Expositiones Mathematicae最新文献

{"title":"A theory of composites perspective on matrix valued Stieltjes functions","authors":"Graeme W. Milton ,&nbsp;Mihai Putinar","doi":"10.1016/j.exmath.2022.12.005","DOIUrl":"10.1016/j.exmath.2022.12.005","url":null,"abstract":"<div><p>A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47108630","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}

{"title":"Splitting fields of Xn−","authors":"Chandrashekhar B. Khare, Alfio Fabio La Rosa, Gabor Wiese","doi":"10.1016/j.exmath.2023.02.007","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.02.007","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45757313","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}

{"title":"Rational points on X0+(","authors":"V. Arul, J. Müller","doi":"10.1016/j.exmath.2023.02.009","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.02.009","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46963822","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}

{"title":"Approximations of the Riley slice","authors":"Alex Elzenaar ,&nbsp;Gaven Martin ,&nbsp;Jeroen Schillewaert","doi":"10.1016/j.exmath.2022.12.002","DOIUrl":"10.1016/j.exmath.2022.12.002","url":null,"abstract":"<div><p>Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify ‘half-space’ neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42835286","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}

{"title":"The product of lattice covolume and discrete series formal dimension: p-adic GL(2)","authors":"L.C. Ruth","doi":"10.1016/j.exmath.2022.09.001","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.09.001","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in <span><math><mrow><mi>P</mi><mi>G</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> and the formal dimension of a discrete series representation of <span><math><mrow><mi>G</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877666","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}

{"title":"An analogue of Furstenberg–Sárközy’s theorem and an alternative solution to Waring’s problem over finite fields","authors":"Yeşi̇m Demi̇roğlu Karabulut","doi":"10.1016/j.exmath.2022.10.003","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.10.003","url":null,"abstract":"<div><p>In this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring’s problem over finite fields, proving that any element of a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> can be written as a sum of <span><math><mi>m</mi></math></span> many <span><math><mrow><mi>k</mi><mtext>th</mtext></mrow></math></span> powers as long as <span><math><mrow><mi>q</mi><mo>&gt;</mo><msup><mrow><mi>k</mi></mrow><mrow><mfrac><mrow><mn>2</mn><mi>m</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></math></span>; and we also compute the smallest positive integers <span><math><mi>m</mi></math></span> such that every element of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> can be written as a sum of <span><math><mi>m</mi></math></span> many <span><math><mrow><mi>k</mi><mtext>th</mtext></mrow></math></span> powers for all <span><math><mi>q</mi></math></span> too small to be covered by the above mentioned results when <span><math><mrow><mn>2</mn><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mn>37</mn></mrow></math></span>.</p><p>In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy’s Theorem) showing that any subset <span><math><mi>E</mi></math></span> of a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for which <span><math><mrow><mrow><mo>|</mo><mi>E</mi><mo>|</mo></mrow><mo>&gt;</mo><mfrac><mrow><mi>q</mi><mi>k</mi></mrow><mrow><msqrt><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></mfrac></mrow></math></span> must contain at least two distinct elements whose difference is a <span><math><mrow><mi>k</mi><mtext>th</mtext></mrow></math></span> power.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877667","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}

{"title":"Thresholds for the monochromatic clique transversal game","authors":"Csilla Bujtás ,&nbsp;Pakanun Dokyeesun ,&nbsp;Sandi Klavžar","doi":"10.1016/j.exmath.2022.11.001","DOIUrl":"10.1016/j.exmath.2022.11.001","url":null,"abstract":"<div><p>We study a recently introduced two-person combinatorial game, the <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span>-monochromatic clique transversal game which is played by Alice and Bob on a graph <span><math><mi>G</mi></math></span>. As we observe, this game is equivalent to the <span><math><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></math></span>-biased Maker–Breaker game played on the clique-hypergraph of <span><math><mi>G</mi></math></span>. Our main results concern the threshold bias <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> that is the smallest integer <span><math><mi>a</mi></math></span> such that Alice can win in the <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-monochromatic clique transversal game on <span><math><mi>G</mi></math></span> if she is the first to play. Among other results, we determine the possible values of <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for the disjoint union of graphs, prove a formula for <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> if <span><math><mi>G</mi></math></span> is triangle-free, and obtain the exact values of <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>□</mo><mspace></mspace><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>□</mo><mspace></mspace><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>□</mo><mspace></mspace><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for all possible pairs <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43977444","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}

{"title":"Noncommutative Ck functions and Fréchet derivatives of operator functions","authors":"Evangelos A. Nikitopoulos","doi":"10.1016/j.exmath.2022.12.004","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.12.004","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46532265","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}

{"title":"Hilbert’s Nullstellensatz for analytic trigonometric polynomials","authors":"Jie Xiao ,&nbsp;Cheng Yuan","doi":"10.1016/j.exmath.2022.09.005","DOIUrl":"10.1016/j.exmath.2022.09.005","url":null,"abstract":"<div><p><span>This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if </span><span><math><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msubsup></math></span> are analytic trigonometric polynomials without common zero in the finite complex plane <span><math><mi>ℂ</mi></math></span> then there are analytic trigonometric polynomials <span><math><msubsup><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><mn>1</mn></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msubsup></math></span> obeying <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msubsup><msub><mrow><mi>f</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></math></span> in <span><math><mi>ℂ</mi></math></span>, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on <span><math><mi>ℂ</mi></math></span>.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42459702","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}

{"title":"Observations about the Lie algebra g2⊂so(7), associative 3-planes, and so(4) subalgebras","authors":"Max Chemtov ,&nbsp;Spiro Karigiannis","doi":"10.1016/j.exmath.2022.10.004","DOIUrl":"10.1016/j.exmath.2022.10.004","url":null,"abstract":"<div><p><span>We make several observations relating the Lie algebra </span><span><math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊂</mo><mi>so</mi><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow></math></span>, associative 3-planes, and <span><math><mrow><mi>so</mi><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></math></span><span> subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element </span><span><math><mrow><mi>X</mi><mo>∈</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span><span> cannot have rank 2, and if it has rank 4 then its kernel is an associative subspace. We prove a canonical form theorem for elements of </span><span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Given an associative 3-plane <span><math><mi>P</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>7</mn></mrow></msup></math></span>, we construct a Lie subalgebra <span><math><mrow><mi>Θ</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mi>so</mi><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>7</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> that is isomorphic to <span><math><mrow><mi>so</mi><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></math></span>. This <span><math><mrow><mi>so</mi><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></math></span> subalgebra differs from other known constructions of <span><math><mrow><mi>so</mi><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></math></span> subalgebras of <span><math><mrow><mi>so</mi><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow></math></span> determined by an associative 3-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"105269634","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}