The main purpose of this paper is to establish the theory of the multi-parameter Hardy spaces H p H^p ( 0 > p ≤ 1 0>pleq 1 ) associated to a class of multi-parameter singular integrals extensively studied in the recent book of B. Street (2014), where the L p L^p ( 1 > p > ∞ ) (1>p>infty ) estimates are proved for this class of singular integrals. This class of multi-parameter singular integrals are intrinsic to the underlying multi-parameter Carnot-Carathéodory geometry, where the quantitative Frobenius theorem was established by B. Street (2011), and are closely related to both the one-parameter and multi-parameter settings of singular Radon transforms considered by Stein and Street (2011, 2012a, 2012b, 2013). More precisely, Street (2014) studied the L p L^p ( 1 > p > ∞ ) (1>p>infty ) boundedness, using elementary operators, of a type of generalized multi-parameter Calderón Zygmund operators on smooth and compact manifolds, which include a certain type of singular Radon transforms. In this work, we are interested in the endpoint estimates for the singular integral operators in both one and multi-parameter settings considered by Street (2014). Actually, using the discrete Littlewood-Paley-Stein analysis, we will introduce the Hardy space H p H^p ( 0 > p ≤ 1 0>pleq 1 ) associated with the multi-parameter structures arising from the multi-parameter Carnot-Carathéodory metrics using the appropriate discrete Littlewood-Paley-Stein square functions, and then establish the Hardy space boundedness of singular integrals in both the single and multi-parameter settings. Our approach is much inspired by the work of Street (2014) where he introduced the notions of elementary operators so that the type of singular integrals under consideration can be decomposed into elementary operators.
{"title":"Multi-Parameter Hardy Spaces Theory and Endpoint Estimates for Multi-Parameter Singular Integrals","authors":"G. Lu, Jiawei Shen, Lu Zhang","doi":"10.1090/memo/1388","DOIUrl":"https://doi.org/10.1090/memo/1388","url":null,"abstract":"The main purpose of this paper is to establish the theory of the multi-parameter Hardy spaces \u0000\u0000 \u0000 \u0000 H\u0000 p\u0000 \u0000 H^p\u0000 \u0000\u0000 (\u0000\u0000 \u0000 \u0000 0\u0000 >\u0000 p\u0000 ≤\u0000 1\u0000 \u0000 0>pleq 1\u0000 \u0000\u0000) associated to a class of multi-parameter singular integrals extensively studied in the recent book of B. Street (2014), where the \u0000\u0000 \u0000 \u0000 L\u0000 p\u0000 \u0000 L^p\u0000 \u0000\u0000 \u0000\u0000 \u0000 \u0000 (\u0000 1\u0000 >\u0000 p\u0000 >\u0000 ∞\u0000 )\u0000 \u0000 (1>p>infty )\u0000 \u0000\u0000 estimates are proved for this class of singular integrals. This class of multi-parameter singular integrals are intrinsic to the underlying multi-parameter Carnot-Carathéodory geometry, where the quantitative Frobenius theorem was established by B. Street (2011), and are closely related to both the one-parameter and multi-parameter settings of singular Radon transforms considered by Stein and Street (2011, 2012a, 2012b, 2013).\u0000\u0000More precisely, Street (2014) studied the \u0000\u0000 \u0000 \u0000 L\u0000 p\u0000 \u0000 L^p\u0000 \u0000\u0000 \u0000\u0000 \u0000 \u0000 (\u0000 1\u0000 >\u0000 p\u0000 >\u0000 ∞\u0000 )\u0000 \u0000 (1>p>infty )\u0000 \u0000\u0000 boundedness, using elementary operators, of a type of generalized multi-parameter Calderón Zygmund operators on smooth and compact manifolds, which include a certain type of singular Radon transforms. In this work, we are interested in the endpoint estimates for the singular integral operators in both one and multi-parameter settings considered by Street (2014). Actually, using the discrete Littlewood-Paley-Stein analysis, we will introduce the Hardy space \u0000\u0000 \u0000 \u0000 H\u0000 p\u0000 \u0000 H^p\u0000 \u0000\u0000 (\u0000\u0000 \u0000 \u0000 0\u0000 >\u0000 p\u0000 ≤\u0000 1\u0000 \u0000 0>pleq 1\u0000 \u0000\u0000) associated with the multi-parameter structures arising from the multi-parameter Carnot-Carathéodory metrics using the appropriate discrete Littlewood-Paley-Stein square functions, and then establish the Hardy space boundedness of singular integrals in both the single and multi-parameter settings. Our approach is much inspired by the work of Street (2014) where he introduced the notions of elementary operators so that the type of singular integrals under consideration can be decomposed into elementary operators.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45728673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peter M. Luthy, H. Šikić, F. Soria, G. Weiss, E. Wilson
