<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">皆様<br class=""><br class="">複素解析幾何セミナー@東大数理のご案内です。<br class=""><br class="">2020年06月29日(月)<br class="">10:30-12:00    Zoom 開催<br class="">日下部佑太 氏 (大阪大学)<br class="">Oka properties of complements of holomorphically convex sets<br class=""><br class=""><div class=""><span style="caret-color: rgb(51, 51, 51); color: rgb(51, 51, 51); font-family: メイリオ, Meiryo, "ヒラギノ角ゴ Pro W3", "Hiragino Kaku Gothic Pro", Osaka, "MS Pゴシック", "MS PGothic", Helvetica, Arial, sans-serif; font-size: 13px; background-color: rgb(239, 239, 239);" class="">[ 講演概要 ]</span><div class="waku" style="margin: 5px 0px 8px; padding: 9px; font-family: メイリオ, Meiryo, "ヒラギノ角ゴ Pro W3", "Hiragino Kaku Gothic Pro", Osaka, "MS Pゴシック", "MS PGothic", Helvetica, Arial, sans-serif; font-size: 13px; font-stretch: inherit; line-height: inherit; vertical-align: baseline; border: 1px solid rgb(204, 204, 204); background-color: rgb(239, 239, 239); caret-color: rgb(51, 51, 51); color: rgb(51, 51, 51);">A complex manifold is called an Oka manifold if the Oka principle for maps from Stein spaces holds. In this talk, we consider the question of when a holomorphically convex set in an Oka manifold has an Oka complement. Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in <span class="MathJax_Preview" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; color: inherit;"></span><span id="MathJax-Element-1-Frame" class="MathJax_CHTML mjx-chtml" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">C</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msup></math>" role="presentation" style="margin: 0px; padding: 1px 0px; font-family: inherit; font-size: 15.989999771118164px; font-variant-caps: inherit; font-stretch: inherit; line-height: 0; vertical-align: baseline; display: inline-block; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;"><span id="MJXc-Node-1" class="mjx-math" aria-hidden="true" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; border-collapse: separate; border-spacing: 0px;"><span id="MJXc-Node-2" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-3" class="mjx-msubsup" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-base" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-4" class="mjx-texatom" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-5" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-6" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-ams-R" style="margin: 0px; padding: 0.441em 0px 0.316em; font-family: MJXc-TeX-ams-R, MJXc-TeX-ams-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">C</span></span></span></span></span><span class="mjx-sup" style="margin: 0px; padding: 0px 0.071em 0px 0px; font-family: inherit; font-size: 11.30492877960205px; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: 0.642em; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-7" class="mjx-texatom" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-8" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-9" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-math-I" style="margin: 0px; padding: 0.191em 0px 0.316em; font-family: MJXc-TeX-math-I, MJXc-TeX-math-Ix, MJXc-TeX-math-Iw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation" style="margin: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-user-select: none; padding: 1px 0px 0px !important; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">C</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msup></math></span></span> <span class="MathJax_Preview" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; color: inherit;"></span><span id="MathJax-Element-2-Frame" class="MathJax_CHTML mjx-chtml" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>&gt;</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="margin: 0px; padding: 1px 0px; font-family: inherit; font-size: 15.989999771118164px; font-variant-caps: inherit; font-stretch: inherit; line-height: 0; vertical-align: baseline; display: inline-block; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;"><span id="MJXc-Node-10" class="mjx-math" aria-hidden="true" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; border-collapse: separate; border-spacing: 0px;"><span id="MJXc-Node-11" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-12" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">(</span></span><span id="MJXc-Node-13" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-math-I" style="margin: 0px; padding: 0.191em 0px 0.316em; font-family: MJXc-TeX-math-I, MJXc-TeX-math-Ix, MJXc-TeX-math-Iw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">n</span></span><span id="MJXc-Node-14" class="MJXc-space3 mjx-mo" style="margin: 0px 0px 0px 0.278em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.253em 0px 0.378em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">></span></span><span id="MJXc-Node-15" class="MJXc-space3 mjx-mn" style="margin: 0px 0px 0px 0.278em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.378em 0px 0.316em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">1</span></span><span id="MJXc-Node-16" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation" style="margin: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-user-select: none; padding: 1px 0px 0px !important; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>></mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span> is Oka. The relative version of the main theorem can also be proved. As an application, we show that the complement <span class="MathJax_Preview" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; color: inherit;"></span><span id="MathJax-Element-3-Frame" class="MathJax_CHTML mjx-chtml" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">C</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msup><mo class="MJX-variant">&#x2216;</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msup></math>" role="presentation" style="margin: 0px; padding: 1px 0px; font-family: inherit; font-size: 15.989999771118164px; font-variant-caps: inherit; font-stretch: inherit; line-height: 0; vertical-align: baseline; display: inline-block; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;"><span id="MJXc-Node-17" class="mjx-math" aria-hidden="true" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; border-collapse: separate; border-spacing: 0px;"><span id="MJXc-Node-18" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-19" class="mjx-msubsup" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-base" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-20" class="mjx-texatom" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-21" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-22" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-ams-R" style="margin: 0px; padding: 0.441em 0px 0.316em; font-family: MJXc-TeX-ams-R, MJXc-TeX-ams-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">C</span></span></span></span></span><span class="mjx-sup" style="margin: 0px; padding: 0px 0.071em 0px 0px; font-family: inherit; font-size: 11.30492877960205px; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: 0.642em; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-23" class="mjx-texatom" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-24" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-25" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-math-I" style="margin: 0px; padding: 0.191em 0px 0.316em; font-family: MJXc-TeX-math-I, MJXc-TeX-math-Ix, MJXc-TeX-math-Iw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">n</span></span></span></span></span></span><span