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All reported errata to the first edition have been addressed in this addition, and a full change log appears in Appendix F Changes. The diff-marked version linked in the frontmatter highlights all changes between the first and second editions. In addition to incorporating errata, the main change in this addition is to recognise that MathML parsing is also specified in [HTML5] and where necessary to note where HTML and XML usage differ.

No matter how successfully MathML achieves its goals as a markup language, it is clear that MathML is useful only if it is implemented well. The W3C Math Working Group has identified a short list of additional implementation goals. These goals attempt to describe concisely the minimal functionality MathML rendering and processing software should try to provide.

A color is specified either by "#" followed by hexadecimal values for the red, green, and blue components, with no intervening whitespace, or by an html-color-name. The color components can be either 1-digit or 2-digit, but must all have the same number of digits; the component ranges from 0 (component not present) to FF (component fully present). Note that, for example, by the digit-doubling rule specified under Colors in [CSS21] #123 is a short form for #112233.

As the previous examples show, to be useful, the concept of MathML conformance frequently involves a judgment about what parts of the language are meaningfully implemented, as opposed to parts that are merely processed in a technically correct way with respect to the definitions of conformance. This requires some mechanism for giving a quantitative statement about which parts of MathML are meaningfully implemented by a given application. To this end, the W3C Math Working Group has provided a test suite.

The test suite consists of a large number of MathML expressions categorized by markup category and dominant MathML element being tested. The existence of this test suite makes it possible, for example, to characterize quantitatively the hypothetical computer algebra interface mentioned above by saying that it is a MathML-input-conformant processor which meaningfully implements MathML content markup, including all of the expressions in the content markup section of the test suite.

Similarly, superscripts are attached to the full expression constituting their base rather than to the just preceding character. This structure permits better-quality rendering of mathematics, especially when details of the rendering environment, such as display widths, are not known ahead of time to the document author. It also greatly eases automatic interpretation of the represented mathematical structures.

Characters can be either represented directly as Unicode character data, or indirectly via numeric or character entity references. See Chapter 7 Characters, Entities and Fonts for a discussion of the advantages and disadvantages of numeric character references versus entity references, and [Entities] for a full list of the entity names available. Also, see Section 7.7 Anomalous Mathematical Characters for a discussion of the appropriate character content to choose for certain applications.

Note that since the rendering context, such as the available width and current font, is not always available to the author of the MathML, a render may ignore the values of these attributes if they result in a line in which the remaining width is too small to usefully display the expression or if they result in a line in which the remaining width exceeds the available linewrapping width.

The value "northeastarrow" is a recommended value to implement because it can be used to implement TeX's \cancelto command. If a renderer implements other arrows for menclose, it is recommended that the arrow names are chosen from the following full set of names for consistancy and standardization among renderers:

A particularly important case for renderers to handle gracefully is the interaction of alignment elements with the matrix content element, since this element may or may not be internally converted to an expression containing an mtable element for rendering. To partially resolve this ambiguity, it is suggested, but not required, that if the matrix element is converted to an expression involving an mtable element, that the mtable element be given the attribute alignmentscope="false", which will make the interaction of the matrix element with the alignment elements no different than that of a generic presentation element (in particular, it will allow it to contain malignmark elements that operate within the alignment scopes created by the columns of an mtable that contains the matrix element in one of its table cells).

In MathML 3, a subset, or profile, of Content MathML is defined: Strict Content MathML. This uses a minimal, but sufficient, set of elements to represent the meaning of a mathematical expression in a uniform structure, while the full Content MathML grammar is backward compatible with MathML 2.0, and generally tries to strike a more pragmatic balance between verbosity and formality.

In MathML 3, formal semantics Content MathML expressions are given by specifying equivalent Strict Content MathML expressions. Since Strict Content MathML expressions all have carefully-defined semantics given in terms of OpenMath Content Dictionaries, all Content MathML expressions inherit well-defined semantics in this way. To make the correspondence exact, an algorithm is given in terms of transformation rules that are applied to rewrite non-Strict MathML constructs into a strict equivalents. The individual rules are introduced in context throughout the chapter. In Section 4.6 The Strict Content MathML Transformation, the algorithm as a whole is described.

