<!-- Mathematical, Greek and Symbolic characters for XHTML -->

<!-- Character entity set. Typical invocation:
     <!ENTITY % HTMLsymbol PUBLIC
        "-//W3C//ENTITIES Symbols for XHTML//EN"
        "http://www.w3.org/TR/xhtml1/DTD/xhtml-symbol.ent">
     %HTMLsymbol;
-->

<!-- Portions (C) International Organization for Standardization 1986:
     Permission to copy in any form is granted for use with
     conforming SGML systems and applications as defined in
     ISO 8879, provided this notice is included in all copies.
-->

<!-- Relevant ISO entity set is given unless names are newly introduced.
     New names (i.e., not in ISO 8879 list) do not clash with any
     existing ISO 8879 entity names. ISO 10646 character numbers
     are given for each character, in hex. values are decimal
     conversions of the ISO 10646 values and refer to the document
     character set. Names are Unicode names. 
-->

<!-- Latin Extended-B -->
<!ENTITY fnof     "&#402;"> <!-- latin small letter f with hook = function
                                    = florin, U+0192 ISOtech -->

<!-- Greek -->
<!ENTITY Alpha    "&#913;"> <!-- greek capital letter alpha, U+0391 -->
<!ENTITY Beta     "&#914;"> <!-- greek capital letter beta, U+0392 -->
<!ENTITY Gamma    "&#915;"> <!-- greek capital letter gamma,
                                    U+0393 ISOgrk3 -->
<!ENTITY Delta    "&#916;"> <!-- greek capital letter delta,
                                    U+0394 ISOgrk3 -->
<!ENTITY Epsilon  "&#917;"> <!-- greek capital letter epsilon, U+0395 -->
<!ENTITY Zeta     "&#918;"> <!-- greek capital letter zeta, U+0396 -->
<!ENTITY Eta      "&#919;"> <!-- greek capital letter eta, U+0397 -->
<!ENTITY Theta    "&#920;"> <!-- greek capital letter theta,
                                    U+0398 ISOgrk3 -->
<!ENTITY Iota     "&#921;"> <!-- greek capital letter iota, U+0399 -->
<!ENTITY Kappa    "&#922;"> <!-- greek capital letter kappa, U+039A -->
<!ENTITY Lambda   "&#923;"> <!-- greek capital letter lamda,
                                    U+039B ISOgrk3 -->
<!ENTITY Mu       "&#924;"> <!-- greek capital letter mu, U+039C -->
<!ENTITY Nu       "&#925;"> <!-- greek capital letter nu, U+039D -->
<!ENTITY Xi       "&#926;"> <!-- greek capital letter xi, U+039E ISOgrk3 -->
<!ENTITY Omicron  "&#927;"> <!-- greek capital letter omicron, U+039F -->
<!ENTITY Pi       "&#928;"> <!-- greek capital letter pi, U+03A0 ISOgrk3 -->
<!ENTITY Rho      "&#929;"> <!-- greek capital letter rho, U+03A1 -->
<!-- there is no Sigmaf, and no U+03A2 character either -->
<!ENTITY Sigma    "&#931;"> <!-- greek capital letter sigma,
                                    U+03A3 ISOgrk3 -->
<!ENTITY Tau      "&#932;"> <!-- greek capital letter tau, U+03A4 -->
<!ENTITY Upsilon  "&#933;"> <!-- greek capital letter upsilon,
                                    U+03A5 ISOgrk3 -->
<!ENTITY Phi      "&#934;"> <!-- greek capital letter phi,
                                    U+03A6 ISOgrk3 -->
<!ENTITY Chi      "&#935;"> <!-- greek capital letter chi, U+03A7 -->
<!ENTITY Psi      "&#936;"> <!-- greek capital letter psi,
                                    U+03A8 ISOgrk3 -->
<!ENTITY Omega    "&#937;"> <!-- greek capital letter omega,
                                    U+03A9 ISOgrk3 -->

