<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="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 15"> <meta name=Originator content="Microsoft Word 15"> <link rel=File-List href="publications_files/filelist.xml"> <link rel=Edit-Time-Data href="publications_files/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>index.html</title> <!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Author>ren@math.utexas.edu</o:Author> <o:Template>Normal</o:Template> <o:LastAuthor>ren@math.utexas.edu</o:LastAuthor> <o:Revision>4</o:Revision> <o:TotalTime>1</o:TotalTime> <o:Created>2017-10-19T22:51:00Z</o:Created> <o:LastSaved>2017-10-20T00:22:00Z</o:LastSaved> <o:Pages>5</o:Pages> <o:Words>1693</o:Words> <o:Characters>9653</o:Characters> <o:Company>Microsoft</o:Company> <o:Lines>80</o:Lines> <o:Paragraphs>22</o:Paragraphs> <o:CharactersWithSpaces>11324</o:CharactersWithSpaces> <o:Version>16.00</o:Version> </o:DocumentProperties> <o:OfficeDocumentSettings> <o:AllowPNG/> </o:OfficeDocumentSettings> </xml><![endif]--> <link rel=themeData href="publications_files/themedata.thmx"> <link rel=colorSchemeMapping href="publications_files/colorschememapping.xml"> <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Print</w:View> <w:SpellingState>Clean</w:SpellingState> <w:GrammarState>Clean</w:GrammarState> <w:TrackMoves>false</w:TrackMoves> <w:TrackFormatting/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-US</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SplitPgBreakAndParaMark/> </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" DefPriority="99" LatentStyleCount="371"> <w:LsdException Locked="false" Priority="0" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" Priority="9" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 6"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 7"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 8"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index 9"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 1"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 2"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 3"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 4"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 5"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 6"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 7"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 8"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" Name="toc 9"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Normal Indent"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="footnote text"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="annotation text"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="header"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="footer"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="index heading"/> <w:LsdException Locked="false" Priority="35" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="caption"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="table of figures"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="envelope address"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="envelope return"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="footnote reference"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="annotation reference"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="line number"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="page number"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="endnote reference"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="endnote text"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="table of authorities"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="macro"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="toa heading"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Bullet"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Number"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Bullet 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Bullet 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Bullet 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Bullet 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Number 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Number 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Number 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Number 5"/> <w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Closing"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Signature"/> <w:LsdException Locked="false" Priority="1" SemiHidden="true" UnhideWhenUsed="true" Name="Default Paragraph Font"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text Indent"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Continue"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Continue 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Continue 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Continue 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="List Continue 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Message Header"/> <w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Salutation"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Date"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text First Indent"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text First Indent 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Note Heading"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text Indent 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Body Text Indent 