Pub Date : 2025-10-09DOI: 10.1016/j.exmath.2025.125731
Frank Sottile , Thomas Yahl
As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic study only occurring in the past 15 years. We discuss the current directions of this study, including open problems and conjectures.
{"title":"Galois groups in enumerative geometry and applications","authors":"Frank Sottile , Thomas Yahl","doi":"10.1016/j.exmath.2025.125731","DOIUrl":"10.1016/j.exmath.2025.125731","url":null,"abstract":"<div><div>As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic study only occurring in the past 15 years. We discuss the current directions of this study, including open problems and conjectures.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125731"},"PeriodicalIF":0.9,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145324319","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}
Pub Date : 2025-09-30DOI: 10.1016/j.exmath.2025.125732
Bei Liu, Meng Ning, Rui Li, Liping Wu, Qingyue Zhang
In this paper, we demonstrate the real-valued bandlimited signals can be recovered (up to global sign) from the magnitude of their fractional wavelet transforms, also called fractional wavelet phase retrieval. We present a new uniqueness result for fractional wavelet phase retrieval. To be precise, we show that any real-valued bandlimited functions can be recovered uniquely (up to global sign) by fractional wavelets which have finitely many vanishing moments. Furthermore, we show that the real-valued bandlimited function can be uniquely recovered up to global sign from the absolute values of the discrete fractional wavelet transform for fixed .
{"title":"Fractional wavelet phase retrieval of bandlimited functions","authors":"Bei Liu, Meng Ning, Rui Li, Liping Wu, Qingyue Zhang","doi":"10.1016/j.exmath.2025.125732","DOIUrl":"10.1016/j.exmath.2025.125732","url":null,"abstract":"<div><div>In this paper, we demonstrate the real-valued bandlimited signals can be recovered (up to global sign) from the magnitude of their fractional wavelet transforms, also called fractional wavelet phase retrieval. We present a new uniqueness result for fractional wavelet phase retrieval. To be precise, we show that any real-valued bandlimited functions can be recovered uniquely (up to global sign) by fractional wavelets which have finitely many vanishing moments. Furthermore, we show that the real-valued bandlimited function can be uniquely recovered up to global sign from the absolute values of the discrete fractional wavelet transform for fixed <span><math><mrow><mi>a</mi><mo>></mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125732"},"PeriodicalIF":0.9,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145262723","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}
Pub Date : 2025-09-24DOI: 10.1016/j.exmath.2025.125730
Akram Aldroubi , Carlos Cabrelli , Ursula Molter
In this paper, we examine a discrete dynamical system defined by , where takes values in a Hilbert space and is a periodic source with values in a fixed closed subspace of . Our goal is to identify conditions on some spatial sampling system of that enable stable recovery of the unknown source term from space–time samples . We provide necessary and sufficient conditions on to ensure stable recovery of any . Additionally, we explicitly construct an operator , dependent on , such that .
{"title":"Periodic source detection in discrete dynamical systems via space–time sampling","authors":"Akram Aldroubi , Carlos Cabrelli , Ursula Molter","doi":"10.1016/j.exmath.2025.125730","DOIUrl":"10.1016/j.exmath.2025.125730","url":null,"abstract":"<div><div>In this paper, we examine a discrete dynamical system defined by <span><math><mrow><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mi>A</mi><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>+</mo><mi>w</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>x</mi></math></span> takes values in a Hilbert space <span><math><mi>H</mi></math></span> and <span><math><mi>w</mi></math></span> is a periodic source with values in a fixed closed subspace <span><math><mi>W</mi></math></span> of <span><math><mi>H</mi></math></span>. Our goal is to identify conditions on some spatial sampling system <span><math><mrow><mi>G</mi><mo>=</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow></math></span> of <span><math><mi>H</mi></math></span> that enable stable recovery of the unknown source term <span><math><mi>w</mi></math></span> from space–time samples <span><math><msub><mrow><mrow><mo>{</mo><mrow><mo>〈</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>〉</mo></mrow><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></math></span>. We provide necessary and sufficient conditions on <span><math><mrow><mi>G</mi><mo>=</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow></math></span> to ensure stable recovery of any <span><math><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow></math></span>. Additionally, we explicitly construct an operator <span><math><mi>R</mi></math></span>, dependent on <span><math><mi>G</mi></math></span>, such that <span><math><mrow><mi>R</mi><msub><mrow><mrow><mo>{</mo><mrow><mo>〈</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>〉</mo></mrow><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mi>w</mi></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125730"},"PeriodicalIF":0.9,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157524","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}
