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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>INSTITUTE FOR ADVANCED STUDY<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>School of Mathematics<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Princeton, NJ 08540<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Mathematics Seminars<o:p></o:p></span></b></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Week of April 30, 2018<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>--------------<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>To view mathematics in titles and abstracts, please click on the talk's link.<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>--------------<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Monday, April 30<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Princeton/IAS Symplectic Geometry Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: Birational Calabi-Yau manifolds have the same small quantum products.<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Mark McLean, Stony Brook University<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 4:00pm - 5:00pm/Simonyi Hall 101<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=135651">http://www.math.ias.edu/seminars/abstract?event=135651</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></b></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Tuesday, May 1<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Locally Symmetric Spaces Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'> No Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: No Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: - <o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'> No Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: No Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: - <o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Joint IAS/Princeton University Number Theory Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: A converse to a theorem of Gross--Zagier, Kolyvagin and Rubin, II<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Ashay Burungale, Universite Paris 13; Member, School of Mathematics<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 4:45pm - 5:45pm/Simonyi Hall 101<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=136745">http://www.math.ias.edu/seminars/abstract?event=136745</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Thursday, May 3<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Working Group on Algebraic Number Theory<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: To Be Announced<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 2:00pm - 4:00pm/1201 Fine Hall, Princeton University<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Joint IAS/Princeton University Number Theory Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: A </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-fareast-language:EN-US'><img width=8 height=17 style='width:.0833in;height:.177in' id="_x0000_i1025" src="cid:image001.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>-adically entire function with integral values on </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=19 height=19 style='width:.1979in;height:.1979in' id="_x0000_i1025" src="cid:image002.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>and </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-fareast-language:EN-US'><img width=8 height=17 style='width:.0833in;height:.177in' id="_x0000_i1025" src="cid:image001.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>-adic Fourier expansions.<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Francesco Baldassarri, University of Padova<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 4:30pm - 5:30pm/Fine Hall 214, Princeton University<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=136754">http://www.math.ias.edu/seminars/abstract?event=136754</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'>1 Birational Calabi-Yau manifolds have the same small quantum products. <br> Mark McLean <o:p></o:p></span></p><p style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:11.0pt'>We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace. <o:p></o:p></span></p><p style='margin-top:3.0pt'><span style='font-size:11.0pt'><a href="http://www.math.ias.edu/seminars/abstract?event=135651">http://www.math.ias.edu/seminars/abstract?event=135651</a><o:p></o:p></span></p><p style='margin-top:3.0pt'><span style='font-size:11.0pt'><br><br>2 A converse to a theorem of Gross--Zagier, Kolyvagin and Rubin, II <br> Ashay Burungale <o:p></o:p></span></p><p style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:11.0pt'>Let </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>E</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:16.5pt;mso-text-raise:-16.5pt;mso-fareast-language:EN-US'><img width=10 height=36 style='width:.1041in;height:.375in' id="_x0000_i1025" src="cid:image007.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'> be a CM elliptic curve over a totally real number field </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>F</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:16.5pt;mso-text-raise:-16.5pt;mso-fareast-language:EN-US'><img width=10 height=36 style='width:.1041in;height:.375in' id="_x0000_i1025" src="cid:image009.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'> and </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:17.5pt;mso-text-raise:-17.5pt;mso-fareast-language:EN-US'><img width=8 height=36 style='width:.0833in;height:.375in' id="_x0000_i1025" src="cid:image010.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'> an odd ordinary prime. If the </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:e><m:sup><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>∞</m:r></span></i></m:sup></m:sSup></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:17.5pt;mso-text-raise:-17.5pt;mso-fareast-language:EN-US'><img width=19 height=36 style='width:.1979in;height:.375in' id="_x0000_i1025" src="cid:image011.