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Block Text"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Hyperlink"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="FollowedHyperlink"/> <w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Document Map"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Plain Text"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="E-mail Signature"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Top of Form"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Bottom of Form"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Normal (Web)"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Acronym"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Address"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Cite"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Code"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Definition"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Keyboard"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Preformatted"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Sample"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Typewriter"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="HTML Variable"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Normal Table"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="annotation subject"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="No List"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Outline List 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Outline List 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Outline List 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Simple 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Simple 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Simple 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Classic 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Classic 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Classic 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Classic 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Colorful 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Colorful 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Colorful 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Columns 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Columns 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Columns 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Columns 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Columns 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 6"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 7"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Grid 8"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 6"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 7"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table List 8"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table 3D effects 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table 3D effects 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table 3D effects 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Contemporary"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Elegant"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Professional"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Subtle 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Subtle 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Web 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Web 2"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Web 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Balloon Text"/> <w:LsdException Locked="false" Priority="39" Name="Table Grid"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" Name="Table Theme"/> <w:LsdException Locked="false" 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" 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:LsdException Locked="false" Priority="41" Name="Plain Table 1"/> <w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/> <w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/> <w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/> <w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/> <w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/> <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/> <w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/> <w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/> <w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/> <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/> <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/> <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/> <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light Accent 1"/> <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/> <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/> <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/> <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/> <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful Accent 1"/> <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful Accent 1"/> <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light Accent 2"/> <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/> <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/> <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/> <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/> <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful Accent 2"/> <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful Accent 