Pub Date : 2025-09-18DOI: 10.1016/j.exmath.2025.125720
Ole Christensen , Hong Oh Kim , Rae Young Kim
The concept of weaving of frames for Hilbert spaces was introduced by Bemrose et al. in 2016. Two frames are woven if the “mixed system” is a frame for each index set that is, processing a signal using two woven frames yields a certain stability against loss of information. The concept easily extends to frames, for any integer Unfortunately it is nontrivial to construct useful woven frames, and the literature is sparse concerning explicit constructions. In this paper we introduce so-called information packets, which contain as well frames as fusion frames as special case. The concept of woven frames immediately generalizes to information packets, and we demonstrate how to construct practically relevant woven information packets based on particular wavelet systems in Interestingly, we show that certain wavelet systems can be split into woven information packets, for any integer We finally consider corresponding questions for Gabor system in and prove that for any fixed we can find a Gabor frame that can be split into woven information packets; however, in contrast to the wavelet case, the density conditions for Gabor system excludes the possibility of finding a single Gabor frame that works simultaneously for all
{"title":"Weaving information packets","authors":"Ole Christensen , Hong Oh Kim , Rae Young Kim","doi":"10.1016/j.exmath.2025.125720","DOIUrl":"10.1016/j.exmath.2025.125720","url":null,"abstract":"<div><div>The concept of weaving of frames for Hilbert spaces was introduced by Bemrose et al. in 2016. Two frames <span><math><mrow><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>,</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></math></span> are woven if the “mixed system” <span><math><mrow><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>σ</mi></mrow></msub><mo>∪</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>I</mi><mo>∖</mo><mi>σ</mi></mrow></msub></mrow></math></span> is a frame for each index set <span><math><mrow><mi>σ</mi><mo>⊂</mo><mi>I</mi><mo>;</mo></mrow></math></span> that is, processing a signal using two woven frames yields a certain stability against loss of information. The concept easily extends to <span><math><mi>N</mi></math></span> frames, for any integer <span><math><mrow><mi>N</mi><mo>></mo><mn>2</mn><mo>.</mo></mrow></math></span> Unfortunately it is nontrivial to construct useful woven frames, and the literature is sparse concerning explicit constructions. In this paper we introduce so-called information packets, which contain as well frames as fusion frames as special case. The concept of woven frames immediately generalizes to information packets, and we demonstrate how to construct practically relevant woven information packets based on particular wavelet systems in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> Interestingly, we show that certain wavelet systems can be split into <span><math><mi>N</mi></math></span> woven information packets, for any integer <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn><mo>.</mo></mrow></math></span> We finally consider corresponding questions for Gabor system in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> and prove that for any fixed <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi></mrow></math></span> we can find a Gabor frame that can be split into <span><math><mi>N</mi></math></span> woven information packets; however, in contrast to the wavelet case, the density conditions for Gabor system excludes the possibility of finding a single Gabor frame that works simultaneously for all <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi><mo>.</mo></mrow></math></span></div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125720"},"PeriodicalIF":0.9,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117694","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}
Pub Date : 2025-09-06DOI: 10.1016/j.exmath.2025.125721
Bruce C. Berndt , Pieter Moree
Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail.
