ÿþ<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns:st1="urn:schemas-microsoft-com:office:smarttags" xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=Content-Type content="text/html; charset=unicode"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 12"> <meta name=Originator content="Microsoft Word 12"> <link rel=File-List href="www_archivos/filelist.xml"> <link rel=Edit-Time-Data href="www_archivos/editdata.mso"> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--> <title>Antonio Montes - Personal Page</title> <o:SmartTagType namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="City"/> <o:SmartTagType namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="country-region"/> <o:SmartTagType namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="metricconverter"/> <o:SmartTagType namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="place"/> <!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Author>MA2</o:Author> <o:Template>Normal</o:Template> <o:LastAuthor>Anton Montes</o:LastAuthor> <o:Revision>4</o:Revision> <o:TotalTime>256</o:TotalTime> <o:Created>2012-05-16T09:15:00Z</o:Created> <o:LastSaved>2012-05-16T09:30:00Z</o:LastSaved> <o:Pages>3</o:Pages> <o:Words>2859</o:Words> <o:Characters>15730</o:Characters> <o:Company>UPC</o:Company> <o:Lines>131</o:Lines> <o:Paragraphs>37</o:Paragraphs> <o:CharactersWithSpaces>18552</o:CharactersWithSpaces> <o:Version>12.00</o:Version> </o:DocumentProperties> </xml><![endif]--> <link rel=themeData href="www_archivos/themedata.thmx"> <link rel=colorSchemeMapping href="www_archivos/colorschememapping.xml"> <!--[if gte mso 9]><xml> <w:WordDocument> <w:TrackMoves>false</w:TrackMoves> <w:TrackFormatting/> <w:HyphenationZone>21</w:HyphenationZone> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>ES</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="&#45;-"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="false" DefSemiHidden="false" DefQFormat="false" LatentStyleCount="267"> <w:LsdException Locked="false" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="caption"/> <w:LsdException Locked="false" QFormat="true" Name="Title"/> <w:LsdException Locked="false" Priority="1" Name="Default Paragraph Font"/> <w:LsdException Locked="false" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" Priority="99" Name="No List"/> <w:LsdException Locked="false" Priority="99" SemiHidden="true" Name="Placeholder Text"/> <w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading"/> <w:LsdException Locked="false" Priority="61" Name="Light List"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/> <w:LsdException Locked="false" Priority="70" Name="Dark List"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/> <w:LsdException Locked="false" Priority="99" SemiHidden="true" Name="Revision"/> <w:LsdException Locked="false" Priority="34" QFormat="true" Name="List Paragraph"/> <w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/> <w:LsdException Locked="false" Priority="30" QFormat="true" Name="Intense Quote"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" SemiHidden="true" UnhideWhenUsed="true" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/> </w:LatentStyles> </xml><![endif]--><!--[if !mso]><object classid="clsid:38481807-CA0E-42D2-BF39-B33AF135CC4D" id=ieooui></object> <style> st1\:*{behavior:url(#ieooui) } </style> <![endif]--> <style> <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:-1610611985 1107304683 0 0 159 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-1610611985 1073750139 0 0 159 0;} @font-face {font-family:t1-gul-regular; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:LMRoman9-Regular; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:LMRomanCaps10-Regular; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-alt:"MS Mincho"; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:0 134676480 16 0 131073 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; color:black;} h2 {mso-style-unhide:no; mso-style-qformat:yes; mso-style-link:"Título 2 Car"; mso-margin-top-alt:auto; margin-right:0cm; mso-margin-bottom-alt:auto; margin-left:0cm; mso-pagination:widow-orphan; mso-outline-level:2; font-size:18.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; color:black; font-weight:bold;} h3 {mso-style-unhide:no; mso-style-qformat:yes; mso-style-link:"Título 3 Car"; mso-margin-top-alt:auto; margin-right:0cm; mso-margin-bottom-alt:auto; margin-left:0cm; mso-pagination:widow-orphan; mso-outline-level:3; font-size:13.5pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; color:black; font-weight:bold;} h4 {mso-style-unhide:no; mso-style-qformat:yes; mso-style-link:"Título 4 Car"; mso-margin-top-alt:auto; margin-right:0cm; mso-margin-bottom-alt:auto; margin-left:0cm; mso-pagination:widow-orphan; mso-outline-level:4; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; color:black; font-weight:bold;} a:link, span.MsoHyperlink {mso-style-unhide:no; color:maroon; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {mso-style-unhide:no; color:olive; text-decoration:underline; text-underline:single;} p {mso-style-unhide:no; mso-margin-top-alt:auto; margin-right:0cm; mso-margin-bottom-alt:auto; margin-left:0cm; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; color:black;} span.Ttulo2Car {mso-style-name:"Título 2 Car"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Título 2"; mso-ansi-font-size:13.0pt; mso-bidi-font-size:13.0pt; font-family:"Cambria","serif"; mso-ascii-font-family:Cambria; mso-ascii-theme-font:major-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:major-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi; color:#4F81BD; mso-themecolor:accent1; font-weight:bold;} span.Ttulo3Car {mso-style-name:"Título 3 Car"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Título 3"; mso-ansi-font-size:12.0pt; mso-bidi-font-size:12.0pt; font-family:"Cambria","serif"; mso-ascii-font-family:Cambria; mso-ascii-theme-font:major-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:major-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi; color:#4F81BD; mso-themecolor:accent1; font-weight:bold;} span.Ttulo4Car {mso-style-name:"Título 4 Car"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Título 4"; mso-ansi-font-size:12.0pt; mso-bidi-font-size:12.0pt; font-family:"Cambria","serif"; mso-ascii-font-family:Cambria; mso-ascii-theme-font:major-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:major-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi; color:#4F81BD; mso-themecolor:accent1; font-weight:bold; font-style:italic;} span.Ttol2Car {mso-style-name:"Títol 