The full algorithm described inSection 4.6 The Strict Content MathML Transformation is complete in the sense that it gives every Content MathML expression a specific meaning in terms of a Strict Content MathML expression. This means it has to give specific strict interpretations to some expressions whose meaning was insufficiently specified in MathML2. The intention of this algorithm is to be faithful to mathematical intuitions. However edge cases may remain where the normative interpretation of the algorithm may break earlier intuitions.

In the text below, descriptions of semantics for predefined MathML symbols refer to the Content Dictionaries developed by the OpenMath Society in conjunction with the W3C Math Working Group. It is important to note, however, that this information is informative, and not normative. In general, the precise mathematical semantics of predefined symbols are not not fully specified by the MathML 3 Recommendation, and the only normative statements about symbol semantics are those present in the text of this chapter. The semantic definitions provided by the OpenMath Content CDs are intended to be sufficient for most applications, and are generally compatible with the semantics specified for analogous constructs in the MathML 2.0 Recommendation. However, in contexts where highly precise semantics are required (e.g. communication between computer algebra systems, within formal systems such as theorem provers, etc.) it is the responsibility of the relevant community of practice to verify, extend or replace definitions provided by OpenMath CDs as appropriate.

External definitions (in OpenMath CDs or elsewhere) may also be specified directly for a csymbol using the definitionURL attribute. When used to reference OpenMath symbol definitions, the abstract triple of (symbol name, CD name, CD base) is mapped to a fully-qualified URI as follows:

The OpenGL ES 3.0 specification[17] was publicly released in August 2012.[18] OpenGL ES 3.0 is backwards compatible with OpenGL ES 2.0, enabling applications to incrementally add new visual features to applications. OpenGL 4.3 provides full compatibility with OpenGL ES 3.0. Version 3.0 is also the basis for WebGL 2.0.[19]The actual is version 3.0.6.[20]

