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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 bgcolor=white lang=EN-US link=blue vlink=purple style='tab-interval:.5in' width=750 topmargin=0 leftmargin=0> <div class=WordSection1> <p><o:p>&nbsp;</o:p></p> <h2 align=center style='text-align:center'><span style='mso-fareast-font-family: "Times New Roman";mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="Picture_x0020_6" o:spid="_x0000_i1027" type="#_x0000_t75" style='width:634.5pt;height:63pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="seminar2012_files/image001.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img width=846 height=84 src="seminar2012_files/image003.jpg" v:shapes="Picture_x0020_6"><![endif]></span><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h2> <h2 align=center style='text-align:center'><span style='font-size:22.0pt; mso-fareast-font-family:"Times New Roman"'>School of Science<o:p></o:p></span></h2> <h2 align=center style='text-align:center'><span style='font-size:22.0pt; mso-fareast-font-family:"Times New Roman"'>Mathematics Seminar 2011-2012<o:p></o:p></span></h2> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div> <div class=MsoNormal><span style='mso-fareast-font-family:"Times New Roman"'> <hr size=1 width="85%" align=left> </span></div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Thursday, April 5<sup>th</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 117 OBS</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Daniel J. <span class=SpellE>Galiffa</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2; mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='mso-margin-top-alt:auto;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Title:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>The <span class=SpellE>Sheffer</span> A-Type 0 Orthogonal Sequences, Part II: Applications.<o:p></o:p></span></p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='text-align:justify'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>We continue the previous talk by addressing several of the applications of the <span class=SpellE>Sheffer</span> A-Type 0 orthogonal sets. These applications <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>include</span></span><span style='mso-fareast-font-family:"Times New Roman"'>; first and second-order differential equations, approximation theory and quantum mechanics. We will show how to develop and prove <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>the</span></span><span style='mso-fareast-font-family:"Times New Roman"'> relevant theorems and also give some specific examples, e.g. a connection with the Schrödinger Equation, which can be used to describe the <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>wave-nature</span></span><span style='mso-fareast-font-family:"Times New Roman"'> of quanta. We also show how several relationships of the <span class=SpellE>Sheffer</span> A-Type 0 orthogonal sets with other branches of mathematics <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>arise</span></span><span style='mso-fareast-font-family:"Times New Roman"'> as a byproduct of the applications.<o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span style='mso-fareast-font-family: "Times New Roman"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-align:justify'><span style='mso-fareast-font-family: "Times New Roman"'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Thursday, March 29<sup>th</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 117 OBS</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Daniel J. <span class=SpellE>Galiffa</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2; mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='mso-margin-top-alt:auto;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Title:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>The <span class=SpellE>Sheffer</span> A-Type 0 Orthogonal Sequences, Part I: Characterizing Theorems.<o:p></o:p></span></p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='text-align:justify'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>Previously, we have discussed a method for obtaining the <span class=SpellE>Sheffer</span> <i style='mso-bidi-font-style:normal'>B-Type 1</i> orthogonal polynomial sequences, which utilized computer <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>algebra</span></span><span style='mso-fareast-font-family:"Times New Roman"'> (<span class=SpellE>Mathematica</span>).<span style='mso-spacerun:yes'>  </span>Recently, the presenter has had his research monograph entitled <i style='mso-bidi-font-style:normal'> On the Higher-Order <span class=SpellE>Sheffer</span> Orthogonal Polynomial <o:p></o:p></i></span></p> <p class=MsoNormal style='text-align:justify'><i style='mso-bidi-font-style: normal'><span style='mso-fareast-font-family:"Times New Roman"'>Sequences </span></i><span style='mso-fareast-font-family:"Times New Roman"'> accepted for publication in the Springer Briefs in Mathematics series. This work contains not only the new results on the <i style='mso-bidi-font-style:normal'>B-Type 1</i> <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>polynomials</span></span><span style='mso-fareast-font-family:"Times New Roman"'>, but also the characterizations of the orthogonal <i style='mso-bidi-font-style:normal'>A-Type 0</i> sets conducted by J. <span class=SpellE>Meixner</span> and I.M. <span class=SpellE>Sheffer</span>, applications of these sets<o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>and</span></span><span style='mso-fareast-font-family:"Times New Roman"'> future research. In the present talk, we cover the aforesaid characterizations of <span class=SpellE>Meixner</span> and <span class=SpellE>Sheffer</span> and supplement these results with some <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>concrete</span></span><span style='mso-fareast-font-family:"Times New Roman"'> examples. We conclude with theorems that characterize the notion of <i style='mso-bidi-font-style:normal'>Type</i>. <o:p></o:p></span></p> <p class=MsoNormal style='text-align:justify'><span style='mso-fareast-font-family: "Times New Roman"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-align:justify'><span style='mso-fareast-font-family: "Times New Roman"'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, February 28<sup>th</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 170 Nick</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. <span class=SpellE>Papiya</span> <span class=SpellE>Bhattacharjee</span></span><span style='font-size:20.0pt; color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Title: <span style='mso-fareast-font-family:Calibri'>Distributive Join <span class=SpellE>Semilattices</span>, with 0</span>.