<!ENTITY alpha    "&#945;"> <!-- greek small letter alpha,
                                    U+03B1 ISOgrk3 -->
<!ENTITY beta     "&#946;"> <!-- greek small letter beta, U+03B2 ISOgrk3 -->
<!ENTITY gamma    "&#947;"> <!-- greek small letter gamma,
                                    U+03B3 ISOgrk3 -->
<!ENTITY delta    "&#948;"> <!-- greek small letter delta,
                                    U+03B4 ISOgrk3 -->
<!ENTITY epsilon  "&#949;"> <!-- greek small letter epsilon,
                                    U+03B5 ISOgrk3 -->
<!ENTITY zeta     "&#950;"> <!-- greek small letter zeta, U+03B6 ISOgrk3 -->
<!ENTITY eta      "&#951;"> <!-- greek small letter eta, U+03B7 ISOgrk3 -->
<!ENTITY theta    "&#952;"> <!-- greek small letter theta,
                                    U+03B8 ISOgrk3 -->
<!ENTITY iota     "&#953;"> <!-- greek small letter iota, U+03B9 ISOgrk3 -->
<!ENTITY kappa    "&#954;"> <!-- greek small letter kappa,
                                    U+03BA ISOgrk3 -->
<!ENTITY lambda   "&#955;"> <!-- greek small letter lamda,
                                    U+03BB ISOgrk3 -->
<!ENTITY mu       "&#956;"> <!-- greek small letter mu, U+03BC ISOgrk3 -->
<!ENTITY nu       "&#957;"> <!-- greek small letter nu, U+03BD ISOgrk3 -->
<!ENTITY xi       "&#958;"> <!-- greek small letter xi, U+03BE ISOgrk3 -->
<!ENTITY omicron  "&#959;"> <!-- greek small letter omicron, U+03BF NEW -->
<!ENTITY pi       "&#960;"> <!-- greek small letter pi, U+03C0 ISOgrk3 -->
<!ENTITY rho      "&#961;"> <!-- greek small letter rho, U+03C1 ISOgrk3 -->
<!ENTITY sigmaf   "&#962;"> <!-- greek small letter final sigma,
                                    U+03C2 ISOgrk3 -->
<!ENTITY sigma    "&#963;"> <!-- greek small letter sigma,
                                    U+03C3 ISOgrk3 -->
<!ENTITY tau      "&#964;"> <!-- greek small letter tau, U+03C4 ISOgrk3 -->
<!ENTITY upsilon  "&#965;"> <!-- greek small letter upsilon,
                                    U+03C5 ISOgrk3 -->
<!ENTITY phi      "&#966;"> <!-- greek small letter phi, U+03C6 ISOgrk3 -->
<!ENTITY chi      "&#967;"> <!-- greek small letter chi, U+03C7 ISOgrk3 -->
<!ENTITY psi      "&#968;"> <!-- greek small letter psi, U+03C8 ISOgrk3 -->
<!ENTITY omega    "&#969;"> <!-- greek small letter omega,
                                    U+03C9 ISOgrk3 -->
<!ENTITY thetasym "&#977;"> <!-- greek theta symbol,
                                    U+03D1 NEW -->
<!ENTITY upsih    "&#978;"> <!-- greek upsilon with hook symbol,
                                    U+03D2 NEW -->
<!ENTITY piv      "&#982;"> <!-- greek pi symbol, U+03D6 ISOgrk3 -->

<!-- General Punctuation -->
<!ENTITY bull     "&#8226;"> <!-- bullet = black small circle,
                                     U+2022 ISOpub  -->
<!-- bullet is NOT the same as bullet operator, U+2219 -->
<!ENTITY hellip   "&#8230;"> <!-- horizontal ellipsis = three dot leader,
                                     U+2026 ISOpub  -->
<!ENTITY prime    "&#8242;"> <!-- prime = minutes = feet, U+2032 ISOtech -->
<!ENTITY Prime    "&#8243;"> <!-- double prime = seconds = inches,
                                     U+2033 ISOtech -->
<!ENTITY oline    "&#8254;"> <!-- overline = spacing overscore,
                                     U+203E NEW -->
<!ENTITY frasl    "&#8260;"> <!-- fraction slash, U+2044 NEW -->