The theory of wavelets has been thoroughly studied by many authors; standard references include books by I. Daubechies, by Y. Meyer, by R. Coifman and Y. Meyer, by C.K. Chui, and by M.V. Wickerhauser. In addition, the development of wavelets influenced the study of various other reproducing function systems. Interestingly enough, some open questions remained unsolved or only partially solved for more than twenty years even in the most basic case of dyadic orthonormal wavelets in a single dimension. These include issues related to the MRA structure (for example, a complete understanding of filters), the structure of the space of negative dilates (in particular, with respect to what is known as the Baggett problem), and the variety of resolution structures that may occur. In this article we offer a comprehensive, yet technically fairly elementary approach to these questions. On this path, we present a multitude of new results, resolve some of the old questions, and provide new advances for some problems that remain open for the future. In this study, we have been guided mostly by the philosophy presented some twenty years ago in a book by E. Hernandez and G. Weiss (one of us), in which the orthonormal wavelets are characterized by four basic equations, so that the interplay between wavelets and Fourier analysis provides a deeper insight into both fields of research. This book has influenced hundreds of researchers, and their effort has produced a variety of new techniques, many of them reaching far beyond the study of one-dimensional orthonormal wavelets. Here we are trying to close the circle in some sense by applying these new techniques to the original subject of one-dimensional wavelets. We are primarily interested in the quality of new results and their clear presentations; for this reason, we keep our study on the level of a single dimension, although we are aware that many of our results can be extended beyond that case. Given ψ psi , a square integrable function on the real line, we want to address the following question: What sort of structures can one obtain from the affine wavelet family { 2 j / 2 ψ ( 2 j x − k ) : j , k ∈ Z } {2^{j/2} psi ( 2^{j}x - k ) : j,kin mathbb Z} associated with ψ psi ? It may be too difficult to directly attack this problem via the function ψ psi . We argue in this article that the appropriate object to study is the principal shift invariant space generated by ψ psi (these spaces were introduced by H.Helson decades ago and applied very successfully in the approximation theory by C. de Boor, R.A. DeVore, and A. Ron, with more recent applications to wavelets introduced by A. Ron and Z. Shen). With this goal in mind, in Chapter 1, we present a very detailed study of principal shift invariant spaces and their generating families. These include the relationships between principal shift invariant spaces, various basis-like
{"title":"One-Dimensional Dyadic Wavelets","authors":"Peter M. Luthy, H. Šikić, F. Soria, G. Weiss, E. Wilson","doi":"10.1090/memo/1378","DOIUrl":"https://doi.org/10.1090/memo/1378","url":null,"abstract":"The theory of wavelets has been thoroughly studied by many authors; standard references include books by I. Daubechies, by Y. Meyer, by R. Coifman and Y. Meyer, by C.K. Chui, and by M.V. Wickerhauser. In addition, the development of wavelets influenced the study of various other reproducing function systems. Interestingly enough, some open questions remained unsolved or only partially solved for more than twenty years even in the most basic case of dyadic orthonormal wavelets in a single dimension. These include issues related to the MRA structure (for example, a complete understanding of filters), the structure of the space of negative dilates (in particular, with respect to what is known as the Baggett problem), and the variety of resolution structures that may occur. In this article we offer a comprehensive, yet technically fairly elementary approach to these questions. On this path, we present a multitude of new results, resolve some of the old questions, and provide new advances for some problems that remain open for the future.\u0000\u0000In this study, we have been guided mostly by the philosophy presented some twenty years ago in a book by E. Hernandez and G. Weiss (one of us), in which the orthonormal wavelets are characterized by four basic equations, so that the interplay between wavelets and Fourier analysis provides a deeper insight into both fields of research. This book has influenced hundreds of researchers, and their effort has produced a variety of new techniques, many of them reaching far beyond the study of one-dimensional orthonormal wavelets. Here we are trying to close the circle in some sense by applying these new techniques to the original subject of one-dimensional wavelets. We are primarily interested in the quality of new results and their clear presentations; for this reason, we keep our study on the level of a single dimension, although we are aware that many of our results can be extended beyond that case.