<span style='color:black'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span>A lattice is a set L with a relation<span class=GramE>,<span style='mso-spacerun:yes'>  </span>d"</span>, such that any pair of elements from L has a least upper bound (<span class=SpellE>supremum</span>) and a greatest lower </p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>bound</span> (<span class=SpellE>infimum</span>). On the other hand, a join <span class=SpellE>semilattice</span> is a set S with a relation d" such that any pair of elements from S has a <span class=SpellE><span class=GramE>supremum</span></span><span class=GramE><span style='mso-spacerun:yes'>  </span>(</span>there is </p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>no</span> <span class=SpellE>infimum</span>). </p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>It is known that the study of distributive lattices is equivalent to the study of algebraic frames satisfying the finite intersection property (FIP).<span style='mso-spacerun:yes'>   </span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Also, the study of distributive join <span class=SpellE>semilattices</span> is equivalent to the study of algebraic frames (without FIP).<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Algebraic frames have been studied for a long period of time and FIP plays an important role. In this research talk the speaker will be focused on </p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>algebraic</span> frames without FIP, from the point of view of join <span class=SpellE>semilattices</span>.<span style='mso-spacerun:yes'>  </span>The speaker will discuss various results that are analogous to lattices.<span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Thursday, February 23<sup>rd</sup>, 2:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 170 Nick</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Mr. <span class=SpellE>Gregor</span> <span class=SpellE>Ol]avský</span> </span><span style='font-size:20.0pt;color:#0F243E; mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Title:<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:11.0pt; mso-fareast-font-family:Calibri'>Integers mod <i style='mso-bidi-font-style: normal'>n</i>: ideals, modules and direct sums</span>.<span style='color:black'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>The purpose of the talk is to exhibit the structure of the ring </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif";mso-fareast-font-family: "Times New Roman"'><m:r>Z</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif";mso-fareast-font-family: "Times New Roman"'><m:r>n</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:12.75pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image004.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=17 height=19 src="seminar2012_files/image005.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-fareast-font-family:"Times New Roman"'><span style='mso-spacerun:yes'> </span>as a direct sum of ideals formed from idempotent elements of the ring.<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-fareast-font-family:"Times New Roman"'>The only prerequisite is the definition of a&nbsp;group.&nbsp; I will show that the ring of <span class=GramE>integers</span> mod <i style='mso-bidi-font-style:normal'>n</i> can be viewed as a module over itself, which means <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>that</span></span><span style='mso-fareast-font-family:"Times New Roman"'> any <span class=SpellE>submodule</span> is also an ideal of the ring.&nbsp; A nice implication of this is that any element <i style='mso-bidi-font-style:normal'>r</i> in the ring can be written in a unique way as the <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>sum</span></span><span style='mso-fareast-font-family:"Times New Roman"'> of <i style='mso-bidi-font-style: normal'>q</i> elements, where <i style='mso-bidi-font-style:normal'>q</i> is the number of distinct prime factors of <i style='mso-bidi-font-style:normal'>n</i>.</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Thursday, February 2<sup>nd</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location TBD</span><span style='font-size: 16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Amos <span class=SpellE>Ong</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span><span class=SpellE>Askey</span>-Wilson Operator on <span class=SpellE>Chebyshev</span> Polynomials of the Second Kind.</p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>This talk will review what <span class=SpellE>Chebyshev</span> Polynomials of the 2nd kind are and define the <span class=SpellE>Askey</span>-Wilson operator.<span style='mso-spacerun:yes'>  </span>The <span class=SpellE>Askey</span>-Wilson operator <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>on</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> <span class=SpellE>Chebyshev</span> Polynomials of the first kind is a constant times a <span class=SpellE>Chebyshev</span> Polynomial of the 2nd kind.<span style='mso-spacerun:yes'>  </span>This will be proved again in this talk.<o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>And then with a clever trick, I will show the result of <span class=SpellE>Askey</span>-Wilson Operator on <span class=SpellE>Chebyshev</span> Polynomials of the 2nd kind.<span style='mso-spacerun:yes'>  </span>If time permits, I will <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>show</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> how to characterize Orthogonal Polynomial Sequences that satisfy:<span style='mso-spacerun:yes'>  </span></span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>D</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>q</m:r></span></i></m:sub></m:sSub><m:d><m:dPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>P</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>n</m:r></span></i></m:sub></m:sSub></m:e></m:d><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>=</m:r><m:r>³</m:r></span></i><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>P</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>n</m:r><m:r>-1</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:4.5pt;mso-text-raise:-4.5pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:80.25pt;height:15.75pt'> <v:imagedata src="seminar2012_files/image006.