2"/> <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light Accent 3"/> <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/> <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/> <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/> <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/> <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful Accent 3"/> <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful Accent 3"/> <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light Accent 4"/> <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/> <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/> <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/> <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/> <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful Accent 4"/> <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful Accent 4"/> <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light Accent 5"/> <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/> <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/> <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/> <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/> <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful Accent 5"/> <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful Accent 5"/> <w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light Accent 6"/> <w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/> <w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/> <w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/> <w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/> <w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful Accent 6"/> <w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful Accent 6"/> <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/> <w:LsdException Locked="false" Priority="47" Name="List Table 2"/> <w:LsdException Locked="false" Priority="48" Name="List Table 3"/> <w:LsdException Locked="false" Priority="49" Name="List Table 4"/> <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/> <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/> <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/> <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light Accent 1"/> <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/> <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/> <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/> <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/> <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful Accent 1"/> <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful Accent 1"/> <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light Accent 2"/> <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/> <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/> <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 2"/> <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 2"/> <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful Accent 2"/> <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful Accent 2"/> <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light Accent 3"/> <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 3"/> <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 3"/> <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 3"/> <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 3"/> <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful Accent 3"/> <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful Accent 3"/> <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light Accent 4"/> <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 4"/> <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 4"/> <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 4"/> <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 4"/> <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful Accent 4"/> <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful Accent 4"/> <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light Accent 5"/> <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 5"/> <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 5"/> <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 5"/> <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 5"/> <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful Accent 5"/> <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful Accent 5"/> <w:LsdException Locked="false" Priority="46" Name="List Table 1 Light Accent 6"/> <w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 6"/> <w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 6"/> <w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 6"/> <w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 6"/> <w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful Accent 6"/> <w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful Accent 6"/> </w:LatentStyles> </xml><![endif]--> <style> <!