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'>-</span><span style='font-size:11.0pt'>Selmer group of</span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r> </m:r><m:r>E</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:16.5pt;mso-text-raise:-16.5pt;mso-fareast-language:EN-US'><img width=13 height=36 style='width:.1354in;height:.375in' id="_x0000_i1025" src="cid:image025.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'> over </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>F</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:16.5pt;mso-text-raise:-16.5pt;mso-fareast-language:EN-US'><img width=10 height=36 style='width:.1041in;height:.375in' id="_x0000_i1025" src="cid:image009.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'> has </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=16 height=38 style='width:.1666in;height:.3958in' id="_x0000_i1025" src="cid:image026.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'>‑corank one, we show that the analytic rank of</span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r> </m:r><m:r>E</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:16.5pt;mso-text-raise:-16.5pt;mso-fareast-language:EN-US'><img width=13 height=36 style='width:.1354in;height:.375in' id="_x0000_i1025" src="cid:image025.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'> over </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>F</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:16.5pt;mso-text-raise:-16.5pt;mso-fareast-language:EN-US'><img width=10 height=36 style='width:.1041in;height:.375in' id="_x0000_i1025" src="cid:image009.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'> is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy. <o:p></o:p></span></p><p style='margin-top:3.0pt'><span style='font-size:11.0pt'><a href="http://www.math.ias.edu/seminars/abstract?event=136745">http://www.math.ias.edu/seminars/abstract?event=136745</a><o:p></o:p></span></p><p style='margin-top:3.0pt'><span style='font-size:11.0pt'><br><br>3 A </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:17.5pt;mso-text-raise:-17.5pt;mso-fareast-language:EN-US'><img width=8 height=36 style='width:.0833in;height:.375in' id="_x0000_i1025" src="cid:image010.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'>-adically entire function with integral values on </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=19 height=38 style='width:.1979in;height:.3958in' id="_x0000_i1025" src="cid:image027.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'>and </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:17.5pt;mso-text-raise:-17.5pt;mso-fareast-language:EN-US'><img width=8 height=36 style='width:.0833in;height:.375in' id="_x0000_i1025" src="cid:image010.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'>-adic Fourier expansions. <br> Francesco Baldassarri<o:p></o:p></span></p><p class=MsoNormal style='line-height:21.3pt'><span style='font-family:"Times New Roman",serif'>We explain the magic of the entire function</span> <!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>Ψ</m:r></span></span></m:e><m:sub><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:sub></m:sSub><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>∈</m:r></span></span><i><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i><m:d><m:dPr><m:begChr m:val="⟦"/><m:endChr m:val="⟧"/><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>x</m:r></span></i></span></m:e></m:d><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>∩</m:r></span></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span style='font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>{</m:r></span></span><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>x</m:r></span></i></span><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>} </m:r></span></span></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=123 height=19 style='width:1.2812in;height:.1979in' id="_x0000_i1025" src="cid:image029.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>defined by the functional equation <o:p></o:p></span></p><p class=MsoNormal align=center style='text-align:center;line-height:21.3pt'><!--[if gte msEquation 12]><m:oMathPara><m:oMath><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>x</m:r></span></i></span><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>=</m:r></span></span><m:nary><m:naryPr><m:chr m:val="∑"/><m:limLoc m:val="undOvr"/><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:naryPr><m:sub><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>j</m:r><m:r>=0</m:r></span></i></span></m:sub><m:sup><i><span style='font-family:"Cambria Math",serif'><m:r>∞</m:r></span></i></m:sup><m:e><m:sSup><m:sSupPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:e><m:sup><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>-</m:r><m:r>j</m:r></span></i></span></m:sup></m:sSup></m:e></m:nary><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>Ψ</m:r></span></span></m:e><m:sub><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:sub></m:sSub><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>(</m:r></span></span><m:sSup><m:sSupPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:e><m:sup><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>j</m:r></span></i></span></m:sup></m:sSup><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>x</m:r></span></i></span><m:sSup><m:sSupPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>)</m:r></span></span></m:e><m:sup><m:sSup><m:sSupPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:e><m:sup><span class=mathjax1><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>j</m:r></span></i></span></m:sup></m:sSup></m:sup></m:sSup><span class=mathjax1><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr> </m:r></span></span></m:oMath></m:oMathPara><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US'><img width=141 height=50 style='width:1.4687in;height:.5208in' id="_x0000_i1025" src="cid:image075.