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=107 height=21 src="seminar2012_files/image007.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>.<o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, January 24<sup>th</sup>, 3:15pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>169 Nick</span><span style='font-size:16.0pt; color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Hope <span class=SpellE>Gerson</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2; mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Brigham Young University</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span>The Role of Authority in the Negotiation of <span class=SpellE>Sociomathematical</span> Norms.</p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>The study of <span class=SpellE>sociomathematical</span> norms initiated by <span class=SpellE>Yackel</span> and Cobb (1996) has become a popular way to make sense of the complexity <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>of</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> mathematical activity in the classroom.<span style='mso-spacerun:yes'>  </span><span class=SpellE>Levenson</span>, <span class=SpellE>Tirosh</span>, and <span class=SpellE>Tsamir</span> (2009) found that teachers and students did not share the same <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>interpretations</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> of teacher-initiated <span class=SpellE>sociomathematical</span> norms.<span style='mso-spacerun:yes'>  </span>In this talk I will share findings from an ongoing study of the role authority plays in <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>the</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> negotiation and legitimization of student-initiated <span class=SpellE>sociomathematical</span> norms.<span style='mso-spacerun:yes'>  </span>In an inquiry-based university honors calculus II course, we found <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>that</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> mathematical authority legitimized through mathematical argument and justification played a major role in the negotiation of <span class=SpellE>sociomathematical</span> <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>norms</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>.<span style='mso-spacerun:yes'>  </span>We suggest that creating an environment where mathematical authority is used to negotiate and legitimize <span class=SpellE>sociomathematical</span> norms will lead <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>to</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> better agreement on the expectation.<o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Thursday, January 19<sup>th</sup>, 2:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>123 OBS</span><span style='font-size:16.0pt; color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Courtney Nagle </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span>Prerequisite Notions for an Introductory Conception of a Functional Limit.</p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>The limit concept is the foundation of calculus.<span style='mso-spacerun:yes'>  </span>Unfortunately, studies have shown that many students lack a strong conceptual <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>understanding</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> of limits and instead have several misconceptions. While past research has examined students development of limit ideas at the <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=SpellE><span class=GramE><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>premathematics</span></span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>, introductory calculus, and formal calculus stages, they have not examined <span class=SpellE>precalculus</span> students understandings.<span style='mso-spacerun:yes'>  </span>In this study, <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>150 <span class=SpellE>precalculus</span> students were engaged in a series of limit tasks.<span style='mso-spacerun:yes'>  </span>Their responses were qualitatively analyzed via grounded theory to identify the <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>underlying</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> knowledge concepts and then statistically analyzed via correlations, factor analysis, simplex models, <span class=SpellE>Guttman</span> <span class=SpellE>scalograms</span> and <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>regressions</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> to examine their relationships.<span style='mso-spacerun:yes'>  </span>Results suggested that several concepts formed a developmental progression of students understanding <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>of</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> limits. The results also showed that students who used the ideas of  closeness and  approaching completed more tasks successfully. These <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>results</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> suggest that pre-calculus emphasis on  closeness and  approaching can aid students understanding of limits.<o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, January 17<sup>th</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>120 OBS</span><span style='font-size:16.0pt; color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Jennifer <span class=SpellE>Cribbs</span><span style='mso-spacerun:yes'>  </span></span><span style='font-size:20.0pt; color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Clemson University</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span>Examining the Impact of Mathematics Identity on the Choice of Engineering Careers for Male and Female Students.</p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>Previous research suggests strong connections between mathematics background and persistence in engineering career aspirations.