<!-- Letterlike Symbols -->
<!ENTITY weierp   "&#8472;"> <!-- script capital P = power set
                                     = Weierstrass p, U+2118 ISOamso -->
<!ENTITY image    "&#8465;"> <!-- black-letter capital I = imaginary part,
                                     U+2111 ISOamso -->
<!ENTITY real     "&#8476;"> <!-- black-letter capital R = real part symbol,
                                     U+211C ISOamso -->
<!ENTITY trade    "&#8482;"> <!-- trade mark sign, U+2122 ISOnum -->
<!ENTITY alefsym  "&#8501;"> <!-- alef symbol = first transfinite cardinal,
                                     U+2135 NEW -->
<!-- alef symbol is NOT the same as hebrew letter alef,
     U+05D0 although the same glyph could be used to depict both characters -->

<!-- Arrows -->
<!ENTITY larr     "&#8592;"> <!-- leftwards arrow, U+2190 ISOnum -->
<!ENTITY uarr     "&#8593;"> <!-- upwards arrow, U+2191 ISOnum-->
<!ENTITY rarr     "&#8594;"> <!-- rightwards arrow, U+2192 ISOnum -->
<!ENTITY darr     "&#8595;"> <!-- downwards arrow, U+2193 ISOnum -->
<!ENTITY harr     "&#8596;"> <!-- left right arrow, U+2194 ISOamsa -->
<!ENTITY crarr    "&#8629;"> <!-- downwards arrow with corner leftwards
                                     = carriage return, U+21B5 NEW -->
<!ENTITY lArr     "&#8656;"> <!-- leftwards double arrow, U+21D0 ISOtech -->
<!-- Unicode does not say that lArr is the same as the 'is implied by' arrow
    but also does not have any other character for that function. So lArr can
    be used for 'is implied by' as ISOtech suggests -->
<!ENTITY uArr     "&#8657;"> <!-- upwards double arrow, U+21D1 ISOamsa -->
<!ENTITY rArr     "&#8658;"> <!-- rightwards double arrow,
                                     U+21D2 ISOtech -->
<!-- Unicode does not say this is the 'implies' character but does not have 
     another character with this function so rArr can be used for 'implies'
     as ISOtech suggests -->
<!ENTITY dArr     "&#8659;"> <!-- downwards double arrow, U+21D3 ISOamsa -->
<!ENTITY hArr     "&#8660;"> <!-- left right double arrow,
                                     U+21D4 ISOamsa -->