{"title":"Sums of two squares and the tau-function: Ramanujan’s trail","authors":"Bruce C. Berndt , Pieter Moree","doi":"10.1016/j.exmath.2025.125721","DOIUrl":"10.1016/j.exmath.2025.125721","url":null,"abstract":"<div><div>Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125721"},"PeriodicalIF":0.9,"publicationDate":"2025-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048954","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}
Pub Date : 2025-08-11DOI: 10.1016/j.exmath.2025.125718
James E. Hanson
<div><div>In this expository paper aimed at a general mathematical audience, we discuss how to combine certain classic theorems of set-theoretic inner model theory and effective descriptive set theory with work on Hilbert’s tenth problem and universal Diophantine equations to produce the following surprising result: There is a specific polynomial <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>70</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> of degree 7 with integer coefficients such that it is independent of <span><math><mi>ZFC</mi></math></span> (and much stronger theories) whether the function <span><span><span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>inf</mo></mrow><mrow><mi>y</mi><mo>∈</mo><mi>R</mi></mrow></munder><munder><mrow><mo>sup</mo></mrow><mrow><mi>z</mi><mo>∈</mo><mi>R</mi></mrow></munder><munder><mrow><mo>inf</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></munder><munder><mrow><mo>sup</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>70</mn></mrow></msup></mrow></munder><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>n</mi><mo>,</mo><mover><mrow><mi>k</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span></span></span>is Lebesgue measurable. We also give similarly defined <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> with the property that the statement “<span><math><mrow><mi>x</mi><mo>↦</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> is measurable for every <span><math><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow></math></span>” has large cardinal consistency strength (and in particular implies the consistency of <span><math><mi>ZFC</mi></math></span>) and <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>h</mi><mrow><mo>(</mo><mn>16</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></span> can consistently be the indicator functions of a Banach–Tarski paradoxical decomposition of the sphere.</div><div>Finally, we discuss some situations in which measurability of analogously defined functions can be concluded by inspection, which touches on model-theoretic o-minimality and the fact that sufficiently strong large cardinal hypotheses (such as Vopěnka’s principle and much weaker assumptions
{"title":"Any function I can actually write down is measurable, right?","authors":"James E. Hanson","doi":"10.1016/j.exmath.2025.125718","DOIUrl":"10.1016/j.exmath.2025.125718","url":null,"abstract":"<div><div>In this expository paper aimed at a general mathematical audience, we discuss how to combine certain classic theorems of set-theoretic inner model theory and effective descriptive set theory with work on Hilbert’s tenth problem and universal Diophantine equations to produce the following surprising result: There is a specific polynomial <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>70</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> of degree 7 with integer coefficients such that it is independent of <span><math><mi>ZFC</mi></math></span> (and much stronger theories) whether the function <span><span><span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>inf</mo></mrow><mrow><mi>y</mi><mo>∈</mo><mi>R</mi></mrow></munder><munder><mrow><mo>sup</mo></mrow><mrow><mi>z</mi><mo>∈</mo><mi>R</mi></mrow></munder><munder><mrow><mo>inf</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></munder><munder><mrow><mo>sup</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>70</mn></mrow></msup></mrow></munder><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>n</mi><mo>,</mo><mover><mrow><mi>k</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span></span></span>is Lebesgue measurable. We also give similarly defined <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> with the property that the statement “<span><math><mrow><mi>x</mi><mo>↦</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> is measurable for every <span><math><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow></math></span>” has large cardinal consistency strength (and in particular implies the consistency of <span><math><mi>ZFC</mi></math></span>) and <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>h</mi><mrow><mo>(</mo><mn>16</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></span> can consistently be the indicator functions of a Banach–Tarski paradoxical decomposition of the sphere.</div><div>Finally, we discuss some situations in which measurability of analogously defined functions can be concluded by inspection, which touches on model-theoretic o-minimality and the fact that sufficiently strong large cardinal hypotheses (such as Vopěnka’s principle and much weaker assumptions","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125718"},"PeriodicalIF":0.9,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144851911","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}
Pub Date : 2025-07-31DOI: 10.1016/j.exmath.2025.125716
Andrés Navas
We provide a large family of examples of affine isometries of the Banach spaces , and that are fixed-point-free despite being recurrent (in particular, they have zero drift). These come from natural cocycles on the group of circle diffeomorphisms, namely the logarithmic, affine and (a variation of the) Schwarzian derivative. Quite interestingly, they arise from diffeomorphisms that are generic in an appropriate context. We also show how to promote these examples in order to obtain families of commuting isometries satisfying the same properties.
{"title":"Some examples of affine isometries of Banach spaces arising from 1-D dynamics","authors":"Andrés Navas","doi":"10.1016/j.exmath.2025.125716","DOIUrl":"10.1016/j.exmath.2025.125716","url":null,"abstract":"<div><div>We provide a large family of examples of affine isometries of the Banach spaces <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> that are fixed-point-free despite being recurrent (in particular, they have zero drift). These come from natural cocycles on the group of circle diffeomorphisms, namely the logarithmic, affine and (a variation of the) Schwarzian derivative. Quite interestingly, they arise from diffeomorphisms that are generic in an appropriate context. We also show how to promote these examples in order to obtain families of commuting isometries satisfying the same properties.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125716"},"PeriodicalIF":0.9,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144757943","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}
Pub Date : 2025-07-19DOI: 10.1016/j.exmath.2025.125715
Matěj Doležálek
A proof of Lagrange’s and Jacobi’s four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an order with a good factorization theory and the condition that all orbits under the action of the group of elements of norm 1 acting by multiplication intersect the suborder corresponding to the quadratic form to be studied. We use recent results on class numbers of quaternion orders and then find all suborders satisfying the orbit condition. Subsequently, we obtain universality and formulas for the number of representations by the corresponding quadratic forms. We also present a quaternionic proof of Götzky’s four-square theorem.