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'><o:p></o:p></span></p><p style='margin-top:0in'><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>which satisfies </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>Ψ</m:r></span></span></m:e><m:sub><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:sub></m:sSub><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub></m:e></m:d><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>⊂</m:r></span></span><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="p"/></m:rPr>Z</m:r></span></span></m:e><m:sub><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=85 height=39 style='width:.8854in;height:.4062in' id="_x0000_i1025" src="cid:image076.png@01D3DE6A.87264C10"></span><![endif]><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'> </span></span><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>and admits an integral addition law. We then use the Artin-Hasse exponential</span><!--[if gte msEquation 12]><m:oMath><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><br></span></i></span></m:oMath><![endif]--><!--[if gte msEquation 12]><m:oMathPara><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>F</m:r></span></i><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>T</m:r></span></i></m:e></m:d><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>=</m:r></span></span><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>exp</m:r></span><m:nary><m:naryPr><m:chr m:val="∑"/><m:limLoc m:val="undOvr"/><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:naryPr><m:sub><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>j</m:r><m:r>=0</m:r></span></i></span></m:sub><m:sup><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>∞</m:r></span></i></m:sup><m:e><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>T</m:r></span></i></span></m:e><m:sup><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:e><m:sup><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>i</m:r></span></i></span></m:sup></m:sSup></m:sup></m:sSup></m:e></m:nary><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:e><m:sup><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>i</m:r></span></i></span></m:sup></m:sSup><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>∈ </m:r></span></i></span><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>(</m:r><m:r>p</m:r><m:r>)</m:r></span></i></m:sub></m:sSub><span class=mathjax1><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r> </m:r></span></i></span><m:d><m:dPr><m:begChr m:val="⟦"/><m:endChr m:val="⟧"/><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>T</m:r></span></i></m:e></m:d></m:oMath></m:oMathPara><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US'><img width=197 height=69 style='width:2.052in;height:.7187in' id="_x0000_i1025" src="cid:image077.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'><o:p></o:p></span></p><p style='margin-top:0in'><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>to deduce from </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>Ψ</m:r></span></span></m:e><m:sub><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></span></m:sub></m:sSub><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r> </m:r></span></i></span></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=21 height=38 style='width:.2187in;height:.3958in' id="_x0000_i1025" src="cid:image078.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>a topological basis </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><m:d><m:dPr><m:begChr m:val="{"/><m:endChr m:val="}"/><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>G</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>q</m:r></span></i></m:sub></m:sSub></m:e></m:d></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>q</m:r><m:r>∈</m:r><m:r>S</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r> </m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:22.0pt;mso-text-raise:-22.0pt;mso-fareast-language:EN-US'><img width=50 height=43 style='width:.5208in;height:.4479in' id="_x0000_i1025" src="cid:image079.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>, where </span><!--[if gte msEquation 12]><m:oMath><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>S</m:r></span></i></span><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>=</m:r></span></span><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i><m:d><m:dPr><m:begChr m:val="["/><m:endChr m:val="]"/><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>1/</m:r><m:r>p</m:r></span></i></span></m:e></m:d><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>∩</m:r></span></span><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span class=mathjax1><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>R</m:r></span></i></span></m:e><m:sub><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="p"/></m:rPr>≥</m:r><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>0</m:r></span></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:17.5pt;mso-text-raise:-17.5pt;mso-fareast-language:EN-US'><img width=108 height=36 style='width:1.125in;height:.375in' id="_x0000_i1025" src="cid:image080.png@01D3DE6A.87264C10"></span><![endif]><span class=mathjax1><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'> </span></span><span style='font-size:11.0pt'>,</span><span style='font-size:11.0pt;font-family:bodoni-urw;color:black'> <span lang=EN>of the Fréchet space of continuous functions </span></span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>→</m:r></span></i><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=58 height=38 style='width:.6041in;height:.3958in' id="_x0000_i1025" src="cid:image081.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>, which consists of entire functions </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>G</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>q</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>:</m:r></span></i><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>C</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>→</m:r></span></i><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>C</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=75 height=38 style='width:.7812in;height:.3958in' id="_x0000_i1025" src="cid:image082.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'> defined over </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>(</m:r><m:r>p</m:r><m:r>)</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=25 height=38 style='width:.2604in;height:.3958in' id="_x0000_i1025" src="cid:image083.