id="MJXc-Node-26" class="MJXc-space2 mjx-mo" style="margin: 0px 0px 0px 0.222em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">∖</span></span><span id="MJXc-Node-27" class="MJXc-space2 mjx-msubsup" style="margin: 0px 0px 0px 0.222em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-base" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-28" class="mjx-texatom" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-29" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-30" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-ams-R" style="margin: 0px; padding: 0.441em 0px 0.316em; font-family: MJXc-TeX-ams-R, MJXc-TeX-ams-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">R</span></span></span></span></span><span class="mjx-sup" style="margin: 0px; padding: 0px 0.071em 0px 0px; font-family: inherit; font-size: 11.30492877960205px; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: 0.615em; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-31" class="mjx-texatom" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-32" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-33" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-math-I" style="margin: 0px; padding: 0.441em 0px 0.316em; font-family: MJXc-TeX-math-I, MJXc-TeX-math-Ix, MJXc-TeX-math-Iw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">k</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation" style="margin: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-user-select: none; padding: 1px 0px 0px !important; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">C</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msup><mo class="MJX-variant">∖</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msup></math></span></span> of a totally real affine subspace is Oka if <span class="MathJax_Preview" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; color: inherit;"></span><span id="MathJax-Element-4-Frame" class="MathJax_CHTML mjx-chtml" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>" role="presentation" style="margin: 0px; padding: 1px 0px; font-family: inherit; font-size: 15.989999771118164px; font-variant-caps: inherit; font-stretch: inherit; line-height: 0; vertical-align: baseline; display: inline-block; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;"><span id="MJXc-Node-34" class="mjx-math" aria-hidden="true" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; border-collapse: separate; border-spacing: 0px;"><span id="MJXc-Node-35" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-36" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-math-I" style="margin: 0px; padding: 0.191em 0px 0.316em; font-family: MJXc-TeX-math-I, MJXc-TeX-math-Ix, MJXc-TeX-math-Iw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">n</span></span><span id="MJXc-Node-37" class="MJXc-space3 mjx-mo" style="margin: 0px 0px 0px 0.278em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.253em 0px 0.378em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">></span></span><span id="MJXc-Node-38" class="MJXc-space3 mjx-mn" style="margin: 0px 0px 0px 0.278em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.378em 0px 0.316em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation" style="margin: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-user-select: none; padding: 1px 0px 0px !important; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math></span></span> and <span class="MathJax_Preview" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; color: inherit;"></span><span id="MathJax-Element-5-Frame" class="MathJax_CHTML mjx-chtml" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>&#x2260;</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></math>" role="presentation" style="margin: 0px; padding: 1px 0px; font-family: inherit; font-size: 15.989999771118164px; font-variant-caps: inherit; font-stretch: inherit; line-height: 0; vertical-align: baseline; display: inline-block; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;"><span id="MJXc-Node-39" class="mjx-math" aria-hidden="true" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; border-collapse: separate; border-spacing: 0px;"><span id="MJXc-Node-40" class="mjx-mrow" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span id="MJXc-Node-41" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">(</span></span><span id="MJXc-Node-42" class="mjx-mi" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-math-I" style="margin: 0px; padding: 0.191em 0px 0.316em; font-family: MJXc-TeX-math-I, MJXc-TeX-math-Ix, MJXc-TeX-math-Iw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">n</span></span><span id="MJXc-Node-43" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: -0.184em 0px 0px; padding: 0px 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">,</span></span><span id="MJXc-Node-44" class="MJXc-space1 mjx-mi" style="margin: 0px 0px 0px 0.167em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-math-I" style="margin: 0px; padding: 0.441em 0px 0.316em; font-family: MJXc-TeX-math-I, MJXc-TeX-math-Ix, MJXc-TeX-math-Iw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">k</span></span><span id="MJXc-Node-45" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">)</span></span><span id="MJXc-Node-46" class="MJXc-space3 mjx-mo" style="margin: 0px 0px 0px 0.278em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">≠</span></span><span id="MJXc-Node-47" class="MJXc-space3 mjx-mo" style="margin: 0px 0px 0px 0.278em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">(</span></span><span id="MJXc-Node-48" class="mjx-mn" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.378em 0px 0.316em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">2</span></span><span id="MJXc-Node-49" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: -0.184em 0px 0px; padding: 0px 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">,</span></span><span id="MJXc-Node-50" class="MJXc-space1 mjx-mn" style="margin: 0px 0px 0px 0.167em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.378em 0px 0.316em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">1</span></span><span id="MJXc-Node-51" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">)</span></span><span id="MJXc-Node-52" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: -0.184em 0px 0px; padding: 0px 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">,</span></span><span id="MJXc-Node-53" class="MJXc-space1 mjx-mo" style="margin: 0px 0px 0px 0.167em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.441em 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">(</span></span><span id="MJXc-Node-54" class="mjx-mn" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.378em 0px 0.316em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">2</span></span><span id="MJXc-Node-55" class="mjx-mo" style="margin: 0px; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: -0.184em 0px 0px; padding: 0px 0px 0.566em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: block; white-space: pre; box-sizing: content-box !important;">,</span></span><span id="MJXc-Node-56" class="MJXc-space1 mjx-mn" style="margin: 0px 0px 0px 0.167em; padding: 0px; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; vertical-align: baseline; display: inline-block; box-sizing: content-box !important;"><span class="mjx-char MJXc-TeX-main-R" style="margin: 0px; padding: 0.378em 0px 0.316em; font-family: MJXc-TeX-main-R, MJXc-TeX-main-Rw; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: inherit; verti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