Time Binning with Option tbinYou can also invoke a time binning style of auto-windowing with option -tbin. The -tbin option is only available in versions of teqc made on or after 18 July 2008. This is a powerful option to create batches of RINEX or BINEX output files in various lengths, e.g. daily, 8-hour, hourly, 30-minute, 15-minute, 5-minute, ..., from one or more input files (manufacturers' formats, RINEX, or BINEX, and as usual, the input files all have to be of the same type). The -tbin option can create smaller output files than the input files (e.g. hourly files from daily files), and it also makes long files from several shorter files. It can use mixed length input files. So, the input might be a list of files of varying lengths, some shorter than a day, and the output could be binned into daily files. Two options are used for time binning. The first and required option is -tbin which has two arguments, such as -tbin 1h myfile. The first and required argument, such as 1h, is the time duration for each bin (output file). The trailing letter specifies time units: s for seconds, m for minutes, h for hours, d for days. You must use d, h, m, or s. Time durations in sub-seconds are allowed: if sub-seconds are indicated, then resulting filenames will expand to include an additional ".ddd" after the seconds value showing the time down to milliseconds. Any value for the time duration is allowed: caveat emptor! If you type 1s and the input data is most of a day long, you are asking for tens of thousands of files to be made. The second argument for -tbin, such as myfile, is the prefix part of the output filenames. The prefix can be a 4-char ID, or some other printable string without whitespace. The output filenames have the form == prefix + doy-of-year, where the filenames themselves would be:a) 0.yyo if delta (bin time duration) in days is an integerb) a.yyo - x.yyo if case (a) doesn't work, but24/(duration in hours) results in an integer hour, with the hour indicated by a letter from a to x:a = start in hour 0, ..., x = start in hour 23c) [a-x]00.yyo - [a-x]59.yyo if cases (a) and (b) don't work,but 60/(delta in minutes) results in an integer00 = start in minute 0, ..., 59 = start in minute 59d) [a-x][00-59]00.yyo - [a-x][00-59]59.yyo if cases (a)-(c)don't work, but 60/(delta in seconds) results in an integer (or maybeall remaining cases as well ...)00 = start in second 0, ..., 59 = start in second 59In a teqc command like teqc -tr d +obs + +nav + -tbin 1h mytbinfile inputdatafile the extra bare + signs after +obs and +nav mean make a sequence of obs and nav files which match the 1 hour time bins made with the -tbin 1h temp arguments. The + argument after the +obs and +nav options only has meaning when using the -tbin option. If you tried that without -tbin you'd end up with a file named + . The option +nav +,+ means make separate NAVSTAR GPS and GLONASS nav files for the time bins, butthis syntax can also be extended to also include SBAS, Galileo, Beidou/Compass, QZSS, and IRNSS -- in that order.For example, a command to translate from Trimble to RINEXteqc -tr d +obs + +nav myfull.nav -tbin 1h myhourly grnr2600.dat will put all the GPS nav messages into one file myfull.nav, but the RINEX obs are tbinned as hourly files:teqc: creating file myhourly260a.97o ...teqc: creating file myhourly260b.97o ...teqc: creating file myhourly260c.97o ...This example may be what you want if the tbin window is shorter than about 2-4 hours. Recall that the -tr d part is not required when the input file is a Trimble DAT/ION/EPH/MES download fileset (dat file) file.There is a second option, -ast, aligned start time, which may be used with -tbin to specify the start time of the first bin. This option is not required for time binning. If no -ast option is specified, time binning uses a default alignment starting at 00:00:00 on the first data (obs, nav, or met). Option -ast - or -ast _ means start alignment to the first epoch that is output; -ast [[[[[[YY]YY]MM]DD]hh]mm]ss[.sssss] means start alignment at the specified time. Sub-seconds may be used with -ast.Note the common and different option -st is reserved to mean when to start the data, which by default is the first epoch found. Don't confuse -ast and -st.Option tbin works for all these cases:N manufacturer's format or BINEX files -> M RINEX filesetsN RINEX files of same type -> M RINEX files of same type as inputN manufacturer's format or BINEX files -> M BINEX filesN RINEX filesets -> M BINEX filesThe RINEX filenaming scheme with tbin is:daily = 0.[onghem]hourly = [a-x].[onghem]minute = [a-x]00.[onghem] - [a-x]59.[onghem]second = [a-x][00-59]00.[onghem] - [a-x][00-59]59.[onghem]subsec = [a-x][00-59]00.000.[onghem] - [a-x][00-59]59.999.[onghem]where -- user supplied == day of year[a-x] for hours 00 - 23 == year modulo 100 [onghem] -- RINEX suffix, e.g. 