2 Car"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Títol 2"; mso-ansi-font-size:13.0pt; mso-bidi-font-size:13.0pt; font-family:"Cambria","serif"; mso-ascii-font-family:Cambria; mso-ascii-theme-font:major-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:major-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi; color:#4F81BD; mso-themecolor:accent1; font-weight:bold;} p.Ttol2, li.Ttol2, div.Ttol2 {mso-style-name:"Títol 2"; mso-style-unhide:no; mso-style-link:"Títol 2 Car"; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; color:black;} span.Ttol3Car {mso-style-name:"Títol 3 Car"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Títol 3"; mso-ansi-font-size:12.0pt; mso-bidi-font-size:12.0pt; font-family:"Cambria","serif"; mso-ascii-font-family:Cambria; mso-ascii-theme-font:major-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:major-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi; color:#4F81BD; mso-themecolor:accent1; font-weight:bold;} p.Ttol3, li.Ttol3, div.Ttol3 {mso-style-name:"Títol 3"; mso-style-unhide:no; mso-style-link:"Títol 3 Car"; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; color:black;} span.Ttol4Car {mso-style-name:"Títol 4 Car"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Títol 4"; mso-ansi-font-size:12.0pt; mso-bidi-font-size:12.0pt; font-family:"Cambria","serif"; mso-ascii-font-family:Cambria; mso-ascii-theme-font:major-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:major-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi; color:#4F81BD; mso-themecolor:accent1; font-weight:bold; font-style:italic;} p.Ttol4, li.Ttol4, div.Ttol4 {mso-style-name:"Títol 4"; mso-style-unhide:no; mso-style-link:"Títol 4 Car"; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; color:black;} p.1, li.1, div.1 {mso-style-name:1; mso-style-priority:99; mso-style-unhide:no; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:none; text-autospace:none; font-size:10.0pt; font-family:"Arial","sans-serif"; mso-fareast-font-family:"Times New Roman"; color:black;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt;} @page WordSection1 {size:595.3pt 841.9pt; margin:70.85pt 3.0cm 70.85pt 3.0cm; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} /* List Definitions */ @list l0 {mso-list-id:619842437; mso-list-template-ids:672404026;} @list l0:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l0:level2 {mso-level-tab-stop:72.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level3 {mso-level-tab-stop:108.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level4 {mso-level-tab-stop:144.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level5 {mso-level-tab-stop:180.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level6 {mso-level-tab-stop:216.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level7 {mso-level-tab-stop:252.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level8 {mso-level-tab-stop:288.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level9 {mso-level-tab-stop:324.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1 {mso-list-id:909996822; mso-list-template-ids:-1890943272;} @list l1:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l1:level2 {mso-level-tab-stop:72.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level3 {mso-level-tab-stop:108.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level4 {mso-level-tab-stop:144.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level5 {mso-level-tab-stop:180.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level6 {mso-level-tab-stop:216.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level7 {mso-level-tab-stop:252.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level8 {mso-level-tab-stop:288.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level9 {mso-level-tab-stop:324.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2 {mso-list-id:1107121333; mso-list-template-ids:-669479536;} @list l2:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l2:level2 {mso-level-tab-stop:72.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2:level3 {mso-level-tab-stop:108.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2:level4 {mso-level-tab-stop:144.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2:level5 {mso-level-tab-stop:180.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2:level6 {mso-level-tab-stop:216.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2:level7 {mso-level-tab-stop:252.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2:level8 {mso-level-tab-stop:288.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l2:level9 {mso-level-tab-stop:324.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l3 {mso-list-id:1254557363; mso-list-template-ids:-2143409718;} @list l3:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l4 {mso-list-id:1980916122; mso-list-template-ids:1353762674;} @list l4:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l5 {mso-list-id:2107266964; mso-list-template-ids:734682566;} @list l5:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; mso-ansi-font-size:10.0pt; font-family:Symbol;} ol {margin-bottom:0cm;} ul {margin-bottom:0cm;} --> </style> <!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} table.Taulanormal {mso-style-name:"Taula normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--><!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="5122"/> </xml><![endif]--><!--[if gte mso 9]><xml> <o:shapelayout v:ext="edit"> <o:idmap v:ext="edit" data="1"/> </o:shapelayout></xml><![endif]--> </head> <body bgcolor="#FFFFE8" lang=ES link=maroon vlink=olive style='tab-interval: 35.4pt' alink="#ff0000"> <div class=WordSection1> <h2><span style='mso-fareast-font-family:"Times New Roman"'>&nbsp;<o:p></o:p></span></h2> <blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'> <p class=MsoNormal><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="Imatge_x0020_2" o:spid="_x0000_s1030" type="#_x0000_t75" alt="Descripción: C:\Users\hmercade\Desktop\montes-small.JPG" style='position:absolute; margin-left:0;margin-top:0;width:150pt;height:195pt;z-index:1;visibility:visible; mso-wrap-style:square;mso-wrap-distance-left:0;mso-wrap-distance-top:0; mso-wrap-distance-right:0;mso-wrap-distance-bottom:0; mso-position-horizontal:left;mso-position-horizontal-relative:text; mso-position-vertical:absolute;mso-position-vertical-relative:line' o:allowoverlap="f"> <v:imagedata src="montes-small.JPG"/> <w:wrap type="square" anchory="line"/> </v:shape><![endif]--><![if !vml]><img width=200 height=260 src=montes-small.JPG align=left alt="Descripción: C:\Users\hmercade\Desktop\montes-small.JPG" v:shapes="Imatge_x0020_2"><![endif]>&nbsp; <b><span style='font-size:18.0pt'><br> &nbsp; </span></b><b><span style='font-size:18.0pt;font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'>Antonio Montes</span></b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'> <o:p></o:p></span></p> <p style='margin-bottom:12.0pt'><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'>&nbsp;&nbsp; Departament de Matemàtica Aplicada II <br> &nbsp;&nbsp; Universitat Politècnica de Catalunya (UPC)<br> &nbsp;&nbsp; Edifici Omega <br> &nbsp;&nbsp; Jordi Girona 1-3<br> &nbsp;&nbsp; 08034 Barcelona<br> &nbsp;&nbsp; Spain <br> <b>&nbsp;&nbsp;&nbsp; e-mail: </b><a href="mailto:antonio.montes@upc.edu">antonio.montes@upc.edu</a> <br> <b>&nbsp;&nbsp;&nbsp; tel: </b>+ 34 93 4131 77 04 (<b>Secretary:</b> +34 93 41376 80) <br> <b>&nbsp;&nbsp;&nbsp; fax: </b>+ 34 93 413 77 01<o:p></o:p></span></p> </blockquote> <div class=MsoNormal align=center style='text-align:center'> <hr size=2 width="100%" noshade style='color:#ACA899' align=center> </div> <h3><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin'>Institutional links<o:p></o:p></span></h3> <ul type=disc> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l1 level1 lfo3;tab-stops:list 36.0pt'><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="http://www-ma2.upc.edu/">Departament de Matemàtica Aplicada II</a><o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l1 level1 lfo3;tab-stops:list 36.0pt'><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="http://www.fib.upc.edu">Facultat d'Informàtica de Barcelona</a><o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l1 level1 lfo3;tab-stops:list 36.0pt'><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="http://www-fme.upc.edu">Facultat de Matemàtiques i Estadística</a><o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l1 level1 lfo3;tab-stops:list 36.0pt'><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="http://www.upc.edu">Universitat Politècnica de Catalunya</a><o:p></o:p></span></li> </ul> <div class=MsoNormal align=center style='text-align:center'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin'> <hr size=2 width="100%" noshade style='color:#ACA899' align=center> </span></div> <h3><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB'>Teaching<o:p></o:p></span></h3> <p class=MsoNormal><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>Courses in the period 1981-2012. <o:p></o:p></span></p> <ul type=disc> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'>Computer Algebra</span></i><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'> (Facultat de Matemàtiques i Estadística). </span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US'>5th course. <o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Calculus</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'> (Facultat d'Informàtica). 1st course.<o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Symbolic Computation Workshop</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB'> (Facultat de Matemàtiques i Estadística).&nbsp; </span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US'>Free in </span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'>UPC.<o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Mathematical Methods 1</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB'> (Facultat de Matemàtiques i Estadística). </span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin'>1st course. <o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'>Mathematics 1</span></i><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'> (Facultat d'Informàtica de Barcelona). </span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US'>1st course.<o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'>Discrete Mathematics</span></i><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'> (Facultat d'Informàtica de Barcelona). 2nd course.<o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'>Algebra</span></i><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'> (Facultat d'Informàtica de Barcelona). 1st course.<o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Algebra and Computing</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB'> (Facultat de Matemàtiques i Estadística). 1st course. </span><span lang=EN-US style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-US'><o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'>Numerical Methods</span></i><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'> (Facultat d'Informàtica de Barcelona). 3rd course.<o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'>Analysis 2</span></i><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'> (Facultat d'Informàtica de Barcelona). 2nd course.<o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US'>Fonaments de Matemàtiques </span></i><span lang=EN-US style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-US;mso-bidi-font-style:italic'>(Facultat d Inform</span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ansi-language: EN-US;mso-bidi-font-style:italic'>à</span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US;mso-bidi-font-style: italic'>tica). 1st course.</span><span lang=EN-US style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-US'><o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l2 level1 lfo6;tab-stops:list 36.0pt'><i><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US'>Matemàtiques 2 </span></i><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US; mso-bidi-font-style:italic'>(Facultat d Inform</span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ansi-language:EN-US; mso-bidi-font-style:italic'>à</span><span lang=EN-US style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-US;mso-bidi-font-style:italic'>tica). 1<sup>st</sup> course<i>.</i></span><span lang=EN-US style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-US'><o:p></o:p></span></li> </ul> <p><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin'>Àlgebra Computacional. Assignatura optativa de la Llicenciatura de Matemàtiques i Estadística de la Facultat de Matemàtiques i Estadística de la UPC.<o:p></o:p></span></p> <ul type=disc> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l0 level1 lfo9;tab-stops:list 36.0pt'><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="Prog2011.pdf">Programa i Bibliografia. (2012).</a><o:p></o:p></span></li> <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto; mso-list:l0 level1 lfo9;tab-stops:list 36.0pt'><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="AC_L_noSol.pdf">Apunts d'Algebra Computacional. Antonio Montes.