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt;font-family:bodoni-urw'> </span><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>taking </span><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:17.5pt;mso-text-raise:-17.5pt;mso-fareast-language:EN-US'><img width=8 height=36 style='width:.0833in;height:.375in' id="_x0000_i1025" src="cid:image010.png@01D3DE6A.87264C10"></span><![endif]><span style='font-size:11.0pt'>-adic </span><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>integral values all over </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-fareast-language:EN-US'><img width=19 height=38 style='width:.1979in;height:.3958in' id="_x0000_i1025" src="cid:image027.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>. They satisfy </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>G</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>0</m:r></span></i></m:sub></m:sSub><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>x</m:r></span></i></m:e></m:d><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>=1</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:17.5pt;mso-text-raise:-17.5pt;mso-fareast-language:EN-US'><img width=63 height=36 style='width:.6562in;height:.375in' id="_x0000_i1025" src="cid:image084.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'>,<o:p></o:p></span></p><p style='margin-top:0in'><!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>G</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>q</m:r></span></i></m:sub></m:sSub><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>+</m:r><m:r>y</m:r></span></i></m:e></m:d><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>=</m:r></span></i><m:nary><m:naryPr><m:chr m:val="∑"/><m:limLoc m:val="undOvr"/><m:supHide m:val="on"/><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:naryPr><m:sub><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>1</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>+</m:r></span></i><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>=</m:r><m:r>q</m:r></span></i></m:sub><m:sup></m:sup><m:e><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>G</m:r></span></i></m:e><m:sub><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>q</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>1</m:r></span></i></m:sub></m:sSub></m:sub></m:sSub><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e></m:d><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>G</m:r></span></i></m:e><m:sub><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>q</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sub></m:sSub></m:sub></m:sSub><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>y</m:r></span></i></m:e></m:d><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr> </m:r><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>and</m:r><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr> </m:r></span><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>G</m:r></span></i></m:e><m:sub><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>pq</m:r></span></i></m:sub></m:sSub><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>px</m:r></span></i></m:e></m:d><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>=</m:r></span></i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr> </m:r></span><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>G</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>q</m:r></span></i></m:sub></m:sSub><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e></m:d><i><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r>, </m:r><m:r>∀</m:r><m:r>q</m:r><m:r>∈</m:r><m:r>S</m:r></span></i></m:e></m:nary><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>.</m:r><m:r> </m:r></span></i></m:oMath></m:oMathPara><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US'><img width=413 height=60 style='width:4.302in;height:.625in' id="_x0000_i1025" src="cid:image085.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'><o:p></o:p></span></p><p class=MsoNormal><span lang=EN style='font-family:bodoni-urw;color:black'>The convergence of the previous sum is uniform on compact subsets of </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span style='font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=19 height=19 style='width:.1979in;height:.1979in' id="_x0000_i1025" src="cid:image002.png@01D3DE6A.87264C10"></span><![endif]><span style='font-family:bodoni-urw'> </span><span lang=EN style='font-family:bodoni-urw;color:black'>along the filter of cofinite subsets of </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-fareast-language:EN-US'><img width=8 height=17 style='width:.0833in;height:.177in' id="_x0000_i1025" src="cid:image086.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>. We identify the</span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r> </m:r><m:r>(</m:r><m:r>p</m:r><m:r>,</m:r><m:r>T</m:r><m:r>)</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-fareast-language:EN-US'><img width=38 height=17 style='width:.3958in;height:.177in' id="_x0000_i1025" src="cid:image087.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>-adic completion </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="script"/><m:sty m:val="i"/></m:rPr>D</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-fareast-language:EN-US'><img width=11 height=17 style='width:.1145in;height:.177in' id="_x0000_i1025" src="cid:image088.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'> of the ring </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>(</m:r><m:r>p</m:r><m:r>)</m:r></span></i></m:sub></m:sSub><m:d><m:dPr><m:begChr m:val="["/><m:endChr m:val="]"/><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSup><m:sSupPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>T</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>1/</m:r></span></i><m:sSup><m:sSupPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>∞</m:r></span></i></m:sup></m:sSup></m:sup></m:sSup></m:e></m:d></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=73 height=21 style='width:.7604in;height:.2187in' id="_x0000_i1025" src="cid:image089.