\u0000\u0000Given \u0000\u0000 \u0000 ψ\u0000 psi\u0000 \u0000\u0000, a square integrable function on the real line, we want to address the following question: What sort of structures can one obtain from the affine wavelet family \u0000\u0000 \u0000 \u0000 {\u0000 \u0000 2\u0000 \u0000 j\u0000 \u0000 /\u0000 \u0000 2\u0000 \u0000 \u0000 ψ\u0000 (\u0000 \u0000 2\u0000 \u0000 j\u0000 \u0000 \u0000 x\u0000 −\u0000 k\u0000 )\u0000 :\u0000 j\u0000 ,\u0000 k\u0000 ∈\u0000 \u0000 Z\u0000 \u0000 }\u0000 \u0000 {2^{j/2} psi ( 2^{j}x - k ) : j,kin mathbb Z}\u0000 \u0000\u0000 associated with \u0000\u0000 \u0000 ψ\u0000 psi\u0000 \u0000\u0000? It may be too difficult to directly attack this problem via the function \u0000\u0000 \u0000 ψ\u0000 psi\u0000 \u0000\u0000. We argue in this article that the appropriate object to study is the principal shift invariant space generated by \u0000\u0000 \u0000 ψ\u0000 psi\u0000 \u0000\u0000 (these spaces were introduced by H.Helson decades ago and applied very successfully in the approximation theory by C. de Boor, R.A. DeVore, and A. Ron, with more recent applications to wavelets introduced by A. Ron and Z. Shen). With this goal in mind, in Chapter 1, we present a very detailed study of principal shift invariant spaces and their generating families. These include the relationships between principal shift invariant spaces, various basis-like","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47625249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. These results are part of a conjectural decomposition of the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program, analogous to the description of Bondal-Orlov (Derived categories of coherent sheaves, 2002) and Kawamata (Derived categories of toric varieties, 2006) of the bounded derived category of coherent sheaves on a compact complex manifold.
{"title":"Floer cohomology and flips","authors":"François Charest, C. Woodward","doi":"10.1090/memo/1372","DOIUrl":"https://doi.org/10.1090/memo/1372","url":null,"abstract":"We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. These results are part of a conjectural decomposition of the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program, analogous to the description of Bondal-Orlov (Derived categories of coherent sheaves, 2002) and Kawamata (Derived categories of toric varieties, 2006) of the bounded derived category of coherent sheaves on a compact complex manifold.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44552873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re. In this work, we show that there is constant 0>c0≪10 > c_0 ll 1, independent of Remathbf {Re}, such that sufficiently regular disturbances of size ϵ≲Re−2/3−δepsilon lesssim mathbf {Re}^{-2/3-delta } for any δ>0delta > 0 exist at least until t=c0
{"title":"Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case","authors":"J. Bedrossian, P. Germain, N. Masmoudi","doi":"10.1090/memo/1377","DOIUrl":"https://doi.org/10.1090/memo/1377","url":null,"abstract":"<p>This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number <bold>Re</bold>. In this work, we show that there is constant <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 greater-than c 0 much-less-than 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>≪<!-- ≪ --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">0 > c_0 ll 1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, independent of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper R bold e\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">R</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">e</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {Re}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, such that sufficiently regular disturbances of size <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon less-than-or-equivalent-to bold upper R bold e Superscript negative 2 slash 3 minus delta\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ϵ<!-- ϵ --></mml:mi>\u0000 <mml:mo>≲<!-- ≲ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">R</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">e</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">epsilon lesssim mathbf {Re}^{-2/3-delta }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for any <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">delta > 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> exist at least until <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t equals c 0 epsilon Superscript negative 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mn>0</mml:m","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47145456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we give explicit formulas of archimedean Whittaker functions on GL(3)GL(3) and GL(2)GL(2). Moreover, we apply those to the calculation of archimedean zeta integrals for GL(3)×GL(2)GL(3)times GL(2), and show that the zeta integral for appropriate Whittaker functions is equal to the associated LL-factors.