&nbsp; (Actualització: 2012 Febrer)</a>. <o:p></o:p></span></li> </ul> <div class=MsoNormal align=center style='text-align:center'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin'> <hr size=2 width="100%" noshade style='color:#ACA899' align=center> </span></div> <h3><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB'>Research<o:p></o:p></span></h3> <p class=MsoNormal><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>Main subject of interest: Computer Algebra <o:p></o:p></span></p> <p><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>See &nbsp;<a href="RedEACA">RedEACA</a>,&nbsp; </span><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="http://medicis.polytechnique.fr/"><span lang=EN-GB style='mso-ansi-language:EN-GB'>http://medicis.polytechnique.fr/</span></a></span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>&nbsp; and&nbsp; </span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin'><a href="http://www.SymbolicNet.org/"><span lang=EN-GB style='mso-ansi-language:EN-GB'>Computer Algebra Internet Resources page</span></a></span><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'> to find information about CA sites, mailing lists, news groups, courses, books, journals, events, and more information about Computer Algebra. <o:p></o:p></span></p> <div class=MsoNormal align=center style='text-align:center'> <hr size=2 width="100%" noshade style='color:#ACA899' align=center> </div> <h4><span style='font-size:14.0pt;font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-hansi-theme-font:minor-latin'>Some papers<o:p></o:p></span></h4> <h4><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin'><a href="Columna.pdf">La Columna de Matemática Computacional : Discusión de sistemas polinómicos con<span style='mso-spacerun:yes'>  </span>parámetros</a>. </span><i style='mso-bidi-font-style:normal'><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-hansi-theme-font:minor-latin;font-weight:normal;mso-bidi-font-weight:bold'>Antonio Montes.</span></i><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;font-weight:normal;mso-bidi-font-weight:bold'> <o:p></o:p></span></h4> <h4><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; font-weight:normal;mso-bidi-font-weight:bold'>La Gaceta de la RSME, 14:3, (2011), 527-544.<o:p></o:p></span></h4> <h4><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; font-weight:normal;mso-bidi-font-weight:bold'>Prólogo de Tomás Recio<o:p></o:p></span></h4> <p class=MsoNormal style='mso-layout-grid-align:none;text-autospace:none'><span style='font-family:"Calibri","sans-serif";mso-bidi-font-family:LMRoman9-Regular; color:windowtext'>Se detalla brevemente el desarrollo histórico de los algoritmos para la discusión de sistemas de ecuaciones polinómicas con parámetros y se dan los elementos esenciales para comprender y poder utilizar el nuevo algoritmo del cubrimiento canónico de Gröbner (</span><span style='font-family:"Calibri","sans-serif";mso-bidi-font-family:LMRomanCaps10-Regular; color:windowtext'>GröbnerCover</span><span style='font-family:"Calibri","sans-serif"; mso-bidi-font-family:LMRoman9-Regular;color:windowtext'>) de un ideal paramétrico, introducido recientemente por el autor y por Michael Wibmer, de la Universidad de Aachen. A fin de que los no especialistas puedan comprender bien su significado e interés, antes de abordar el tema se hace una<o:p></o:p></span></p> <p class=MsoNormal style='mso-layout-grid-align:none;text-autospace:none'><span style='font-family:"Calibri","sans-serif";mso-bidi-font-family:LMRoman9-Regular; color:windowtext'>introducción elemental a la teoría de las bases de Gröbner, que subyace a este nuevo algoritmo. Posteriormente se dan ejemplos donde puede apreciarse su utilidad para resolver problemas en los que aparecen ecuaciones polinómicas con parámetros.</span><span style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin'><o:p></o:p></span></p> <p class=1 style='mso-pagination:lines-together;page-break-after:avoid'><b style='mso-bidi-font-weight:normal'><span lang=PT style='color:#984806; mso-ansi-language:PT'><o:p>&nbsp;</o:p></span></b></p> <p class=1 style='mso-pagination:lines-together;page-break-after:avoid'><b style='mso-bidi-font-weight:normal'><span lang=PT style='font-size:12.0pt; font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin;color:#984806;mso-ansi-language:PT'><a href="p180-montes.pdf">ISSAC 2011. Software for computing the Gröbner cover</a></span></b><span lang=PT style='font-size:12.0pt;mso-ansi-language:PT'>.</span><span lang=PT style='mso-ansi-language:PT'> <i style='mso-bidi-font-style:normal'>Antonio Montes</i><o:p></o:p></span></p> <h4><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-fareast-font-family: "Times New Roman";color:windowtext;mso-ansi-language:EN-US;font-weight:normal; mso-bidi-font-weight:bold'>ACM Communications in Computer Algebra</span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-US;font-weight:normal; mso-bidi-font-weight:bold'>, 45, 180-182, (2011).<span style='mso-spacerun:yes'>  </span></span><span lang=PT style='font-family:"Calibri","sans-serif"; mso-fareast-font-family:"Times New Roman";mso-ansi-language:PT;font-weight: normal;mso-bidi-font-weight:bold'>DOI: 10:1145/2110170.2110178. </span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-fareast-font-family: "Times New Roman";mso-ansi-language:EN-US;font-weight:normal;mso-bidi-font-weight: bold'>ISSAC-2011.</span><span lang=PT style='font-family:"Calibri","sans-serif"; mso-fareast-font-family:"Times New Roman";mso-ansi-language:PT;font-weight: normal;mso-bidi-font-weight:bold'><o:p></o:p></span></h4> <p class=MsoNormal style='mso-layout-grid-align:none;text-autospace:none'><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-bidi-font-family:CMR10;color:windowtext; mso-ansi-language:EN-US'>The objective of this software presentation is to show the behavior and applications of the Singular library<o:p></o:p></span></p> <p class=MsoNormal style='mso-layout-grid-align:none;text-autospace:none'><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-bidi-font-family:CMR10;color:windowtext; mso-ansi-language:EN-US'>grobcov.lib that we have recently developed (Singular web) to compute the Gröbner cover of a parametric ideal</span><span lang=EN-US style='font-size:10.5pt;font-family:CMR10;mso-bidi-font-family:CMR10; color:windowtext;mso-ansi-language:EN-US'>.<o:p></o:p></span></p> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB'><a href="ISSAC_2011_Slides.pdf"><span lang=EN-US style='mso-ansi-language:EN-US'>ISSAC_2011_Slides.pdf</span></a></span><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-US'>.