&nbsp; This <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>study</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> expands on this research by examining how mathematics identity impacts the choice of engineering careers for male and female students.&nbsp; <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>The data used in this study were drawn from the Factors Influencing College Success in Mathematics (FICS-Math) project. Results indicate that <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>factors</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> of mathematics interest, mathematics competence and performance, and mathematics recognition are significant predictors of the choice of <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>an</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> engineering career, even after controlling for SAT/ACT math scores and demographics such as parental education.&nbsp; Moreover, the interaction <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>between</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> mathematics recognition and gender is also significant with the estimate indicating that females who are recognized in mathematics are <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>more</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> likely to choose engineering careers.<o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Thursday, December 8<sup>th</sup>, 1:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>169 Nick</span><span style='font-size:16.0pt; color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Amos <span class=SpellE>Ong</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span>Polynomials <!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>T</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>n</m:r></span></i></m:sub></m:sSub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>(</m:r><m:r>x</m:r><m:r>)</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:28.5pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image008.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=38 height=19 src="seminar2012_files/image009.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span><span class=GramE>from </span><!--[if gte msEquation 12]><m:oMath><m:func><m:funcPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:funcPr><m:fName><span style='font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>cos</m:r></span></m:fName><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>n¸</m:r></span></i></m:e></m:func></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:33pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image010.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=44 height=19 src="seminar2012_files/image011.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]>, Part 2.</p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>In this talk, I will complete the proof of the three term recurrence relation <span class=GramE>of </span></span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>T</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>n</m:r></span></i></m:sub></m:sSub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>(</m:r><m:r>x</m:r><m:r>)</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:28.5pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image008.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=38 height=19 src="seminar2012_files/image009.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>.<span style='mso-spacerun:yes'>  </span></span><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>I will also talk about the <span class=SpellE>Chebyshev</span> Polynomials of <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>the</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> second kind, </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>U</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>n</m:r></span></i></m:sub></m:sSub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>(</m:r><m:r>x</m:r><m:r>)</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:30.75pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image012.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=41 height=19 src="seminar2012_files/image013.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>.<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>With these two kinds of polynomials, I will investigate other neat relations that can be proved using elementary<o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>trigonometric</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> identities and simple differentiation and integration.<o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Thursday, December 1<sup>st</sup>, 1:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>169 Nick</span><span style='font-size:16.0pt; color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Amos <span class=SpellE>Ong</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span>Polynomials <!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>T</m:r></span></i></m:e><m:sub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>n</m:r></span></i></m:sub></m:sSub><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>(</m:r><m:r>x</m:r><m:r>)</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:28.5pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image008.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=38 height=19 src="seminar2012_files/image009.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>from <!--[if gte msEquation 12]><m:oMath><m:func><m:funcPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:funcPr><m:fName><span style='font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>cos</m:r></span></m:fName><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>n¸</m:r><m:r>.</m:r></span></i></m:e></m:func></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:35.25pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image014.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=47 height=19 src="seminar2012_files/image015.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]></p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'>In this talk, we will discuss many neat elementary aspects of the <span class=SpellE>Chebyshev</span> polynomials starting from expressing </span><!