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'> with a topological Hopf algebra of </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=16 height=19 style='width:.1666in;height:.1979in' id="_x0000_i1025" src="cid:image090.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>-valued measures on the uniformly open subsets of </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span style='font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=19 height=19 style='width:.1979in;height:.1979in' id="_x0000_i1025" src="cid:image002.png@01D3DE6A.87264C10"></span><![endif]><span style='font-family:bodoni-urw'>, </span><span lang=EN style='font-family:bodoni-urw;color:black'>equipped with the topology of uniform convergence on the families of balls of equal radius, in such a way that, for any </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>i</m:r><m:r> </m:r><m:r><span class=mathjax1><i>∈</i></span></m:r><m:r><span class=mathjax1><i> </i></span></m:r><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr><span class=mathjax1><i>Z</i></span></m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-fareast-language:EN-US'><img width=38 height=17 style='width:.3958in;height:.177in' id="_x0000_i1025" src="cid:image091.png@01D3DE6A.87264C10"></span><![endif]><span class=mathjax1><span lang=EN style='font-family:bodoni-urw;color:black'>, <o:p></o:p></span></span></p><p class=MsoNormal><span class=mathjax1><span lang=EN style='font-family:bodoni-urw;color:black'><o:p> </o:p></span></span></p><p style='margin-top:0in'><!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:func><m:funcPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:funcPr><m:fName><m:limLow><m:limLowPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:limLowPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>lim</m:r></span></i></m:e><m:lim><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>n</m:r><m:r>→</m:r><m:r>+</m:r><m:r>∞</m:r></span></i></m:lim></m:limLow></m:fName><m:e><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>F</m:r></span></i><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>T</m:r></span></i></m:e><m:sup><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>i</m:r><m:r>-</m:r><m:r>n</m:r></span></i></m:sup></m:sSup></m:sup></m:sSup></m:e></m:d></m:e><m:sup><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>n</m:r></span></i></m:sup></m:sSup></m:sup></m:sSup><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>=</m:r></span></i><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>∆</m:r></span></i></m:e><m:sub><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>i</m:r></span></i></m:sup></m:sSup></m:sub></m:sSub></m:e></m:func></m:oMath></m:oMathPara><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US'><img width=149 height=51 style='width:1.552in;height:.5312in' id="_x0000_i1025" src="cid:image092.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'><o:p></o:p></span></p><p class=MsoNormal><span lang=EN style='font-family:bodoni-urw;color:black'>is the Dirac mass at </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>i</m:r></span></i></m:sup></m:sSup></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-fareast-language:EN-US'><img width=13 height=18 style='width:.1354in;height:.1875in' id="_x0000_i1025" src="cid:image093.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>. So, for the inverse series </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>E</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-fareast-language:EN-US'><img width=10 height=17 style='width:.1041in;height:.177in' id="_x0000_i1025" src="cid:image094.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'> of </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>F</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-fareast-language:EN-US'><img width=10 height=17 style='width:.1041in;height:.177in' id="_x0000_i1025" src="cid:image095.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>, let<o:p></o:p></span></p><p style='margin-top:0in'><!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>μ</m:r></span></i></m:e><m:sub><span style='font-size:11.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>can</m:r></span></m:sub></m:sSub><m:box><m:boxPr><m:opEmu m:val="on"/><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:boxPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>∶=</m:r></span></i></m:e></m:box><m:func><m:funcPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:funcPr><m:fName><m:limLow><m:limLowPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:limLowPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>lim</m:r></span></i></m:e><m:lim><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>n</m:r><m:r>→</m:r><m:r>+</m:r><m:r>∞</m:r></span></i></m:lim></m:limLow></m:fName><m:e><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>E</m:r></span></i><m:d><m:dPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>∆</m:r></span></i></m:e><m:sub><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>-</m:r><m:r>n</m:r></span></i></m:sup></m:sSup></m:sub></m:sSub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>-</m:r></span></i><m:sSub><m:sSubPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>∆</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>0</m:r></span></i></m:sub></m:sSub></m:e></m:d></m:e><m:sup><m:sSup><m:sSupPr><span style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:e><m:sup><i><span lang=EN style='font-size:11.0pt;font-family:"Cambria Math",serif;color:black'><m:r>n</m:r></span></i></m:sup></m:sSup></m:sup></m:sSup></m:e></m:func></m:oMath></m:oMathPara><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US'><img width=184 height=48 style='width:1.9166in;height:.5in' id="_x0000_i1025" src="cid:image096.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-size:11.0pt;font-family:bodoni-urw;color:black'><o:p></o:p></span></p><p class=MsoNormal><span lang=EN style='font-family:bodoni-urw;color:black'>be the </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Z</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=16 height=19 style='width:.1666in;height:.1979in' id="_x0000_i1025" src="cid:image090.