratic equation. That format could be used for special symbols. These days, an OpenType font could cause a correct glyph to be displayed, even if the glyph is not a regular Unicode character.</p></div></div></div></blockquote><span class="" style="float: none; display: inline !important;">I think that it would </span><b class="">not</b><span class="" style="float: none; display: inline !important;"> be a good idea to request or or expect that of any font. Any character escapes should be interpreted before any font is involved.</span><br class=""><blockquote type="cite" class=""><div dir="ltr" class=""><div class="auto-created-dir-div" dir="auto" style="unicode-bidi: embed;"><div class=""><p class=""><br class=""></p><p class="">I suggested that an integral be expressed using three capital I letters, one above the other in three lines of text.</p><p class=""><br class=""></p><p class="">I</p><p class="">I</p><p class="">I</p><p class=""><br class=""></p><p class="">that then allows upper and lower limits to be expressed for definite integrals.</p><p class=""><br class=""></p><p class="">For example</p><p class=""><br class=""></p><p class="">I t=1</p><p class="">I exp(-t).dt</p><p class="">I t=0</p><p class=""><br class=""></p><p class="">Then summation could be expressed as follows.</p><p class=""><br class=""></p><p class="">S n=5</p><p class="">S n^2</p><p class="">S n=1</p><p class=""><br class=""></p><p class="">and product similarly using three P characters.</p><p class=""><br class=""></p><p class="">P</p><p class="">P</p><p class="">P</p><p class=""><br class=""></p></div></div></div></blockquote><span class="" style="float: none; display: inline !important;">nroff with eqn did output to a typewriter-like terminal (fixed width), needing multiple character daisy-wheels some with mathematical characters. This terminal was capable of partial line up/down. But changing daisy wheels had to be done by hand. The output was a bit crude, but did have a fair semblance of normal mathematical typesetting. (troff produced better output, for typesetting machines of the day.) This was before TeX came along.</span><div class=""><br class=""></div><div class="">/Kent K</div><div class=""><br class=""></div><div class=""><br class=""><blockquote type="cite" class=""><div dir="ltr" class=""><div class="auto-created-dir-div" dir="auto" style="unicode-bidi: embed;"><div class=""><p class="">I hope this helps.</p><p class=""><br class=""></p><p class="">Best regards,</p><p class=""><br class=""></p><p class="">William Overington</p><p class=""><br class=""></p><p class="">Tuesday 13 December 2022</p><p class=""><br class=""></p><p class=""><br class=""></p><p class=""><br class=""></p><br class=""><blockquote class="" style="margin: 0px auto; padding: 0px 2em; border-left-width: 2px; border-left-style: solid; border-left-color: rgb(0, 173, 229); white-space: pre-wrap;"><br class=""><br class="">------ Original Message ------<br class="">From: "Kent Karlsson via Unicode" <<a href="mailto:unicode@corp.unicode.org" class="">unicode@corp.unicode.org</a>><br class="">To: <a href="mailto:unicode@corp.unicode.org" class="">unicode@corp.unicode.org</a><br class="">Sent: Tuesday, 2022 Dec 13 At 11:36<br class="">Subject: My suggestions for Unicode based math expression format(s)<br class=""><br class="">(Hoping that this goes through ok; I did have some problems with the sum sign when copying this text…)<br class="">
<br class="">
I've deviced a (or rather, several) new format(s) for representing math expressions.<br class="">
Why, you may wonder... Isn't MathML the answer to everyting math? Well, not quite.<br class="">
<br class="">
After more than 20 years since the first version of MathML, it is still not a great<br class="">
success. I think there are several reasons for that. One is the obvious: it is too<br class="">
verbose. Another is that (much due to the verbosity) that one really need authoring<br class="">
tools to be able to write any math expression in the MathML representation. The<br class="">
advantage of TeX math (or even old eqn) expressions is that users can with relative<br class="">
ease type the expression they want on the keyboard. Ordinary cut-paste-modifyViaKeyboard<br class="">
works. Authoring tools are less straight-forward to use. Furhter, not everything is<br class="">
HTML (or XML). One may even want to have math expressions in what is otherwise plain<br class="">
text; for instance for cut and paste, loosing styling per se (colour, bold/..., size)<br class="">
but not the math expressions.<br class="">
<br class="">
But what about typability, directly from the keyboard, without using a special authoring<br class="">
tool? Are eqn or TeX the only options? Well, there is AsciiMath and UnicodeMath...<br class="">
However, those do common parenthesis parsing that is undesirable, among other things.<br class="">
And, apart from UnicodeMath, they were created long before Unicode, so they are not well<br class="">
adapted to using Unicode characters.<br class="">
<br class="">
OMML (Office Math ML, also XML based) is just as verbose as MathML, if not worse.<br class="">
<br class="">
Using {} (a convention borrored from TeX; and using \{ and \} for literal {}) and some<br class="">
other special "mark-down" and character escape inspired notations, we can make a surface<br class="">
form of a math expression representation (encoding if you like) that is typable on a Latin based<br class="">
keyboard; except that ∑ and π in the example here may need some further escape notation,<br class="">
like \sum, \pi, to be fully keyboard typable (similarly to TeX, eqn, UnicodeMath, etc.). Not-so-common<br class="">
symbols will still need to be picked from some kind of menu, or use Unicode charater escapes,<br class="">
\uxxxx, \Uxxxxxx. Here is an example, using the same expression as is used as the lead example<br class="">
in the MathML Core specification; it looks a little bit like TeX, intentionally, due to the selection<br class="">
of {}^_ as meta-characters for certain math expression controls, but isn't TeX:<br class="">
<br class="">
${∑$/{n=1}$\{+∞}{1\/n^2}={π^2\/6}}<br class="">
<br class="">
There is also a HTML/XML compatible form proposed, that is fully equivalent in expressivity<br class="">
with the other forms/variants proposed. Though it is not MathML, but it is using XML tags,<br class="">
so it is a bit longer than the above (read "me" as "math expresion"):<br class="">
<br class="">
<me>∑<blw/><me>n=1</me><abv/><me>+∞</me><me>1<dv/>n<rsp/>2</me>=<me>π<rsp/>2<dv/>6</me></me><br class="">
<br class="">
Or with some more whitespace/linebreaks:<br class="">
<me><br class="">
∑<blw/><me>n=1</me><abv/><me>+∞</me> <me>1<dv/>n<rsp/>2</me><br class="">