<span style='mso-spacerun:yes'>  </span></span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; color:windowtext;mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight: bold'>Download the slides of the talk</span><span lang=EN-GB style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-fareast-font-family: "Times New Roman";mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>, </span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'>June 2011.</span><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'><o:p></o:p></span></h4> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight: bold'>Slides presenting a Tutorial of the Singular grobcov.lib library at ISSAC-2011 in San José (California), June 2011.<o:p></o:p></span></h4> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;color:red;mso-ansi-language:EN-GB'><a href="grobcov.zip">NEW software. Download the Beta version of the new Singular library grobcov.lib</a></span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'><span style='mso-spacerun:yes'>  </span>(November 2011) (new release)<o:p></o:p></span></h4> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight: bold'>The new Singular grobcov.lib<span style='mso-spacerun:yes'>  </span>library implementing the algorithms in A. Montes, M. Wibmer, Gröbner Bases for Polynomial Systems with parameters, JSC 45 (2010) 1391-1425.<o:p></o:p></span></h4> <h4 style='mso-pagination:widow-orphan lines-together;page-break-after:avoid'><span class=MsoHyperlink><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'><a href="YJSCO1182.pdf">Gröbner Bases for Polynomial Systems with parameters.</a></span></span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB'> </span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-US;font-weight:normal;mso-bidi-font-weight:bold'>Antonio Montes, Michael Wibmer.</span></i><span lang=EN-US style='mso-ansi-language: EN-US;font-weight:normal;mso-bidi-font-weight:bold'><o:p></o:p></span></h4> <p class=MsoNormal style='mso-pagination:widow-orphan lines-together; page-break-after:avoid;mso-layout-grid-align:none;text-autospace:none'><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:11.0pt; font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin;mso-bidi-font-family:t1-gul-regular;color:windowtext;mso-ansi-language: EN-US'><span style='mso-spacerun:yes'> </span>Journal of Symbolic Computation</span></i><span lang=EN-US style='font-size:11.0pt;font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-bidi-font-family: t1-gul-regular;color:windowtext;mso-ansi-language:EN-US'><span style='mso-spacerun:yes'>   </span>45 (2010) 1391-1425.<span style='mso-spacerun:yes'>     </span>doi:10.1016/j.jsc.2010.06.017</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:11.0pt; font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-US'><o:p></o:p></span></i></p> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight: bold'>Gröbner bases are the computational method par excellence for studying polynomial systems. In the case of parametric polynomial systems one has to determine the reduced Gröbner basis in dependence of the values of the parameters. In this article we present the algorithm <i style='mso-bidi-font-style: normal'>GröbnerCover</i> which has as input a finite set of parametric polynomials and outputs a finite partition of the parameter space into locally closed subsets together with polynomial data, from which the reduced Gröbner basis for a given parameter point can immediately be determined. The partition of the parameter space is intrinsic and particularly simple if the system is homogeneous. The Singular library can be downloaded at the top at this web.<o:p></o:p></span></h4> <h4><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin'><a href="gb2.handout.pdf"><span lang=EN-GB style='mso-ansi-language:EN-GB; mso-bidi-font-weight:normal'>Gröbner Bases Tutorial: A Sampler of Recent Developments.</span><span lang=EN-GB style='mso-ansi-language:EN-GB'> </span></a></span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB'><span style='mso-spacerun:yes'> </span></span><i style='mso-bidi-font-style:normal'><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB;font-weight:normal; mso-bidi-font-weight:bold'>David Cox<o:p></o:p></span></i></h4> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight: bold'>Professor David Cox presented this Tutorial at ISSAC 2007. It contains an overview of the paper </span><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:Calibri; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'>“</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'>Automatic Discovery of geometric theorems using MCCGS</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:Calibri;mso-ansi-language:EN-GB;font-weight:normal; mso-bidi-font-weight:bold'>”</span><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB;font-weight:normal; mso-bidi-font-weight:bold'> by A. Montes and T. Recio.<o:p></o:p></span></h4> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB'><a href="48690113.pdf">Automatic discovery of geometry theorems using minimal canonical comprehensive Groebner systems.</a> </span><i style='mso-bidi-font-style:normal'><span lang=EN-GB style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-fareast-font-family: "Times New Roman";mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB; font-weight:normal;mso-bidi-font-weight:bold'>Antonio Montes, Tomás Recio.<o:p></o:p></span></i></h4> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight: bold'>ADG 2006, <i style='mso-bidi-font-style:normal'>LNAI</i> 4869, pp. 113-138, 2007. <i style='mso-bidi-font-style:normal'><o:p></o:p></i></span></h4> <h4><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight: bold'>The main idea in this paper is merging two techniques that have been recently developed. On the one hand, we consider MCCGS, standing for Minimal Canonical Comprehensive Groebner Systems, a recently introduced computational tool yielding </span><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:Calibri; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'>“</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'>good</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:Calibri;mso-ansi-language:EN-GB; font-weight:normal;mso-bidi-font-weight:bold'>”</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'> bases for ideals of polynomials over a field <i style='mso-bidi-font-style:normal'>depending</i> on several parameters,<span style='mso-spacerun:yes'>  </span>that specialize </span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:Calibri;mso-ansi-language:EN-GB; font-weight:normal;mso-bidi-font-weight:bold'>“</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'>well</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:Calibri;mso-ansi-language:EN-GB; font-weight:normal;mso-bidi-font-weight:bold'>”</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman";mso-hansi-theme-font:minor-latin; mso-ansi-language:EN-GB;font-weight:normal;mso-bidi-font-weight:bold'>, for instance, regarding the number of solutions for the given ideal, for different values of the parameters. The second ingredient concerns automatic theorem discovery in elementary geometry.