--a:link {so-language: zxx ;} /* Font Definitions */ @font-face {font-family:Courier; panose-1:2 7 4 9 2 2 5 2 4 4; mso-font-charset:0; mso-generic-font-family:modern; mso-font-format:other; mso-font-pitch:fixed; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:Wingdings; panose-1:5 0 0 0 0 0 0 0 0 0; mso-font-charset:2; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:0 268435456 0 0 -2147483648 0;} @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:1; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:0 0 0 0 0 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; 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;} a:link, span.MsoHyperlink {mso-style-noshow:yes; mso-style-priority:99; color:#0000EE; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {mso-style-noshow:yes; mso-style-priority:99; color:#551A8B; text-decoration:underline; text-underline:single;} p {mso-style-noshow:yes; mso-style-priority:99; mso-margin-top-alt:auto; margin-right:0in; margin-bottom:5.75pt; margin-left:0in; mso-pagination:widow-orphan; 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;} p.msonormal0, li.msonormal0, div.msonormal0 {mso-style-name:msonormal; mso-style-noshow:yes; mso-style-priority:99; mso-style-unhide:no; mso-margin-top-alt:auto; margin-right:0in; margin-bottom:5.75pt; margin-left:0in; mso-pagination:widow-orphan; 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;} span.SpellE {mso-style-name:""; mso-spl-e:yes;} span.GramE {mso-style-name:""; mso-gram-e:yes;} .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:8.5in 11.0in; margin:1.0in 1.0in 1.0in 1.0in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} /* List Definitions */ @list l0 {mso-list-id:40986662; mso-list-template-ids:-2008109482;} @list l0:level1 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l0:level2 {mso-level-number-format:bullet; mso-level-text:o; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:"Courier New"; mso-bidi-font-family:"Times New Roman";} @list l0:level3 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:1.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l0:level4 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:2.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l0:level5 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:2.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l0:level6 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:3.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l0:level7 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:3.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l0:level8 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:4.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l0:level9 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:4.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l1 {mso-list-id:369300246; mso-list-template-ids:-1451313726;} @list l1:level1 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l1:level2 {mso-level-number-format:bullet; mso-level-text:o; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:"Courier New"; mso-bidi-font-family:"Times New Roman";} @list l1:level3 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:1.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l1:level4 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:2.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l1:level5 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:2.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l1:level6 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:3.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l1:level7 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:3.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l1:level8 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:4.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l1:level9 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:4.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l2 {mso-list-id:981427240; mso-list-template-ids:1203289696;} @list l2:level1 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l2:level2 {mso-level-number-format:bullet; mso-level-text:o; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:"Courier New"; mso-bidi-font-family:"Times New Roman";} @list l2:level3 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:1.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l2:level4 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:2.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l2:level5 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:2.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l2:level6 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:3.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l2:level7 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:3.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l2:level8 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:4.0in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} @list l2:level9 {mso-level-number-format:bullet; mso-level-text:; mso-level-tab-stop:4.5in; mso-level-number-position:left; text-indent:-.25in; mso-ansi-font-size:10.0pt; font-family:Wingdings;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> </style> <!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman",serif;} </style> <![endif]--> <meta name=author content="Kui Ren"> <meta name=created content="00:00:00"> <meta name=changed content="2017-09-23T12:03:35.655478500"> <meta name=created content="00:00:00"> <meta name=changed content="2017-09-04T11:52:52.454027234"> <meta name=created content="00:00:00"> <meta name=changed content="2017-07-22T21:57:19.530529229"> <meta name=created content="00:00:00"> <meta name=changed content="2017-07-22T21:16:16.700620617"> <meta name=created content="00:00:00"> <meta name=changed content="2017-07-22T21:08:17.743119742"> <meta name=changed content="2017-02-10T16:09:21.428889378"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <meta name=CHANGEDBY content="Kui Ren"> <!