=<br class="">
<me>π<rsp/>2<dv/>6</me><br class="">
</me><br class="">
<br class="">
This shows that having math expressions in an XML compatible format does not need to have<br class="">
clay feet. There are several key reasons for this relative light-footedness. The reasons<br class="">
include using: default styles, short tag/attribute names (for the XML variant) and short<br class="">
controls/markup for the other variants, and the use of a level of structural parsing,<br class="">
uncommon for XML (but otherwise common, also for math, in e.g. eqn and TeX). Details in<br class="">
the spec referenced below.<br class="">
<br class="">
It also shows that equivalent representations can be even more light-footed than the XML/HTML<br class="">
variant, as well as the possiblilty of having variant surface representation that fits with<br class="">
at least some other contexts (than XML/HTML).<br class="">
<br class="">
In addition, the respresentations (all variants) can still be general enough to allowRTL math<br class="">
expressions in a reliable way (in particular, reliable direction of arrows, which in math expressions<br class="">
almost always refer to the left and right side "arguments”, not an external physical direction),<br class="">
as well as chemical reaction formulas (math-like, not graphical) and the like. Re. arrows: see<br class="">
<a target="_blank" href="https://www.unicode.org/L2/L2022/22026r-non-bidi-mirroring.pdf" class="">https://www.unicode.org/L2/L2022/22026r-non-bidi-mirroring.pdf</a>.<br class="">
<br class="">
You can find the proposed format(s) specification at<br class="">
<a target="_blank" href="https://github.com/kent-karlsson/control/blob/main/math-layout-controls-2022-C.pdf" class="">https://github.com/kent-karlsson/control/blob/main/math-layout-controls-2022-C.pdf</a>.<br class="">
<br class="">
There is absolutely no claim that this covers everyting w.r.t. math expressions;<br class="">
very likely it does not. But it does cover more than I set out to cover. There is no attempt<br class="">
to be compatible with MathML (sorry, but that would have killed the idea).<br class="">
<br class="">
Comments are welcome.<br class="">
<br class="">
Happy Lucia!*<br class="">
/Kent Karlsson<br class="">
<br class="">
* <a target="_blank" href="https://en.wikipedia.org/w/index.php?title=Saint_lucia%27s_day" class="">https://en.wikipedia.org/w/index.php?title=Saint_lucia%27s_day</a></blockquote></div></div></div></blockquote></div></div></blockquote><div class=""><div class=""><div class=""><blockquote type="cite" class=""><div dir="ltr" class=""><div class="auto-created-dir-div" dir="auto" style="unicode-bidi: embed;"><div class=""><br class=""></div></div></div></blockquote></div></div></div><div><br class=""><blockquote type="cite" class=""><div class="">27 dec. 2022 kl. 10:11 skrev William_J_G Overington via Unicode <<a href="mailto:unicode@corp.unicode.org" class="">unicode@corp.unicode.org</a>>:</div><br class="Apple-interchange-newline"><div class=""><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="display: inline !important;" class=""><br class="Apple-interchange-newline">Hi</span><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I have never used the various existing packages that have been mentioned.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">May I make three observations please?</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">1. I consider that using control codes to specify layout is a problem. A way to express things without using control codes is needed.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">2. Would a test be that what one wants to typeset can be typeset in Microsoft WordPad? One might need to copy and paste characters from a WordPad file that has one of each character in it, as if it were a typecase. For the avoidance of doubt I am not suggesting that all typesetting should be done in WordPad, not at all, but I am saying that if it cannot be typeset in WordPad then a format may be too complicated or too expensive or too inaccessible for widespread use.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">3. Back in the early 1990s I was involved in a discussion of how to express mathematical equations using just 7-bit ASCII characters in a monospaced display typical of mainframe visual display units terminals at the time.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Some of my suggestions might be relevant here.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I suggested using :pom: to express a 'plus or minus' sign as used in the general solution formula for a quadratic equation. That format could be used for special symbols. These days, an OpenType font could cause a correct glyph to be displayed, even if the glyph is not a regular Unicode character.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I suggested that an integral be expressed using three capital I letters, one above the other in three lines of text.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">that then allows upper and lower limits to be expressed for definite integrals.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">For example</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I t=1</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I exp(-t).dt</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I t=0</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Then summation could be expressed as follows.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">S n=5</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">S n^2</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">S n=1</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">and product similarly using three P characters.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">P</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">P</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">P</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">This system could be used to some extent immediately without any additional software being needed. An OpenType font could be used to substitute a 'plus or minus' sign for :pom: and for other symbols. Hopefully software could be written to substitute the three capital I letters with a single integral sign.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">I hope this helps.</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Best regards,</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">William Overington</div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""></div><div style="margin: 0px; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Tuesday 27 December 2022</div></div></blockquote></div><br class=""></div></div></body></html>