<span style='mso-spacerun:yes'>  </span>Automatic discovery aims to obtain complementary hypotheses for a (generally false) geometric statement to become true. The paper shows how to use MCCGS for automatic discovering of theorems and gives relevant examples.<o:p></o:p></span></h4> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><u><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin'><a href="Manubens_2009_Journal-of-Symbolic-Computation.pdf"><span lang=EN-GB style='mso-ansi-language:EN-GB'>Minimal Canonical Comprehensive Gröbner System</span></a></span></u></b><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>.<span style='mso-spacerun:yes'>  </span><st1:place w:st="on"><i style='mso-bidi-font-style: normal'>Montserrat</i></st1:place><i style='mso-bidi-font-style:normal'> Manubens, Antonio Montes</i>.<o:p></o:p></span></p> <p class=MsoNormal><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:11.0pt;font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:t1-gul-regular; color:windowtext;mso-ansi-language:EN-US'>Journal of Symbolic Computation</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'> 44 (2009) 463-478.<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>This is the continuation of Montes' paper ``On the canonical discussion of polynomial systems with parameters&quot;. In this paper we define the Minimal Canonical Comprehensive Gröbner System of a parametric ideal and fix under which hypothesis it exists and is computable. An algorithm to obtain a canonical description of the segments of the Minimal Canonical CGS is given, completing so the whole MCCGS algorithm (implemented in Maple and Singular). We show its high utility for applications, like automatic theorem proving and discovering, and compare it with other existing methods. A way to detect a counterexample to deny its existence is outlined, although the high number of tests done give evidence of the existence of the Minimal Canonical CGS.<o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>This paper describes the principal new improvements in the DISPGB algorithm implemented in release 7.4 of dpgb74.mpl and in Singular.<o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="0601674v6.pdf"><b><span lang=EN-GB style='mso-ansi-language:EN-GB'>On the canonical discussion of polynomial systems with parameters</span></b></a></span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>. <i>Antonio Montes</i>.<br> Preprint arXiv: AC/0601674.<span style='mso-spacerun:yes'>    </span>Preprint MA2-IR-06-00006<br> <br> Given a parametric polynomial ideal I, the algorithm DISPGB,&nbsp; introduced by the author in 2002, builds up a binary tree describing a dichotomic discussion of the different reduced Groebner bases depending on the values of the parameters. An improvement using a discriminant ideal to rewrite the tree was described by&nbsp; Manubens and the author in <st1:metricconverter ProductID="2005. In" w:st="on">2005. In</st1:metricconverter> this paper we describe how to iterate the use of discriminants to rebuild the tree and show that this leads to an ascending discriminant chain of ideals, and the corresponding descending chain of varieties in the parameter space providing a diff-specification of the cases. From it we show that it is possible to construct canonical specifications of each diff-specification. We also prove the conjectures formulated in the previous paper.<br> <br> This paper describes the principal new improvements in the DISPGB algorithm implemented in the next release 5.0 of dpgb50.mpl. This release contains further improvements that are described in a new preprint in preparation.<br> <br> </span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/0509157.pdf"><b><span lang=EN-GB style='mso-ansi-language:EN-GB'>Some Composition Determinants</span></b></a></span><b><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>&nbsp;</span></b><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'> <i>Josep M. Brunat, Christian Krattenthaler, Alain Lascoux, Antonio Montes</i>. <br> <i>Linear Algebra and Applications</i>. <b style='mso-bidi-font-weight:normal'>416</b> (2006) 355-364. Preprint in arXiv: CO/0509157.<span style='mso-spacerun:yes'>   </span>Preprint MA2-IR-06-00005<br> <br> We compute two parametric determinants in which rows and columns are indexed by compositions, <br> where in one determinant the entries are products of binomial coefficients, while in the other the entries <br> are products of powers. These results generalize previous determinant evaluations due to the first and <br> third author [<st1:place w:st="on"><st1:country-region w:st="on"><i>SIAM</i></st1:country-region></st1:place><i> J. Matrix Anal. and Appl.</i> <b>23</b> (2001), 459-471] and [``A polynomial <br> generalization of the power-compositions determinant,&quot; {<i>Linear and Multilinear Algebra</i> (to appear)], <br> and they prove two conjectures of the second author [``Advanced determinant calculus: a complement,&quot; <br> preliminary version].<br> <br> <a href="LAMA050767revised.pdf">&nbsp;<b>A polynomial generalization of the power-compositions determinant</b></a><b>. &nbsp;</b> <i>Josep M. Brunat, Antonio Montes</i>.<br> <i>Linear and Multilinear Algebra</i> <b style='mso-bidi-font-weight:normal'>55</b>-4 (2007) 303-313. arXiv: math.CO/0601756.<span style='mso-spacerun:yes'>  </span>Preprint MA2-IR-06-00005<br> <br> Let C(<i>n,p</i>) be the set of <i>p</i>-compositions of an integer <i>n</i>, i.e., the set of <i>p</i>-tuples <b><i>alpha</i></b> = (<i>alpha_</i>1<i>, .. , alpha_p</i>) of nonnegative integers such that <i><br> alpha</i>_1 + .. + <i>alpha</i>_<i>p</i>&nbsp; =&nbsp; <i>n</i>, and <b><i>x</i></b> = (<i>x</i>_1, .. ,<i>x_p</i>) a vector of indeterminates.