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="1026"/> </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 lang=EN-US link="#0000EE" vlink="#551A8B" style='tab-interval:.5in'> <div class=WordSection1> <div style='mso-element:para-border-div;border:none;border-bottom:double silver 2.25pt; padding:0in 0in 1.0pt 0in'> <p style='margin-bottom:12.0pt;border:none;mso-border-bottom-alt:double silver 2.25pt; padding:0in;mso-padding-alt:0in 0in 1.0pt 0in'><o:p>&nbsp;</o:p></p> </div> <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 width="100%" style='width:100.0%;mso-cellspacing:0in;mso-yfti-tbllook:1184;mso-padding-alt: 0in 0in 0in 0in'> <col width="131*"><col width="125*"> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width="51%" style='width:51.0%;padding:0in 0in 0in 0in'> <p class=MsoNormal style='mso-margin-top-alt:auto;margin-bottom:5.75pt'><span style='font-size:36.0pt'>Kui Ren</span><br> <br> <a href="http://www.math.utexas.edu/"><span style='font-size:14.0pt'>Department of Mathematics</span></a>&nbsp; and<br> <a href="http://www.ices.utexas.edu/"><span style='font-size:14.0pt'>Institute of Computational Engineering and Sciences</span></a><br> <span style='font-size:14.0pt'>The University of Texas, Austin, TX 78712</span><br> <br> <span style='font-size:14.0pt'>Office: RLM 10.170</span> <br> <span style='font-size:14.0pt'>Phone: 512-471-3152</span> <br> <span style='font-size:14.0pt'>Fax: 512-471-9038</span> <br> <span style='font-size:14.0pt'>Email: <a href="mailto:ren@math.utexas.edu">ren@math.utexas.edu</a></span></p> </td> <td width="49%" style='width:49.0%;padding:0in 0in 0in 0in'> <p class=MsoNormal align=center style='mso-margin-top-alt:auto;margin-bottom: 5.75pt;text-align:center'><span style='font-size:24.0pt'><a href="Ren-Research-Summary.pdf">Research</a><br> <a href="../public_html">Publications</a><br> <a href="teaching.html">Teaching</a><br> <a href="Ren-Vita.pdf">Vita</a></span><span style='font-size:18.0pt'><br> <br> </span><a href="http://www.math.utexas.edu/cna/SeminarNow.html"><span style='font-size:14.0pt'>Math/ICES Center of Numerical Analysis Seminar</span></a></p> </td> </tr> </table> <div class=MsoNormal align=center style='text-align:center'><span style='mso-fareast-font-family:"Times New Roman"'> <hr size=4 width="100%" align=center> </span></div> <p><span style='font-size:36.0pt'>Publications:</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>An implicit boundary integral method for computing electric potential of macromolecules in solvent (with R. Tsai and Y. <span class=SpellE>Zhong</span>), <i>Preprint, </i>2017. [<a href="Publications/Zhong-Tsai-Ren-IBIM-FMM-PB-2017.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Surface effects in dense random graphs with sharp edge constraint (with C. <span class=SpellE>Radin</span> and L. <span class=SpellE>Sadun</span>), <i>Preprint, </i>2017. [<a href="Publications/Radin-Ren-Sadun-Surface-Effect-2017.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Quantitative photo-acoustic imaging of two-photon absorption, (with P. <span class=SpellE>Bardsley</span> and R. Zhang), <i>Submitted</i>, 2017. [<a href="Publications/TP-PAT-Num.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>A global stability estimate for the photo-acoustic inverse problem in layered media, (with F. <span class=SpellE>Triki</span>), <i>Submitted</i>, 2017. [<a href="Publications/Ren-Triki-PAT-Global-Stability-2017.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>The phases of large networks with edge and triangle constraints, (with R. Kenyon, C. <span class=SpellE>Radin</span> and L. <span class=SpellE>Sadun</span>), <i>J. Phys. A: Math. <span class=SpellE>Theor</span>.</i>, 50, 435001, 2017. [<a href="Publications/Kenyon-Radin-Ren-Sadun-Phases-of-Large-Networks-2016.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>A symmetry breaking transition in the edge/triangle network model (with C. <span class=SpellE>Radin</span> and L. <span class=SpellE>Sadun</span>), <i>Submitted</i>, 2016. [<a href="Publications/Radin-Ren-Sadun-arXiv16-Symmetry-Breaking.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>A fast algorithm for radiative transport in isotropic media (with R. Zhang and Y. <span class=SpellE>Zhong</span>), <i>Submitted</i>, 2016. [<a href="Publications/Ren-Zhang-Zhong-FMM-ERT-2016.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Nonlinear quantitative photoacoustic tomography with two-photon absorption (with R. Zhang), <i>SIAM J. Appl. Math</i>, In Press, 2017. [<a href="Publications/Ren-Zhang-TP-PAT-2016.pdf">pdf</a>].</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Numerical algorithms based on <span class=SpellE>Galerkin</span> methods for the modeling of reactive interfaces in <span class=SpellE>photoelectrochemical</span> (PEC) solar cells (with M. Harmon and I. M. <span class=SpellE>Gamba</span>), <i>J. <span class=SpellE>Comput</span>. Phys.</i>, 327, 140-167, 2016. [<a href="Publications/Harmon-Gamba-Ren-2016-Semi-Elec-Interf-DG.pdf">pdf</a></span> <span style='font-family:Courier'>]</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Inverse transport problems in quantitative PAT for molecular imaging</span> <span style='font-family:Courier'>(with R. Zhang and Y. <span class=SpellE>Zhong</span>), <i>Inverse Problems,</i></span> <span style='font-family:Courier'>31, 125012, 2015. <a href="Publications/QfPAT-ERT.pdf">[pdf]</a>.