{"title":"Archimedean zeta integrals for 𝐺𝐿(3)×𝐺𝐿(2)","authors":"Miki Hirano, Taku Ishii, Tadashi Miyazaki","doi":"10.1090/memo/1366","DOIUrl":"https://doi.org/10.1090/memo/1366","url":null,"abstract":"<p>In this article, we give explicit formulas of archimedean Whittaker functions on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L left-parenthesis 3 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GL(3)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L left-parenthesis 2 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GL(2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Moreover, we apply those to the calculation of archimedean zeta integrals for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L left-parenthesis 3 right-parenthesis times upper G upper L left-parenthesis 2 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GL(3)times GL(2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and show that the zeta integral for appropriate Whittaker functions is equal to the associated <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-factors.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44582696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by symmetric monoidal 2-functors which are defined over non-rigid 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes, and which take values in 2-categories of complete graded linear categories. In particular, our construction extends the family of graded 2+1-TQFTs defined for the unrolled version of quantum s l 2 mathfrak {sl}_2 by Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The non-semisimplicity of the theory is witnessed by the presence of non-semisimple graded linear categories associated with critical 1-manifolds.
{"title":"Non-semisimple extended topological quantum field theories","authors":"Marco De Renzi","doi":"10.1090/memo/1364","DOIUrl":"https://doi.org/10.1090/memo/1364","url":null,"abstract":"We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by symmetric monoidal 2-functors which are defined over non-rigid 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes, and which take values in 2-categories of complete graded linear categories. In particular, our construction extends the family of graded 2+1-TQFTs defined for the unrolled version of quantum \u0000\u0000 \u0000 \u0000 \u0000 s\u0000 l\u0000 \u0000 2\u0000 \u0000 mathfrak {sl}_2\u0000 \u0000\u0000 by Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The non-semisimplicity of the theory is witnessed by the presence of non-semisimple graded linear categories associated with critical 1-manifolds.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60559655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study embeddings of PSL2(pa)mathrm {PSL}_2(p^a) into exceptional groups G(pb)G(p^b) for G=F4,E6,2E6,E7G=F_4,E_6,{}^2!E_6,E_7, and pp a prime with a,
我们研究了在G = f4, e6, 2,e6时，P s2 (p2a) mathm {PSL}_2(P ^a)在例外群G(p2b) G(P ^b)中的嵌入。E 7 g = f_4, e_6，{}²!E_6 E_7和p p a '和a b a b正整数。除了一些可能的例外，我们证明了任何具有集合pssl 2(P a) mathm {PSL}_2(P ^a)的几乎简单群，在Lie型的几乎简单例外群f4f_4, e6e_6, 2e6 {}^2!E_6和E_7 E_7，是代数群内对应的a1a_1型最大闭子群的Frobenius映射下的不动点。结合Burness和Testerman关于p p (Coxeter数+ 1)的最新结果，证明了在这些有限几乎单群中，除三种可能的例外(p a = 7,8，25p ^a=7,8,25对于e7 E_7)。在剩下的三种情况中，我们提供了关于潜在最大子群的大量信息。
{"title":"Maximal 𝑃𝑆𝐿₂ Subgroups of Exceptional Groups of Lie Type","authors":"David A. Craven","doi":"10.1090/memo/1355","DOIUrl":"https://doi.org/10.1090/memo/1355","url":null,"abstract":"<p>We study embeddings of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper L Subscript 2 Baseline left-parenthesis p Superscript a Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">P</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">S</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mi>a</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathrm {PSL}_2(p^a)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> into exceptional groups <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G left-parenthesis p Superscript b Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mi>b</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">G(p^b)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G equals upper F 4 comma upper E 6 comma squared upper E 6 comma upper E 7\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:mn>6</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mspace width=\"negativethinmathspace\" />\u0000 <mml:msub>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:mn>6</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:mn>7</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">G=F_4,E_6,{}^2!E_6,E_7</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> a prime with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a comma b\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>a</mml:mi>\u0000 <mml:mo>,</mm","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41511337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 22” presentation for the group of FF-rational points of an arbitrary exceptional simple group of FF-rank at least 44 and to determine defining relations for the group of FF-rational points of an an arbitrary group of FF-rank 11 and absolute type D4D_4,
我们引入了Tits多边形的概念，推广了Moufang多边形的概念，并证明了Tits多边形是由满足Moufang条件的任意球形建筑物上的抛物子群的某些构型自然产生的。我们建立了Tits多边形的许多基本性质，并用Jordan代数刻画了一大类Tits六边形。我们应用这一分类给出了F F -秩至少为4 4的任意例外简单群F F -有理点群的“秩2”表示，并确定了F F -秩1 1的任意群F F -有理点群与绝对类型D 4 D_4, E 6 E_6，e7e_7或e8e_8与Dynkin图中唯一的不与最高根正交的顶点相关联。所有这些结果都是在一个具有任意特征的场上得到的。
{"title":"Tits polygons","authors":"B. Mühlherr, R. Weiss, Holger P. Petersson","doi":"10.1090/memo/1352","DOIUrl":"https://doi.org/10.1090/memo/1352","url":null,"abstract":"<p>We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>” presentation for the group of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rational points of an arbitrary exceptional simple group of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rank at least <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"4\">\u0000 <mml:semantics>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and to determine defining relations for the group of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rational points of an an arbitrary group of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rank <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\u0000 <mml:semantics>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and absolute type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D 4\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">D_4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xml","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41725629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.