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>-valued measure on </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span style='font-family:"Cambria Math",serif'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=19 height=19 style='width:.1979in;height:.1979in' id="_x0000_i1025" src="cid:image002.png@01D3DE6A.87264C10"></span><![endif]><span style='font-family:bodoni-urw'> </span><span lang=EN style='font-family:bodoni-urw;color:black'>corresponding to </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>T</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-fareast-language:EN-US'><img width=9 height=17 style='width:.0937in;height:.177in' id="_x0000_i1025" src="cid:image097.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'>. Then, for any </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>q</m:r><m:r> </m:r><m:r><span class=mathjax1><i>∈</i></span></m:r><m:r><span class=mathjax1><i>S</i></span></m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-fareast-language:EN-US'><img width=37 height=17 style='width:.3854in;height:.177in' id="_x0000_i1025" src="cid:image098.png@01D3DE6A.87264C10"></span><![endif]><span class=mathjax1><span lang=EN style='font-family:bodoni-urw;color:black'>, </span></span><span lang=EN style='font-family:bodoni-urw;color:black'>the measure </span><!--[if gte msEquation 12]><m:oMath><m:sSubSup><m:sSubSupPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubSupPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>μ</m:r></span></i></m:e><m:sub><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>can</m:r></span></m:sub><m:sup><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>q</m:r></span></i></m:sup></m:sSubSup></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-fareast-language:EN-US'><img width=27 height=19 style='width:.2812in;height:.1979in' id="_x0000_i1025" src="cid:image099.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'> exists in</span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r> </m:r><m:r><m:rPr><m:scr m:val="script"/><m:sty m:val="i"/></m:rPr>D</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-fareast-language:EN-US'><img width=14 height=17 style='width:.1458in;height:.177in' id="_x0000_i1025" src="cid:image100.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'> and any continuous function </span><!--[if gte msEquation 12]><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>f</m:r></span></i><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r> : </m:r></span></i><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>→</m:r></span></i><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=82 height=19 style='width:.8541in;height:.1979in' id="_x0000_i1025" src="cid:image101.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'> admits the following generalized Amice-Fourier expansion <o:p></o:p></span></p><p class=MsoNormal align=center style='text-align:center;line-height:21.3pt'><!--[if gte msEquation 12]><m:oMathPara><m:oMath><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>f</m:r></span></i><m:d><m:dPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>-</m:r></span></i></m:e></m:d><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>=</m:r></span></i><m:nary><m:naryPr><m:chr m:val="∑"/><m:limLoc m:val="undOvr"/><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:ctrlPr></m:ctrlPr></span></m:naryPr><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>q</m:r><m:r>∈</m:r><m:r>S</m:r></span></i></m:sub><m:sup></m:sup><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>(</m:r></span></i><m:nary><m:naryPr><m:limLoc m:val="subSup"/><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:naryPr><m:sub><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r><m:rPr><m:scr m:val="double-struck"/><m:sty m:val="i"/></m:rPr>Q</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black'><m:r>p</m:r></span></i></m:sub></m:sSub></m:sub><m:sup></m:sup><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r> </m:r></span></i><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>f</m:r></span></i><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r> </m:r></span></i><m:sSubSup><m:sSubSupPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubSupPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>μ</m:r></span></i></m:e><m:sub><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>can</m:r></span></m:sub><m:sup><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>q</m:r></span></i></m:sup></m:sSubSup></m:e></m:nary><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>)</m:r></span></i><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>G</m:r></span></i></m:e><m:sub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>q</m:r></span></i></m:sub></m:sSub><i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r>(-)</m:r></span></i><span lang=EN style='font-family:"Cambria Math",serif;color:black;border:none windowtext 1.0pt;padding:0in'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>,</m:r><m:r><i> </i></m:r></span></m:e></m:nary></m:oMath></m:oMathPara><![endif]--><![if !msEquation]><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US'><img width=187 height=53 style='width:1.9479in;height:.552in' id="_x0000_i1025" src="cid:image102.png@01D3DE6A.87264C10"></span><![endif]><span lang=EN style='font-family:bodoni-urw;color:black'><o:p></o:p></span></p><p class=MsoPlainText><span style='font-family:"Times New Roman",serif'>where the series converges as above.<o:p></o:p></span></p><p class=MsoPlainText><a href="http://www.math.ias.edu/seminars/abstract?event=136754">http://www.math.ias.edu/seminars/abstract?event=136754</a><o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><span style='font-family:"Times New Roman",serif'><o:p> </o:p></span></p><p class=MsoPlainText><span style='font-family:"Times New Roman",serif'>--------------------------------------<o:p></o:p></span></p><p class=MsoPlainText><span style='font-family:"Times New Roman",serif'>IAS Math Seminars Home Page:<o:p></o:p></span></p><p class=MsoPlainText><span style='font-family:"Times New Roman",serif'><a href="http://www.math.ias.edu/seminars">http://www.math.ias.edu/seminars</a><o:p></o:p></span></p></div></body></html>