<!-- Mathematical Operators -->
<!ENTITY forall   "&#8704;"> <!-- for all, U+2200 ISOtech -->
<!ENTITY part     "&#8706;"> <!-- partial differential, U+2202 ISOtech  -->
<!ENTITY exist    "&#8707;"> <!-- there exists, U+2203 ISOtech -->
<!ENTITY empty    "&#8709;"> <!-- empty set = null set, U+2205 ISOamso -->
<!ENTITY nabla    "&#8711;"> <!-- nabla = backward difference,
                                     U+2207 ISOtech -->
<!ENTITY isin     "&#8712;"> <!-- element of, U+2208 ISOtech -->
<!ENTITY notin    "&#8713;"> <!-- not an element of, U+2209 ISOtech -->
<!ENTITY ni       "&#8715;"> <!-- contains as member, U+220B ISOtech -->
<!ENTITY prod     "&#8719;"> <!-- n-ary product = product sign,
                                     U+220F ISOamsb -->
<!-- prod is NOT the same character as U+03A0 'greek capital letter pi' though
     the same glyph might be used for both -->
<!ENTITY sum      "&#8721;"> <!-- n-ary summation, U+2211 ISOamsb -->
<!-- sum is NOT the same character as U+03A3 'greek capital letter sigma'
     though the same glyph might be used for both -->
<!ENTITY minus    "&#8722;"> <!-- minus sign, U+2212 ISOtech -->
<!ENTITY lowast   "&#8727;"> <!-- asterisk operator, U+2217 ISOtech -->
<!ENTITY radic    "&#8730;"> <!-- square root = radical sign,
                                     U+221A ISOtech -->
<!ENTITY prop     "&#8733;"> <!-- proportional to, U+221D ISOtech -->
<!ENTITY infin    "&#8734;"> <!-- infinity, U+221E ISOtech -->
<!ENTITY ang      "&#8736;"> <!-- angle, U+2220 ISOamso -->
<!ENTITY and      "&#8743;"> <!-- logical and = wedge, U+2227 ISOtech -->
<!ENTITY or       "&#8744;"> <!-- logical or = vee, U+2228 ISOtech -->
<!ENTITY cap      "&#8745;"> <!-- intersection = cap, U+2229 ISOtech -->
<!ENTITY cup      "&#8746;"> <!-- union = cup, U+222A ISOtech -->
<!ENTITY int      "&#8747;"> <!-- integral, U+222B ISOtech -->
<!ENTITY there4   "&#8756;"> <!-- therefore, U+2234 ISOtech -->
<!ENTITY sim      "&#8764;"> <!-- tilde operator = varies with = similar to,
                                     U+223C ISOtech -->
<!-- tilde operator is NOT the same character as the tilde, U+007E,
     although the same glyph might be used to represent both  -->
<!ENTITY cong     "&#8773;"> <!-- approximately equal to, U+2245 ISOtech -->
<!ENTITY asymp    "&#8776;"> <!-- almost equal to = asymptotic to,
                                     U+2248 ISOamsr -->
<!ENTITY ne       "&#8800;"> <!-- not equal to, U+2260 ISOtech -->
<!ENTITY equiv    "&#8801;"> <!-- identical to, U+2261 ISOtech -->
<!ENTITY le       "&#8804;"> <!-- less-than or equal to, U+2264 ISOtech -->
<!ENTITY ge       "&#8805;"> <!-- greater-than or equal to,
                                     U+2265 ISOtech -->
<!ENTITY sub      "&#8834;"> <!-- subset of, U+2282 ISOtech -->
<!ENTITY sup      "&#8835;"> <!-- superset of, U+2283 ISOtech -->
<!ENTITY nsub     "&#8836;"> <!-- not a subset of, U+2284 ISOamsn -->
<!ENTITY sube     "&#8838;"> <!-- subset of or equal to, U+2286 ISOtech -->
<!ENTITY supe     "&#8839;"> <!-- superset of or equal to,
                                     U+2287 ISOtech -->
<!ENTITY oplus    "&#8853;"> <!-- circled plus = direct sum,
                                     U+2295 ISOamsb -->
<!ENTITY otimes   "&#8855;"> <!-- circled times = vector product,
                                     U+2297 ISOamsb -->
<!ENTITY perp     "&#8869;"> <!-- up tack = orthogonal to = perpendicular,
                                     U+22A5 ISOtech -->
<!ENTITY sdot     "&#8901;"> <!-- dot operator, U+22C5 ISOamsb -->
<!-- dot operator is NOT the same character as U+00B7 middle dot -->

<!-- Miscellaneous Technical -->
<!ENTITY lceil    "&#8968;"> <!-- left ceiling = APL upstile,
                                     U+2308 ISOamsc  -->
<!ENTITY rceil    "&#8969;"> <!-- right ceiling, U+2309 ISOamsc  -->
<!ENTITY lfloor   "&#8970;"> <!-- left floor = APL downstile,
                                     U+230A ISOamsc  -->
<!ENTITY rfloor   "&#8971;"> <!-- right floor, U+230B ISOamsc  -->
<!ENTITY lang     "&#9001;"> <!-- left-pointing angle bracket = bra,
                                     U+2329 ISOtech -->
<!-- lang is NOT the same character as U+003C 'less than sign' 
     or U+2039 'single left-pointing angle quotation mark' -->
<!ENTITY rang     "&#9002;"> <!-- right-pointing angle bracket = ket,
                                     U+232A ISOtech -->
<!-- rang is NOT the same character as U+003E 'greater than sign' 
     or U+203A 'single right-pointing angle quotation mark' -->

<!-- Geometric Shapes -->
<!ENTITY loz      "&#9674;"> <!-- lozenge, U+25CA ISOpub -->

<!-- Miscellaneous Symbols -->
<!ENTITY spades   "&#9824;"> <!-- black spade suit, U+2660 ISOpub -->
<!-- black here seems to mean filled as opposed to hollow -->
<!ENTITY clubs    "&#9827;"> <!-- black club suit = shamrock,
                                     U+2663 ISOpub -->
<!ENTITY hearts   "&#9829;"> <!-- black heart suit = valentine,
                                     U+2665 ISOpub -->
<!ENTITY diams    "&#9830;"> <!-- black diamond suit, U+2666 ISOpub -->