{"title":"Generalizing Hurwitz’s quaternionic proof of Lagrange’s and Jacobi’s four-square theorems","authors":"Matěj Doležálek","doi":"10.1016/j.exmath.2025.125715","DOIUrl":"10.1016/j.exmath.2025.125715","url":null,"abstract":"<div><div>A proof of Lagrange’s and Jacobi’s four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an order with a good factorization theory and the condition that all orbits under the action of the group of elements of norm 1 acting by multiplication intersect the suborder corresponding to the quadratic form to be studied. We use recent results on class numbers of quaternion orders and then find all suborders satisfying the orbit condition. Subsequently, we obtain universality and formulas for the number of representations by the corresponding quadratic forms. We also present a quaternionic proof of Götzky’s four-square theorem.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125715"},"PeriodicalIF":0.8,"publicationDate":"2025-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144711948","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}
Pub Date : 2025-07-17DOI: 10.1016/j.exmath.2025.125713
Elcio Lebensztayn, Lucas Sousa Santos
We investigate the generalization of the Maki–Thompson model for the spreading of a rumor through a finite population in which each spreader stops transmitting the rumor right after being involved in unsuccessful telling interactions. We prove that the proportion of people unaware of the rumor at the end of the process converges in probability to a constant, as the population size goes to . The proof relies on an application of the martingale stopping theorem and is based upon the case established by Sudbury (1985), but our approach for proving the convergence is simpler, reducing technicalities.
{"title":"The law of large numbers for stochastic rumor models","authors":"Elcio Lebensztayn, Lucas Sousa Santos","doi":"10.1016/j.exmath.2025.125713","DOIUrl":"10.1016/j.exmath.2025.125713","url":null,"abstract":"<div><div>We investigate the generalization of the Maki–Thompson model for the spreading of a rumor through a finite population in which each spreader stops transmitting the rumor right after being involved in <span><math><mi>k</mi></math></span> unsuccessful telling interactions. We prove that the proportion of people unaware of the rumor at the end of the process converges in probability to a constant, as the population size goes to <span><math><mi>∞</mi></math></span>. The proof relies on an application of the martingale stopping theorem and is based upon the case <span><math><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></math></span> established by Sudbury (1985), but our approach for proving the convergence is simpler, reducing technicalities.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125713"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144663132","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}
Pub Date : 2025-06-25DOI: 10.1016/j.exmath.2025.125711
Marisa Gaetz
The primary goal of this paper is to explicitly write down all semisimple dual pairs in the complex exceptional Lie algebras. (A dual pair in a reductive Lie algebra is a pair of subalgebras such that each member equals the other’s centralizer in .) In a 1994 paper, H. Rubenthaler outlined a process for generating a complete list of candidate dual pairs in each of the complex exceptional Lie algebras. However, the process of checking whether each of these candidate dual pairs is in fact a dual pair is not easy: it requires the development of several distinct methods and the adaptation of results from multiple sources, some of which are not readily available online. In this paper, we carry out this process and explain the relevant concepts as we go. We also give plenty of examples with the hopes of making Rubenthaler’s 1994 result not only more complete but more usable and understandable.
{"title":"An explicit classification of dual pairs in exceptional Lie algebras","authors":"Marisa Gaetz","doi":"10.1016/j.exmath.2025.125711","DOIUrl":"10.1016/j.exmath.2025.125711","url":null,"abstract":"<div><div>The primary goal of this paper is to explicitly write down all semisimple <em>dual pairs</em> in the complex exceptional Lie algebras. (A <em>dual pair</em> in a reductive Lie algebra <span><math><mi>g</mi></math></span> is a pair of subalgebras such that each member equals the other’s centralizer in <span><math><mi>g</mi></math></span>.) In a 1994 paper, H. Rubenthaler outlined a process for generating a complete list of candidate dual pairs in each of the complex exceptional Lie algebras. However, the process of checking whether each of these candidate dual pairs is in fact a dual pair is not easy: it requires the development of several distinct methods and the adaptation of results from multiple sources, some of which are not readily available online. In this paper, we carry out this process and explain the relevant concepts as we go. We also give plenty of examples with the hopes of making Rubenthaler’s 1994 result not only more complete but more usable and understandable.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125711"},"PeriodicalIF":0.8,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596615","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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