To this end, we turn to degenerate elliptic equations. Let Γ⊂RnGamma subset mathbb {R}^n be an Ahlfors regular set of dimension d>n−1d>n-1 (not necessarily integer) and Ω=Rn∖ΓOmega = mathbb {R}^n setminus Gamma. Let L=−divA∇L = - operatorname {div} Anabla be a degenerate elliptic operator with measurable coeffi
集合的许多几何性质和解析性质取决于椭圆测度的性质，而这在高协维集合中是众所周知的缺失。这个手稿的目的是发展一个版本的椭圆理论，与线性偏微分方程相关，最终产生一个类似于谐波测度的概念，对于余维数高于1的集合。为此，我们转向简并椭圆方程。设Γ∧R n Gammasubsetmathbbr{ ^n是维数d>n−1 d>n-1(不一定是整数)的Ahlfors正则集，并且Ω = R n≠Γ }Omega = mathbbr{ ^n }setminusGamma。∇L = - operatornamediv{ A }nabla是一个简并的椭圆算子，它具有可测量的系数，使得矩阵A A的椭圆常数从上到下由一个倍的dist (Γ) d + 1−n operatornamedist{ (}cdot, Gamma)^ {d+1-n}。我们定义弱解;在适当的加权Sobolev空间中证明迹定理和可拓定理;建立了极大值原理、De Giorgi-Nash-Moser估计、Harnack不等式、解(边界内和边界处)的Hölder连续性。我们定义了Green函数，并提供了Green函数及其梯度的点向和/或lpl ^p估计的基本集合。在此基础上，我们定义了与ll相关的调和测度，建立了它的倍性、非简并性、极变公式，最后给出了局部解的比较原理。在即将出现的另一篇文章中，我们将证明当Γ Gamma是具有小Lipschitz常数的Lipschitz函数的图时，我们可以找到一个椭圆算子L L，对于它，这里给出的调和测度相对于Γ Gamma上的d d -Hausdorff测度是绝对连续的，反之亦然。从而将Dahlberg定理推广到余维数大于1的一些集合。
{"title":"Elliptic Theory for Sets with Higher Co-dimensional Boundaries","authors":"G. David, J. Feneuil, S. Mayboroda","doi":"10.1090/memo/1346","DOIUrl":"https://doi.org/10.1090/memo/1346","url":null,"abstract":"<p>Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.</p>\u0000\u0000<p>To this end, we turn to degenerate elliptic equations. Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma subset-of double-struck upper R Superscript n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Gamma subset mathbb {R}^n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be an Ahlfors regular set of dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than n minus 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">d>n-1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (not necessarily integer) and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega equals double-struck upper R Superscript n Baseline minus normal upper Gamma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo class=\"MJX-variant\">∖<!-- ∖ --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Omega = mathbb {R}^n setminus Gamma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L equals minus d i v upper A nabla\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>div</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">L = - operatorname {div} Anabla</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a degenerate elliptic operator with measurable coeffi","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41447701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.
{"title":"Ergodicity of Markov Processes via Nonstandard Analysis","authors":"Haosui Duanmu, J. Rosenthal, W. Weiss","doi":"10.1090/memo/1342","DOIUrl":"https://doi.org/10.1090/memo/1342","url":null,"abstract":"The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43809879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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