&nbsp; For <b><i>alpha</i></b> and <b><i>beta</i></b> two <i>p</i>-compositions of <i>n</i>, define <br> ( <b><i>x</i></b> + <b><i>alpha</i></b>)^<b><i>beta =&nbsp; </i></b>(<i>x</i>_1+<i>alpha</i>_1)^beta_1 ·.. ·(<i>x_p</i>+<i>alpha_p</i>)^<i>beta_p</i>.&nbsp; In this paper we prove an explicit formula for the determinant<br> det (<b><i>x</i></b>+<b><i>alpha</i></b>)^<b><i>beta</i></b>. In the case <i>x</i>_1 = .. = <i>x_p</i> the formula gives a positive answer to a conjecture&nbsp; by C.&nbsp; Krattenthaler.<br> <br> </span><b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="yjsco_873.pdf"><span lang=EN-GB style='mso-ansi-language:EN-GB'>Improving DISPGB algorithm using the discriminant ideal.</span></a></span></b><span lang=EN-GB style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB'>&nbsp; <st1:place w:st="on"><i>Montserrat</i></st1:place><i> Manubens, Antonio Montes.<br> </i><i style='mso-bidi-font-style:normal'>Journal of Symbolic Computation</i>. <b style='mso-bidi-font-weight:normal'>41</b> (2006) 1245-1263. Extended Abstract in <i>A3L-2005 Proceedings </i>(2005).&nbsp; Preprint: arXiv: math.AC/0601763.<span style='mso-spacerun:yes'>  </span>Preprint MAII-IR-04-00015, (2004).<br> <br> &nbsp;In 1992, V. Weispfenning proved the existence of Comprehensive Gröbner Bases (CGB) and gave an algorithm to compute one. That algorithm was not very efficient and not canonical. Using his suggestions, A. Montes obtained in <st1:metricconverter ProductID="2002 a" w:st="on">2002 a</st1:metricconverter> more efficient algorithm (DISPGB) for Discussing Parametric Gröbner Bases. Inspired in its philosophy, V. Weispfenning defined, in 2002, how to obtain a Canonical Comprehensive Gröbner Basis (CCGB) for parametric polynomial ideals, and provided a constructive method.<br> In this paper we use Weispfenning's CCGB ideas to make substantial improvements on Montes DISPGB algorithm. It now includes rewriting of the discussion tree using the Discriminant Ideal and provides a compact and effective discussion. We also describe the new algorithms in the DPGB library containing&nbsp; the improved DISPGB as well as new routines to check whether a given basis is a CGB or not, and to obtain a CGB. Examples and tests are also provided.<br> <br> </span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/dispgb20.pdf"><b><span lang=EN-GB style='mso-ansi-language:EN-GB'>Improving DisPGB algorithm for parametric Gröbner bases</span></b></a></span><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>&nbsp; <i>Montserrat Manubens, Antonio Montes.</i><br> Extended Abstract in<i> Actas de EACA-2004.</i><br> <br> We present important improvements and a thorough redesign of the algorithm DisPGB in the new release 2.1 of the&nbsp; DPGB <i>Maple</i> library for discussing Gröbner bases with parameters. DisPGB20 provides a more compact tree discussion, avoiding incompatible branches, and producing simpler output bases. The new software is more efficient and robust and can increase the speed up to 20 times with respect to the old release.<br> <br> </span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/0601731.pdf"><b><span lang=EN-GB style='mso-ansi-language:EN-GB'>On polynomial digraphs</span></b></a></span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>&nbsp; <i>Josep M. Brunat and Antonio Montes</i>.<br> <i>Discrete Mathematics</i><span style='mso-spacerun:yes'>  </span><b style='mso-bidi-font-weight:normal'>306-</b>4 (2006),<span style='mso-spacerun:yes'>  </span>401-412 . </span><span style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin'><a href="doi:%2010.1016/j.disc.2006.01.001"><span lang=EN-GB style='mso-ansi-language:EN-GB'>doi: </span></a><a href="doi:%2010.1016/j.disc.2006.01.001"><span lang=EN-GB style='mso-ansi-language: EN-GB'>10.1016/j.disc.2006.01.001</span></a></span><i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'> ,&nbsp;</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'> arXiv: math.AC/0601731. Preprint MAII-IR-04-00007, (Mai 2004). Extended Abstract in<i> Actas de EACA-2004</i>, <st1:City w:st="on"><st1:place w:st="on">Santander</st1:place></st1:City>.&nbsp; <br> <br> Let <i>Phi</i>(<i>x,y</i>) be a bivariate polynomial with complex coefficients. The zeroes of <i>Phi</i>(<i>x,y</i>) are given a combinatorial structure by&nbsp; considering them as arcs of a directed graph <i>G</i>(<i>Phi</i>). This paper studies the relationship between&nbsp; the polynomial <i>Phi</i>(<i>x,y</i>) and the structure of <i>G</i>(<i>Phi</i>).<br> <br> </span><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/0601733.pdf"><b><span lang=EN-GB style='mso-ansi-language:EN-GB'>The Characteristic Ideal of a Finite, Connected Regular Graph</span></b></a></span><i><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>&nbsp; Josep M. Brunat,<span style='mso-spacerun:yes'>  </span>Antonio Montes</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>.<br> <i>ISSAC 2004 Proceedings</i>, ACM (2004) 50-57, Santander. arXiv: math.AC/0601733<br> <br> Let&nbsp; <i>Phi</i>(<i>x,y</i>)&nbsp; be&nbsp; be a symmetric polynomial in&nbsp; <i>C</i>[<i>x,y</i>] of partial&nbsp; degree <i>d</i>. The &nbsp; graph <i>G</i>(<i>Phi</i>) is defined by taking <i>C</i> as set of vertices&nbsp; and the points of&nbsp;&nbsp; <b>V</b>(<i>Phi</i>(<i>x,y</i>)) as edges. We study the following&nbsp; problem: given a finite, connected, <i>d</i>-regular graph <i>H</i>, find the&nbsp; polynomials <i>Phi</i>(<i>x,y</i>) such that <i>G</i>(<i>Phi</i>) has some connected &nbsp; component isomorphic to <i>H</i> and, in this case, if <i>G</i>(<i>Phi</i>) has (almost)&nbsp; all components isomorphic to <i>H</i>. The problem is solved by associating to&nbsp; <i>H</i> a characteristic ideal which offers a new perspective to the conjecture formulated in a previous paper, and allows to reduce its scope.&nbsp; In the second part, we determine the characteristic ideal&nbsp; for cycles of lengths less than 6 and for complete graphs of order 6. This results provide new evidence for the conjecture.<br> <br> </span><b><u><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/NewAlgDPGB.pdf"><span lang=EN-GB style='mso-ansi-language:EN-GB'>A new algorithm for discussing Gröbner bases with parameters.