</span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><span style='font-family: Courier'>Bipodal</span></span><span style='font-family:Courier'> structure in oversaturated random graphs (with R. Kenyon, C. <span class=SpellE>Radin</span> and L. <span class=SpellE>Sadun</span>), <i>Int. Math. Res. Notices</i>, 2016, 1-36, 2016. [<a href="http://arxiv.org/pdf/1509.05370v1.pdf">pdf</a>]. </span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo4; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>A one-step reconstruction algorithm for quantitative photoacoustic imaging (with T. Ding and S. <span class=SpellE>Vallelian</span>), <i>Inverse Problems</i>, 31, 095005, 2015. [<a href="Publications/QPAT-OneStep-IP.pdf">pdf</a>]. </span><span style='font-family:Courier;color:red'>This paper was selected as by Inverse Problems as one of the papers in its collection: </span><a href="http://iopscience.iop.org/journal/0266-5611/page/highlights-of-2015"><span style='font-family:Courier;text-decoration:none;text-underline:none'>Highlight of 2015</span></a> </p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo6; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><span style='font-family: Courier'>Multipodal</span></span><span style='font-family:Courier'> Structure and Phase Transitions in Large Constrained Graphs (with R. Kenyon, C. <span class=SpellE>Radin</span> and L. <span class=SpellE>Sadun</span>), <i>J. Stat. Phys., </i>168, 233-258<i>, </i>2017. <a href="Publications/Kenyon-Radin-Ren-Sadun-arXiv14-Multipodality-in-Constrained-Graphs.pdf">[pdf]</a></span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo6; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>The <span class=SpellE>asymptotics</span> of large constrained graphs (with C. <span class=SpellE>Radin</span> and L. <span class=SpellE>Sadun</span>), <em><span style='font-family:Courier'>J. Phys. A: Math. <span class=SpellE>Theor</span>.</span></em></span> <span style='font-family:Courier'>47<b>,</b></span> <span style='font-family: Courier'>175001<i>, </i>2014. [<a href="Publications/Radin-Ren-Sadun-JPA-2014.pdf">pdf]</a>.</span> <span style='font-family:Courier;color:red'>One figure in the paper appears on the <a href="Publications/Radin-Ren-Sadun-JPA-2014-Cover-Picture.pdf">cover</a> of the journal</span><span style='font-family:Courier'>. </span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo6; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Quantitative photoacoustic imaging in the radiative transport regime (with A. <span class=SpellE>Mamonov</span>), <span class=SpellE><i>Commun</i></span><i>. Math. Sci.,</i></span> <span style='font-family:Courier'>12, 201-234, 2014. <a href="Publications/qPAT-ERT.pdf">[pdf]</a></span></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo6; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Quantitative fluorescence photoacoustic tomography (with H. Zhao), <i>SIAM J. <span class=SpellE>Imag</span>. Sci.</i>, 6, 2404-2429, 2013. </span><a href="Publications/QfPAT-Diff.pdf"><span style='font-family:Courier;text-decoration: none;text-underline:none'>[pdf].</span></a></p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo6; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Inverse transport calculations in optical imaging with subspace optimization algorithms (with T. Ding), <i>J. <span class=SpellE>Comput</span>. Phys., </i>273, 212-226, 2014 [<a href="Publications/Ding-Ren-SOM-ERT.pdf">pdf].</a></span></p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>On the modeling and simulation of reaction-transfer dynamics in semiconductor-electrolyte solar cells (with Y. He, I. M. <span class=SpellE>Gamba</span> and H. Lee), <i>SIAM J. Appl. Math.,</i></span> <span style='font-family:Courier'>75, 2515-2539, 2015.</span> <span style='font-family:Courier'><a href="Publications/He-Gamba-Lee-Ren-JCP13-Semi-Elec-Interf.pdf">[pdf]</a>.</span></p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>A hybrid reconstruction method for quantitative PAT (with H. Gao and H. Zhao), <i>SIAM J. <span class=SpellE>Imag</span>. Sci.</i>, 6, 32-55, 2013. <a href="Publications/QPAT-Hybrid-Num.pdf">[pdf]</a></span></p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>On multi-spectral quantitative photoacoustic tomography</span> <span style='font-family:Courier'>(with G. Bal), <i>Inverse Problems</i>, 28, 025010, 2012. <a href="Publications/qPAT-Spectra-OL11.pdf">[pdf]</a>. </span><span style='font-family:Courier;color:red'>This paper was selected as a <a href="http://iopscience.iop.org/0266-5611/28/2/025010">featured article</a> by Inverse Problems and included in its <a href="http://iopscience.iop.org/0266-5611/page/Highlights%20of%202012">Highlight of 2012</a> collection</span><span style='font-family:Courier'>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Quantitative Thermo-acoustics and related problems</span> <span style='font-family:Courier'>(with G. Bal, G. <span class=SpellE>Uhlmann</span> and T. Zhou), <i>Inverse Problems</i>, 27, 055007, 2011. <a href="Publications/TAT-ISCAT-10.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Multiple-source quantitative photoacoustic tomography</span> <span style='font-family:Courier'>(with G. Bal), <i>Inverse Problems</i>, 27, 075003, 2011.</span> <span style='font-family:Courier'><a href="Publications/Quant-Pat-11.pdf">[pdf]</a>. </span><span style='font-family:Courier;color:red'>Read the <a href="http://iopscience.iop.org/0266-5611/labtalk-article/46460">news</a> about this paper from Inverse Problems.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Non-uniqueness result for a hybrid inverse problem</span> <span style='font-family:Courier'>(with G. Bal), <i>Contemporary Mathematics</i>, 559, 29-38, 2011.</span> <span style='font-family:Courier'><a href="Publications/Non-Uniq-AMS-10.