--[if gte msEquation 12]><m:oMath><m:func><m:funcPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:funcPr><m:fName><span style='mso-bidi-font-size:10.0pt;font-family:"Cambria Math","serif"; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>cos</m:r></span></m:fName><m:e><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>n¸</m:r></span></i></m:e></m:func></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:33pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image010.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=44 height=19 src="seminar2012_files/image011.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><span style='mso-spacerun:yes'> </span>in terms of </span><!--[if gte msEquation 12]><m:oMath><m:func><m:funcPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:funcPr><m:fName><span style='mso-bidi-font-size:10.0pt;font-family:"Cambria Math","serif"; mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Courier New"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>cos</m:r></span></m:fName><m:e><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>¸</m:r><m:r>.</m:r></span></i></m:e></m:func></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:28.5pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image016.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=38 height=19 src="seminar2012_files/image017.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>They form a set of orthogonal polynomials with <span class=GramE>weights </span></span><!--[if gte msEquation 12]><m:oMath><m:f><m:fPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>1</m:r></span></i></m:num><m:den><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-family:"Cambria Math","serif";mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";mso-bidi-font-family: "Courier New";font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>1-</m:r></span></i><m:sSup><m:sSupPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Courier New"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><m:r>2</m:r></span></i></m:sup></m:sSup></m:e></m:rad></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:7.0pt;mso-text-raise:-7.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:26.25pt;height:20.25pt'> <v:imagedata src="seminar2012_files/image018.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=35 height=27 src="seminar2012_files/image019.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>.<span style='mso-spacerun:yes'>  </span>We will also derive the three term recurrence relation, the Differential Equation, and <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-family:"Courier New"'>see</span></span><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'> how we can/should define the <span class=SpellE>Chebyshev</span> polynomials of the second kind from there.<span style='mso-spacerun:yes'>  </span>This talk is suitable for undergraduates with only knowledge <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'>of simple trigonometric identities and a little bit of calculus.<o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-bidi-font-size:10.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, November 15<sup>th</sup>, 1:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 22 Witkowski</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Mr. Paul Olson </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span>What if<span style='mso-spacerun:yes'>    </span><!--[if gte msEquation 12]><m:oMath><m:f><m:fPr><span style='font-family: "Cambria Math","serif";mso-ascii-font-family:"Cambria Math";mso-hansi-font-family: "Cambria Math";font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>A</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>1 -</m:r><m:r>R</m:r><m:r> </m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r><span style='mso-spacerun:yes'>  </span>=<span style='mso-spacerun:yes'>    </span></m:r><m:r>A</m:r><m:r>+</m:r><m:r>A</m:r><m:r> </m:r><m:r>R</m:r><m:r> +</m:r><m:r>A</m:r><m:r> </m:r></span></i><m:sSup><m:sSupPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>R</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r> +</m:r><m:r>A</m:r><m:r><span style='mso-spacerun:yes'>  </span></m:r></span></i><m:sSup><m:sSupPr><span style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>R</m:r><m:r> </m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>3</m:r></span></i></m:sup></m:sSup></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:7.0pt;mso-text-raise:-7.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:179.25pt;height:20.25pt'> <v:imagedata src="seminar2012_files/image021.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=239 height=27 src="seminar2012_files/image022.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'>  </span>+<span style='mso-spacerun:yes'>   </span>. . .<span style='mso-spacerun:yes'>       </span>?<span style='mso-spacerun:yes'>  </span></p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span>The idea of infinite series can be found in a number of our mathematics courses, including calculus, differential equations, real analysis and </p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>complex</span> analysis.<span style='mso-spacerun:yes'>  </span><i style='mso-bidi-font-style:normal'>Geometric series </i>provide convenient examples for both <i style='mso-bidi-font-style:normal'>convergent series </i>and <i style='mso-bidi-font-style:normal'>divergent series.</i><span style='mso-spacerun:yes'>  </span>By focusing on (<i style='mso-bidi-font-style: normal'>convergent) <o:p></o:p></i></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><i style='mso-bidi-font-style:normal'>geometric</i></span><i style='mso-bidi-font-style:normal'> series </i>and <i style='mso-bidi-font-style: normal'>partial fraction decomposition</i>, we have a technique that allows us to generate <span class=SpellE>Maclaurin</span> series (and Taylor series) for certain </p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>rational</span> functions <i style='mso-bidi-font-style:normal'>without </i>having to<i style='mso-bidi-font-style:normal'> </i>use the definition of the coefficients for these series.<span style='font-family:"Times","serif"'><o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-family:"Times","serif"'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, November 8<sup>th</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 117 OBS</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Antonio <span class=SpellE>Mastroberardino</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2; mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; mso-layout-grid-align:none;text-autospace:none'>Title:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>Mixed Convection Boundary Layer Flow of a Non-Newtonian Fluid.