</span></a></span></u></b><span lang=EN-GB style='font-family: "Calibri","sans-serif";mso-ascii-theme-font:minor-latin;mso-hansi-theme-font: minor-latin;mso-ansi-language:EN-GB'>&nbsp;<b>&nbsp;</b><i> Antonio Montes</i>. <br> <i style='mso-bidi-font-style:normal'>Journal of Symbolic Computation<span style='mso-bidi-font-style:italic'> </span></i><span style='mso-spacerun:yes'> </span><b>33-</b>2 (2002) 183-208 <o:p></o:p></span></p> <p><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Let <i>F</i> be a set of polynomials in the variables <i>x_1,...,x_n</i>&nbsp; with coefficients in <i>R</i>[<i>a_1,..a_n</i>], where <i>R</i> is a <br> UFD and&nbsp; <i>a_1,..,a_m</i> a set of parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all the cases over the parameters leading to different reduced Gröbner basis, when the parameters in <i>F</i> are substituted in an extension field <i>K</i> of <i>R</i>. This new algorithm improves Weispfenning&nbsp; CGB algorithm,&nbsp; obtaining a reduced set of compatible and disjoint cases. A final improvement determines the minimal singular variety outside of which the Gröbner basis specializes properly (generic case). These constructive methods provide a very satisfactory discussion with rich geometrical interpretation in the applications. <o:p></o:p></span></p> <p><b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/36914.ps"><span lang=EN-GB style='mso-ansi-language:EN-GB'>The power-composition determinant and its application to global optimization.</span></a></span></b><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>&nbsp; <i>Josep M. Brunat</i>, <i>Antonio Montes</i>. <br> <st1:country-region w:st="on"><st1:place w:st="on"><i>SIAM</i></st1:place></st1:country-region><i> J. Matrix Anal. Appl.</i>&nbsp; <b>23-</b>2 (2001)&nbsp; 459-471. <br> Preprint in <i>Actas de EACA-2000</i>, <st1:City w:st="on"><st1:place w:st="on">Barcelona</st1:place></st1:City>, September, 2000. <o:p></o:p></span></p> <p><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Let <i>C</i>(<i>n,p</i>) be the set of <i>p</i>-compositions of an integer <i>n</i>, i.e., the set of <i>p</i>-tuples <i>alpha = </i>(<i>alpha_1,..,alpha_p</i>) of nonnegative integers such that <i>alpha_1 </i>+..+ a<i>lpha_p </i>= <i>n</i>. The main result of this paper is an explicit formula for the determinant of the matrix whose entries are <i>alpha</i>^{<i>beta</i>} = a<i>lpha_1</i>^{<i>beta_1</i>} .. <i>alpha_p</i>^{<i>beta_p</i>} where alpha,beta&nbsp; belong to <i>C</i>(<i>n,p</i>). The formula shows that the determinant is positive and has a nice factorization. As an application, it is <br> shown that the polynomials <i>p_alpha</i>(<i>x</i>) = (<i>alpha_1 x_1+ .. + alpha_p x_p</i>)^<i>n</i>&nbsp; with&nbsp; alpha in <i>C</i>(<i>n,p</i>) form a basis of the vector space <i>H_n</i>[<i>x_1, .. ,x_p</i>] of homogeneous polynomials of degree <i>n</i>&nbsp; in <i>p</i>&nbsp; variables. The result is of interest in the context of global optimization because it allows an explicit representation of polynomials as a difference of convex functions. &nbsp; <o:p></o:p></span></p> <p><b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/PGB.ps"><span lang=EN-GB style='mso-ansi-language:EN-GB'>Basic Algorithms for Specialization in Groebner Bases.</span></a></span></b><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>&nbsp;<b>&nbsp;</b> <i>Antonio Montes </i>. (1999).<br> <i>Actas de EACA-99</i>. <st1:place w:st="on">Tenerife</st1:place>, September 1999. <o:p></o:p></span></p> <p><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Some basic algorithms for dealing with Groebner bases with parametres are given. They allow to construct a general algorithm for a general discussion of polynomial systems with parametrs that improves the Comprehensive Groebner Basis Algorithm of Weisfenning. The complete algorithm will be given in a forthcoming paper. <o:p></o:p></span></p> <p><b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/Lf4n.ps"><span lang=EN-GB style='mso-ansi-language:EN-GB'>Algebraic solution of the load-flow problem for a 4-nodes electrical network.</span></a></span></b><b><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>&nbsp; </span></b><i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Antonio Montes</span></i><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>. <br> <i>Mathematics &amp; Computer in Simulation</i>, <b>45 </b>(1998) 163-174. <o:p></o:p></span></p> <p><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Using algebraic techniques to triangulate polynomial systems of equations we are able to do it for the system describing a complete four-nodes electrical network with all the paramters. The advantages of the algebraic solution compared to the numerical one are discussed. <o:p></o:p></span></p> <p><b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/Lf3g.ps"><span lang=EN-GB style='mso-ansi-language:EN-GB'>Solving the Load Flow Problem Using the Groebner Basis.</span></a></span></b><span lang=EN-GB style='font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>&nbsp;<b>&nbsp; </b><i>Antonio Montes, Jordi Castro.</i> <br> <i>Sigsam Bulletin</i>, <b>29-</b>1 (1995) 1-13. <o:p></o:p></span></p> <p><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>The load-flow problem must be solved in Electrical Engeenering very often with different values of the parameters. Theoretically, it is interessting to obtain a general solution for all values of the parameters using Gröbner basis. In this paper we compute it for a 3-nodes general network and compare the results and conditioning with usual numerical computations. <o:p></o:p></span></p> <p><b><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin'><a href="http://www-ma2.upc.es/%7Emontes/NumCond.ps"><span lang=EN-GB style='mso-ansi-language:EN-GB'>Numerical Conditioning of a System of Algebraic Equations with a Finite Number of Solutions Using Groebner Bases.</span></a></span></b><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>&nbsp;<b>&nbsp;&nbsp;</b> <i>Antonio Montes</i>. <br> <i>Sigsam Bulletin</i>, <b>27-</b>1 (1993) 12-19. <o:p></o:p></span></p> <p><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'>Numerical conditioning for the problem and for the algorithm in computing the roots of a 0-dimensional polynomial system are given and applied to examples triangularized using Groebner bases. <o:p></o:p></span></p> <div class=MsoNormal align=center style='text-align:center'><span style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin'> <hr size=2 width="100%" noshade style='color:#ACA899' align=center> </span></div> <p><span lang=EN-GB style='font-size:10.0pt;font-family:"Calibri","sans-serif"; mso-ascii-theme-font:minor-latin;mso-hansi-theme-font:minor-latin;mso-ansi-language: EN-GB'>(<i>Last update of this page: November 23, 2010</i>)</span><span lang=EN-GB style='font-family:"Calibri","sans-serif";mso-ascii-theme-font:minor-latin; mso-hansi-theme-font:minor-latin;mso-ansi-language:EN-GB'> <o:p></o:p></span></p> </div> </body> </html>