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Recovering doping profiles in semiconductor devices with the Boltzmann-Poisson model (with Y. Cheng and I. M. <span class=SpellE>Gamba</span>), <i>J. <span class=SpellE>Comput</span>. Phys.,</i></span> <span style='font-family:Courier'>230, 3391-3412, 2011<i>.</i></span> <span style='font-family:Courier'><a href="Publications/InverseDoping.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Recent development in numerical techniques for transport-based medical imaging methods, <span class=SpellE><i>Commun</i></span><i>. <span class=SpellE>Comput</span>. Phys., </i>8, 1-50, 2010<i>.</i></span> <span style='font-family:Courier'><a href="Publications/ReviewInvTransport-CCP-10.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Parametric image reconstruction using the discrete cosine transform for optical tomography</span> <span style='font-family:Courier'>(with X. <span class=SpellE>Gu</span>, J. <span class=SpellE>Masciotti</span> and A. H. <span class=SpellE>Hilescher</span>), <i>J. Biomed. Opt., </i>14, 064003, 2009<i>.</i></span> <span style='font-family: Courier'><a href="Publications/GuRenMasciottiHielscher-JBO09-ParametricRecon.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Physics-based models for measurement correlations. Application to an inverse Sturm-<span class=SpellE>Liouville</span> problem</span> <span style='font-family:Courier'>(with G. Bal), <i>Inverse Problems, </i>25, 055006, 2009<i>.</i></span> <span style='font-family:Courier'><a href="Publications/RandomFluct-IP-08.pdf">[pdf</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Transport-based imaging in random media</span> <span style='font-family:Courier'>(with G. Bal), <i>SIAM Applied Math., </i>68, 1738-1762, <span class=GramE>2008 <i>.</i></span></span> <span style='font-family:Courier'><a href="Publications/TransportImaging.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Experimental validation of a transport-based imaging method in highly scattering environments</span> <span style='font-family:Courier'>(with G. Bal, L. <span class=SpellE>Carin</span> and D. Liu), <i>Inverse Problems, </i>23, 2527-2539, 2007<i>.&nbsp;</i></span> <span style='font-family:Courier'><a href="Publications/ExpValidTranspImag.pdf">[pdf]</a>.</span>&nbsp; </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Transport- and diffusion-based optical tomography in small domain: A comparative study</span> <span style='font-family:Courier'>(with G. Bal and A. H. <span class=SpellE>Hilescher</span>), <i>Applied Optics, </i>46, 6669-6679, 2007<i>.</i></span> <span style='font-family:Courier'><a href="Publications/CompDiffTransport.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Frequency-domain sensitivity analysis for small imaging domains using the equation of radiative transfer</span> <span style='font-family:Courier'>(with X. <span class=SpellE>Gu</span> and A. <span class=SpellE>Hielscher</span>), Appl. Optics, 46, 1624-1632, 2007. <a href="Publications/FreqSensitivity.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Frequency domain optical tomography based on the equation of radiative transfer (with G. Bal and A. H. <span class=SpellE>Hielscher</span>), <i>SIAM J. Sci. <span class=SpellE>Comput</span>., </i>28, 1463-1489, 2006<i>.</i></span> <span style='font-family:Courier'><a href="Publications/FreqERTInverse.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Reconstruction of singular surfaces by shape sensitivity analysis and level set method</span> <span style='font-family:Courier'>(with G. Bal), Math. Models Methods Appl. Sci. (M3AS)<i>, </i>16, 1347-1373, 2006<i>.</i></span> <span style='font-family: Courier'><a href="Publications/GeneralizedDiffInverse.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Inverse Problems in Transport and Diffusion Theory with Applications in Optical Tomography, PhD thesis, Columbia University, New York, 2006.</span> <span style='font-family:Courier'><a href="Publications/Ren-PhD-Thesis-2006.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Optical tomographic imaging as a PDE-constrained optimization problem</span> <span style='font-family:Courier'>(with G. <span class=SpellE>Abdoulaev</span> and A. H. <span class=SpellE>Hielscher</span>), <i>Inverse Problems, </i>21, 1507-1530, 2005.</span> <span style='font-family:Courier'><a href="Publications/PDEConstrained.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Atmospheric concentration profile reconstructions from radiation measurements</span> <span style='font-family:Courier'>(with G. Bal), <i>Inverse Problems, </i>21, 153-168, 2005.</span> <span style='font-family:Courier'><a href="Publications/AtmosRecon.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Frequency-domain tomography in small animals with the equation of radiative transfer (with B. Moa-Anderson, <span class=SpellE>G.~Bal</span>, X.~<span class=SpellE>Gu</span>, and A. H. <span class=SpellE>Hielscher</span>), <i>Optical Tomography and Spectroscopy of Tissue VI </i>(Edited by B. Chance, R. R. <span class=SpellE>Alfano</span>, B. J. <span class=SpellE>Tromberg</span>, M. Tamura, and E. M. <span class=SpellE>Sevick-Muraca</span>), 111-120. SPIE, 2005, </span><u><span style='font-family:Courier;color:blue'>[pdf]</span></u><span style='font-family: Courier'>. </span></p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Algorithm <span class=GramE>for&nbsp; solving</span> the equation of radiative transfer in the frequency domain</span> <span style='font-family:Courier'>(G. S. <span class=SpellE>Abdoulaev</span>, G. Bal and A. H. <span class=SpellE>Hilescher</span>), <i>Optics Lett., </i>29, 578-580, 2004<i>.