</span></p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='font-family:"Times","serif"'>The study of the fluid mechanics due to a stretching surface in a fluid at rest is an important aspect of various industrial applications.<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-family:"Times","serif"'>For instance, in polymer extrusion processes, the extruded material passes between two solid blocks into a region containing fluid at rest that cools <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='font-family:"Times","serif"'>the</span></span><span style='font-family:"Times","serif"'> sheet and alters its mechanical properties.<span style='mso-spacerun:yes'>  </span>In this talk, I will present an analysis of a mathematical model describing the aforementioned process <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='font-family:"Times","serif"'>in</span></span><span style='font-family:"Times","serif"'> which the ambient fluid is non-Newtonian and the effects of mixed convection are taken into consideration.<span style='mso-spacerun:yes'>     </span><o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, October 25<sup>th</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 120 OBS</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. <span class=SpellE>Papiya</span> <span class=SpellE>Bhattacharjee</span></span><span style='font-size:20.0pt; color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Title: p-Extensions of Commutative Rings.<span style='color:black'><o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span>If <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>R</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:8.25pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image023.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=11 height=19 src="seminar2012_files/image024.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>and <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:6.75pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image025.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=9 height=19 src="seminar2012_files/image026.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>are two commutative rings with identity, and <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>f</m:r><m:r>:</m:r><m:r>R</m:r><m:r>’!</m:r><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:43.5pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image027.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=58 height=19 src="seminar2012_files/image028.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>is an injective ring homomorphism, then we can consider <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>R</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:8.25pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image023.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=11 height=19 src="seminar2012_files/image024.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>as a </p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>subring</span> of <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:6.75pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image025.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=9 height=19 src="seminar2012_files/image026.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]>, and we say that <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>R</m:r><m:r>’!</m:r><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:31.5pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image029.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=42 height=19 src="seminar2012_files/image030.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>is a ring extension. An extension of rings <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>R</m:r><m:r>’!</m:r><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:31.5pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image029.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=42 height=19 src="seminar2012_files/image030.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>is a p-extension if given <span class=GramE>any </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>s</m:r><m:r>"</m:r><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:26.25pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image031.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=35 height=19 src="seminar2012_files/image032.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]>, there exists an <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math","serif"'><m:r>r</m:r><m:r>"</m:r><m:r>R</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:27.75pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image033.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=37 height=19 src="seminar2012_files/image034.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>such</span> that <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>sS</m:r><m:r>=</m:r><m:r>rS</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:39.75pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image035.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=53 height=19 src="seminar2012_files/image036.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]>; that is, the principal ideals in <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>S</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:6.75pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image025.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=9 height=19 src="seminar2012_files/image026.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='mso-spacerun:yes'> </span>are actually generated by<span style='mso-spacerun:yes'>  </span>elements in <!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>R</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:minor-fareast;position:relative; top:3.0pt;mso-text-raise:-3.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:8.25pt;height:14.25pt'> <v:imagedata src="seminar2012_files/image023.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=11 height=19 src="seminar2012_files/image024.gif" v:shapes="_x0000_i1025"><![endif]></span><![endif]>.