Transactions of the American Mathematical Society, Series B最新文献

{"title":"A group theoretic perspective on entanglements of division fields","authors":"Harris B. Daniels, J. Morrow","doi":"10.1090/btran/95","DOIUrl":"https://doi.org/10.1090/btran/95","url":null,"abstract":"<p>In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and a subgroup <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G subset-of-or-equal-to upper G upper L 2 left-parenthesis double-struck upper Z slash n double-struck upper Z right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo>⊆<!-- ⊆ --></mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Gsubseteq GL_2( mathbb {Z}/nmathbb {Z})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with surjective determinant, we provide a definition for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to represent an <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis a comma b right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>a</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>b</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(a,b)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-entanglement and give additional criteria for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to represent an explained or unexplained <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis a comma b right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>a</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>b</mml:mi>\u0000 <mml:m","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128134151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Separable integer partition classes","authors":"G. Andrews","doi":"10.1090/btran/87","DOIUrl":"https://doi.org/10.1090/btran/87","url":null,"abstract":"A classical method for partition generating function is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 copies of \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 are presented.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123418039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rank varieties and 𝜋-points for elementary supergroup schemes","authors":"D. Benson, S. Iyengar, H. Krause, J. Pevtsova","doi":"10.1090/btran/74","DOIUrl":"https://doi.org/10.1090/btran/74","url":null,"abstract":"<p>We develop a support theory for elementary supergroup schemes, over a field of positive characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than-or-slanted-equals 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>⩾<!-- ⩾ --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">pgeqslant 3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, starting with a definition of a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi\">\u0000 <mml:semantics>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">pi</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi\">\u0000 <mml:semantics>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">pi</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k left-bracket t comma tau right-bracket slash left-parenthesis t Superscript p Baseline minus tau squared right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>τ<!-- τ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>τ<!-- τ --></mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">k[t,tau ]/(t^p-tau ^2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has even degree and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"tau\">\u0000 <mml:semantics>\u0000 <mml:mi>τ<!-- τ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">tau</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has odd degree. The strength of the theory is demonstrated by classifying the pa","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130659597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uniqueness in Cauchy problems for diffusive real-valued strict local martingales","authors":"U. Çetin, Kasper Larsen","doi":"10.1090/btran/141","DOIUrl":"https://doi.org/10.1090/btran/141","url":null,"abstract":"For a real-valued one dimensional diffusive strict local martingale, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution under a local \u0000\u0000 \u0000 \u0000 1\u0000 2\u0000 \u0000 frac 12\u0000 \u0000\u0000-Hölder condition. Under the weaker Engelbert-Schmidt conditions, we provide a set in which the Cauchy problem has a unique weak solution. We exemplify our results using quadratic normal volatility models and the two dimensional Bessel process.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"152 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116095982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dini derivatives and regularity for exchangeable increment processes","authors":"O. A. Hernández, Gerónimo Uribe Bravo","doi":"10.1090/btran/44","DOIUrl":"https://doi.org/10.1090/btran/44","url":null,"abstract":". Let X be an exchangeable increment (EI) process whose sample paths are of infinite variation. We prove that for any fixed t almost surely, limsup h Ñ 0 ˘ p X t ` h ´ X t q{ h “ 8 and limsup h Ñ 0 ˘ p X t ` h ´ X t q{ h “ ´8 . This extends a celebrated result of Rogozin for L´evy processes obtained in 1968 and completes the known picture for finite-variation EI processes. Applications are numerous. For example, we deduce that both half-lines p´8 , 0 q and p 0 , 8q are visited immediately for infinite variation EI processes (called upward and downward regularity). We also generalize the zero-one law of Mil- lar for L´evy processes by showing continuity of X when it reaches its minimum in the infinite variation EI case; an analogous result for all EI processes links right and left continuity at the minimum with upward and downward regularity. We also consider results of Durrett, Iglehart, and Miller on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion considered in [DIM77] and broadened to a subclass of L´evy processes and EI processes by Chaumont and the second author. We prove it here for all infinite variation EI processes. We furthermore obtain a description of the convex minorant known for L´evy processes found in [Ann. Prob. 40 (2012), pp. 1636–1674] and extend it to non-piecewise linear EI processes. Our main tool to study the Dini derivatives is a change of measure for EI processes which extends the Esscher transform for L´evy processes.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127473180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Overgroups of regular unipotent elements in simple algebraic groups","authors":"G. Malle, D. Testerman","doi":"10.1090/btran/72","DOIUrl":"https://doi.org/10.1090/btran/72","url":null,"abstract":"We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134380818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distribution of Kloosterman paths to high prime power moduli","authors":"Djordje Mili'cevi'c, Sichen Zhang","doi":"10.1090/btran/98","DOIUrl":"https://doi.org/10.1090/btran/98","url":null,"abstract":"We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power \u0000\u0000 \u0000 \u0000 p\u0000 n\u0000 \u0000 p^n\u0000 \u0000\u0000 of a fixed odd prime \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000, a pure depth-aspect analogue of theorems of Kowalski–Sawin and Ricotta–Royer–Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as \u0000\u0000 \u0000 \u0000 n\u0000 →\u0000 ∞\u0000 \u0000 nto infty\u0000 \u0000\u0000 to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000 converges in law as \u0000\u0000 \u0000 \u0000 p\u0000 →\u0000 ∞\u0000 \u0000 pto infty\u0000 \u0000\u0000, and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"184 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122867422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality","authors":"Jun Hu, Zhankui Xiao","doi":"10.1090/btran/84","DOIUrl":"https://doi.org/10.1090/btran/84","url":null,"abstract":"<p>In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a quasi-hereditary algebra with a simple preserving duality and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a faithful tilting <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-module, then <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has the double centralizer property with respect to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for which <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A equals upper E n d Subscript upper E n d Sub Subscript upper A Subscript left-parenthesis upper T right-parenthesis Baseline left-parenthesis upper T right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 ","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114233800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New bounds for Ryser’s conjecture and related problems","authors":"Peter Keevash, A. Pokrovskiy, B. Sudakov, Liana Yepremyan","doi":"10.1090/btran/92","DOIUrl":"https://doi.org/10.1090/btran/92","url":null,"abstract":"<p>A Latin square of order <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n times n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n times n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> array filled with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser-Brualdi-Stein from 60s which says that every Latin square of order <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n times n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ntimes n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> contains a transversal of order <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n minus 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\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\">n-1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this paper we prove the existence of a transversal of order <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n minus upper O left-parenthesis log n slash log log n right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>O</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>log</mml:mi>\u0000 <mml:mo>⁡<!-- ⁡ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>n</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>log</mml:mi>\u0000 <mml:mo>⁡<!-- ⁡ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>log</mml:mi>\u0000 <mml:mo>⁡<!-- ⁡ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-OR","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116703118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Chudnovsky’s conjecture and the stable Harbourne–Huneke containment","authors":"Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thai Thanh Nguyen","doi":"10.1090/btran/103","DOIUrl":"https://doi.org/10.1090/btran/103","url":null,"abstract":"We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129581116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}