</i></span> <span style='font-family: Courier'><a href="Publications/FreqERTForward.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Generalized diffusion model in optical tomography with clear layers</span> <span style='font-family:Courier'>(with G. Bal), <i>J. Opt. Soc. Am. A, </i>20, 2355-2364, 2003<i>.</i></span> <span style='font-family:Courier'><a href="Publications/GeneralizedDiffForward.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>The most intensive fluctuation in chaotic time series and relativity principle</span> <span style='font-family:Courier'>(with S-D Liu, S-K Liu and Z-T Fu), <i>Chaos Soliton <span class=SpellE>Fract</span>., </i>15, 627-630, 2003. </span>&nbsp;<span style='font-family:Courier'><a href="Publications/MostIntensive.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Hierarchical similarity in an inhomogeneous turbulent wake flow behind multi-rectangular pillars (with Z. Zou et al), <span class=SpellE><i>Acta</i></span><i> <span class=SpellE>Mechanica</span> <span class=SpellE>Sinica</span>,</i></span> <span style='font-family:Courier'>35, 519-523, 2003. <a href="Publications/WakeTurbulence.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Scaling equation for invariant measure (with S-K Liu, Z-T Fu and S-D Liu), <span class=SpellE><i>Commun</i></span><i>. <span class=SpellE>Theor</span>. Phys.,</i></span> <span style='font-family:Courier'>39, 295-296, 2003. <a href="Publications/InvariantMeasure.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Several problems in the studying of nonlinear dynamics: A review</span> <span style='font-family:Courier'>(with S-D Liu, S-K Liu, S. Liang and Z-T <span class=GramE>Fu )</span>, <span class=SpellE><i>Prog</i></span><i>. Nat. Sci., </i>12, 1-7, 2002.</span>&nbsp;<span style='font-family:Courier'><a href="Publications/NonlinearDynReview.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>The <span class=SpellE>homoclinic</span> orbit solution for functional equation</span> <span style='font-family:Courier'>(S-D Liu, Z-T Fu and S-K Liu), <span class=SpellE><i>Commun</i></span><i>. <span class=SpellE>Theor</span>. Phys.,</i></span> <span style='font-family: Courier'>38, 553-554, 2002. <a href="Publications/HomoclinicFuncEqn.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Multi-scaling hierarchical structure analysis on the sequence of E-coli complete genome</span> <span style='font-family:Courier'>(with J. Wang, Q. Zhang and Z-S She), <i>Chinese Sci. Bull.,</i></span> <span style='font-family:Courier'>46, 1988-1992, 2001. </span>&nbsp;<span style='font-family:Courier'><a href="Publications/EcoliAnalysis.pdf">[pdf]</a>.</span> </p> <p style='margin-top:5.0pt;margin-right:0in;margin-bottom:0in;margin-left:.5in; margin-bottom:.0001pt;text-indent:-.25in;mso-list:l0 level1 lfo6;tab-stops: list .5in'><![if !supportLists]><span style='font-size:10.0pt;mso-bidi-font-size: 12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family: Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><span style='font-family: Courier'>Scalings</span></span><span style='font-family:Courier'> and structures in turbulent <span class=SpellE>Couette</span>-Taylor flow</span> <span style='font-family:Courier'>(with Z-S She, G. S. Lewis and H. L. <span class=SpellE>Swinney</span>), <i>Phys. Rev. E, </i>64, 016308, 2001. </span>&nbsp;<span style='font-family:Courier'><a href="Publications/TaylorCouette.pdf">[pdf]</a>.</span> </p> <p style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo6; tab-stops:list .5in'><![if !supportLists]><span style='font-size:10.0pt; mso-bidi-font-size:12.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-family:Courier'>Hierarchical structure analysis of multi-scale fluctuations from highly nonlinear physical systems, Master Thesis, Peking University, Beijing, 2001.</span> <span style='font-family:Courier'><a href="Publications/Ren-Master-Thesis-2001.pdf">[pdf]</a>.</span> <o:p></o:p></p> <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 width="100%" style='width:100.0%;mso-cellspacing:0in;mso-yfti-tbllook:1184;mso-padding-alt: 0in 0in 0in 0in'> <col width="49*"><col width="64*"><col width="73*"><col width="70*"> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width="19%" style='width:19.0%;padding:0in 0in 0in 0in'> <p class=MsoNormal style='mso-margin-top-alt:auto;margin-bottom:5.75pt'><a name=table1></a>Citations: </p> </td> <td width="25%" style='width:25.0%;padding:0in 0in 0in 0in'> <p class=MsoNormal align=center style='mso-margin-top-alt:auto;margin-bottom: 5.75pt;text-align:center'><a href="http://scholar.google.com/citations?user=fqE6Bk8AAAAJ">Google Scholar</a></p> </td> <td width="28%" style='width:28.0%;padding:0in 0in 0in 0in'> <p class=MsoNormal align=center style='mso-margin-top-alt:auto;margin-bottom: 5.75pt;text-align:center'><a href="http://portal.isiknowledge.com/portal.cgi?SID=A22nE5GMhHG@p39@Onf&amp;Func=Menu&amp;mnu_xs=XS_FORM+WOS+DIIDW+MEDLINE+ZOOREC&amp;mnu=JCR&amp;perscode=UML+PCT&amp;DestApp=WOS" target="_blank"><span style='color:blue'>Scientific Citation Index</span></a></p> </td> <td width="27%" style='width:27.0%;padding:0in 0in 0in 0in'> <p class=MsoNormal align=center style='mso-margin-top-alt:auto;margin-bottom: 5.75pt;text-align:center'><a href="http://www.ams.org/msnmain?fn=130&amp;fmt=hl&amp;pg1=IID&amp;s1=711179&amp;v1=Ren%2C%20Kui"><span class=SpellE><span style='color:blue'>MathSciNet</span></span></a></p> </td> </tr> </table> <p style='margin-bottom:12.0pt'><o:p>&nbsp;</o:p></p> </div> </body> </html>