</p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>In this talk, Dr. <span class=SpellE>Bhattacharjee</span> will discuss p-extensions of general commutative rings and provide various examples. Furthermore, she will focus </p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE>on</span> a special ring, the ring of real-valued continuous functions, and discuss p-extensions in this context.<span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-fareast-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, October 18<sup>th</sup>, 2:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 120 OBS</span><span style='font-size:16.0pt;color:#002060'> <o:p></o:p></span></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Daniel J. <span class=SpellE>Galiffa</span> </span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2; mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span></span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p class=MsoNormal style='mso-margin-top-alt:auto;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Title:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>An Elementary Approach to Characterizing the <span class=SpellE>Sheffer</span> <i>A-Type 0</i> Orthogonal Polynomial Sequences</span>.<span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <pre><span style='font-size:12.0pt;font-family:"Times New Roman","serif"'><o:p>&nbsp;</o:p></span></pre> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Abstract:<span style='mso-spacerun:yes'>  </span><span style='mso-fareast-font-family:"Times New Roman"'>In 1939, I.M. <span class=SpellE>Sheffer</span> published  Some properties of polynomial sets of type zero Duke Math J. 5, p. 590-622, which has been <o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>regarded</span></span><span style='mso-fareast-font-family:"Times New Roman"'> as an indispensable paper in the theory of orthogonal polynomials, in particular characterization theory. Therein, <span class=SpellE>Sheffer</span> basically <o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>proved</span></span><span style='mso-fareast-font-family:"Times New Roman"'> that every polynomial sequence can be classified as belonging to exactly one <i>Type</i>. In addition to various interesting and important <o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>characterizing</span></span><span style='mso-fareast-font-family:"Times New Roman"'> theorems, one of <span class=SpellE>Sheffer's</span> most prominent results pertained to completely characterizing all of the polynomial sequences that <o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>were</span></span><span style='mso-fareast-font-family:"Times New Roman"'> of the most basic type, entitled <i>A-Type 0</i>, and subsequently establishing which of these sets were also orthogonal. However, <span class=SpellE>Sheffer's</span> <o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>elegant</span></span><span style='mso-fareast-font-family:"Times New Roman"'> analysis relied heavily on various functional relationships, formal power series and a generalized degree-lowering operator <i>J</i>. In this <o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-fareast-font-family:"Times New Roman"'>talk, we show how all of the <span class=SpellE>Sheffer</span> <i>A-Type 0</i> orthogonal polynomial sequences can be characterized by using <i>only</i> the generating function <o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span class=GramE><span style='mso-fareast-font-family:"Times New Roman"'>that</span></span><span style='mso-fareast-font-family:"Times New Roman"'> defines this class.<o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Tuesday, September 27<sup>th</sup>, 3:30pm</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><o:p></o:p></span></p> <p style='margin-top:0in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'>Location: 120 OBS</span><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128'><span style='mso-spacerun:yes'>   </span><o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Dr. Joseph P. <span class=SpellE>Previte</span><o:p></o:p></span></p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;color:#002060'>Penn State <span class=SpellE>Behrend</span><o:p></o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Title:<span style='mso-spacerun:yes'>  </span>Growth Rate of Self Similar Trees<br> <br> <span style='color:black'>Abstract:<span style='mso-spacerun:yes'>  </span></span>In this talk, I will explain the growth degree,&nbsp;a measure used to quantify the growth of an infinite graph.<br> Next, I will compute the associated growth degree for trees generated by vertex replacements.<span style='color:black'> <o:p></o:p></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:20.0pt;color:#0F243E;mso-themecolor:text2;mso-themeshade:128; background:aqua;mso-highlight:aqua'><o:p>&nbsp;</o:p></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><b style='mso-bidi-font-weight:normal'><span style='font-size:16.0pt'><a href="seminar2011.html">Mathematics Seminar 2010-2011</a><o:p></o:p></span></b></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> <h2 style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman"'><a href="seminar.html">Mathematics Seminar 2009-2010</a><o:p></o:p></span></h2> <h2 style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></h2> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1026" type="#_x0000_t75" style='width:148.5pt;height:101.25pt; visibility:visible;mso-wrap-style:square'> <v:imagedata src="seminar2012_files/image020.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=198 height=135 src="seminar2012_files/image038.jpg" v:shapes="Picture_x0020_2"><![endif]></span></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> <p style='margin-bottom:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Antonio Mastroberardino</p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Assistant Professor of Mathematics</p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Penn State Erie, The <span class=SpellE>Behrend</span> College<br> School of Science<br> 4205 College Drive <br> Erie, PA 16563-0203</p> <p style='margin:0in;margin-bottom:.0001pt;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><a href="http://www.personal.psu.edu/axm62/">http://www.personal.psu.edu/axm62/</a></p> <p style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><o:p>&nbsp;</o:p></p> </div> </body> </html>