'o' for RINEX obs file, etc.Note that teqc should select the coarsest filenaming binning in order to do what you've asked. Which filenaming binning is used depends on the -tbin unit selected or the -ast time unit. For example if using the default -ast (i.e. not explicitly specified) and using -tbin 30m test, then the minute filenaming binning is used; but if -tbin 1h test or -tbin 60m test, then the hourly filenaming binning is used. For creating RINEX with time-binning, you have pretty good control over which files end up being created with time-binned names; e.g.:+nav temp.gps,temp.glo +obs + -tbin 1h temp == all GPS nav messages go into RINEX file temp.gps and all GLONASS nav messages go into RINEX file temp.glo (in other words, neither are time binned), but all RINEX obs files are time-binned (to 1-hour in this case)+nav +,temp.glo +obs + -tbin 60m temp == all GLONASS nav messages go into RINEX file temp.glo, but all RINEX GPS nav files and all RINEX obs files are time-binned (again, to 1-hour)+nav +,+ +obs + +met + -tbin 3600s temp == all RINEX (GPS and GLONASS nav, obs, and met) files are time-binned (and again, to 1-hour)+tbin 3600s temp is a short-hand for the previous command (notice the + in +tbin), i.e., all RINEX files are time-binned.The only difference to create time-binned BINEX is to use the +binex option (which takes its usual argument), and the resulting BINEX files will be named differently from time-binned RINEX: = prefix + GPS week + '_' + day_of_week (Sun=0,...,Sat=6)daily = .bnxhourly = [a-x].bnxminute = [a-x]00.bnx - [a-x]59.bnxsecond = [a-x][00-59]00.bnx - [a-x][00-59]59.bnxsubsec = [a-x][00-59]00.ddd.bnx - [a-x][00-59]59.ddd.bnxExamples of tbin usage:1) You want the input broken up into daily files and the created obs filenames to start with test:teqc -tr d +obs + -tbin 1d test input.obsIf, say, the data started on day 2007:165, the output files would be named:test1650.07otest1660.07otest1670.07otest1670.07o...No nav files are made.2) You want the input broken up into files of 1/12 sidereal days with filenames starting with sidx, and you want the first file to be time aligned to 00h30m09s (GPS time, as usual) of the first day of data, andyou want the data to start at 01h30m10s in the first file; use:teqc -tr d +obs + -tbin 7180.3409 sidx -ast 00:30:09.000 -st 01:30:10 input.obsIf the data started on day 2007:165, then the output files would be named:sidx165a3009.07o (but data here doesn't start in file until 01:30:10)sidx165c2950.07osidx165e2930.07o...Note that a sidereal day is 23h 56m 4.091s, so 1/12th is 7180.3409 seconds.3) With an input of one Topcon TPS file, cp_1p.tps the commandteqc cp_1p.tpstranslates the input to RINEX as a single stream to stdout.To create hourly time-binned RINEX obs files, and distinct nav files for GPS and GLONASS, use:teqc +nav +,+ +obs + -tbin 1h temp cp_1p.jpscreating temp027m.03n ...creating temp027k.03o ...creating temp027k.03g ...creating temp027l.03o ...creating temp027l.03g ...creating temp027o.03n ...creating temp027m.03o ...creating temp027m.03g ...4) If you want the nav messages in a single RINEX nav file (temp0270.03n) for GPS and a single RINEX nav file (temp0270.03g) for GLONASS, with time binning making 1 hour obs files:teqc +nav temp0270.03n,temp0270.03g +obs + -tbin 1h temp cp_1p.jpscreating temp027k.03o ...creating temp027l.03o ...creating temp027m.03o ...creating temp027n.03o ...creating temp027o.03o ...creating temp027p.03o ...5) Use of the +tbin short-handteqc +tbin 1h mytbinfile grnr2600.datis a short-hand for teqc +nav +,+ +obs + +met + -tbin 1h mytbinfile grnr2600.dat Notice the +tbin instead of -tbin.6) BINEX example with +tbinteqc +binex 0x7f-03 +tbin 15m tmp input.obs! Notice ! using RINEX OBS default observable list! Notice ! using RINEX MET default observable listteqc: creating file tmp1488_1a00.bnx ...teqc: creating file tmp1488_1a15.bnx ...teqc: creating file tmp1488_1a30.bnx ...teqc: creating file tmp1488_1a45.bnx ...The options +obs, +nav, or +met will not be required with the -tbin option in new versions of teqc, if the input target files are RINEX (but if the input is more than one file, the input files still all have to be of the same type). This simplified command syntax choice will be in the next full release after September 2009.Splicing with teqcSection 14.Recall that executing the commandteqc fbar0010.97obasically spews the contents of fbar0010.97o back out to stdout. Supposeyou have the RINEX OBS files fbar0010.97ofor 1 Jan 1997 and fbar0020.97ofor 2 Jan 1997 and you want to combine them into a single RINEX OBS file.It would have been easy if the RINEX standard had been written so that twoRINEX files could be simply concatenated to one another to produce a newvalid RINEX file, a la the UNIX cat system command:cat fbar0010.97o fbar0020.97o > oops0010.97oBut, alas, the RINEX standard does not allow this sort of obvious simplicityand thus the file oops0010.97o is generally useless.However, teqc takes care of the RINEX-idiosyncraticboundary between the two files. Thusteqc fbar0010.97o fbar002