Pub Date : 2023-11-29DOI: 10.1007/s44007-023-00073-y
Jane H. Long
{"title":"Cohomology Classes of the Qd(p) Groups","authors":"Jane H. Long","doi":"10.1007/s44007-023-00073-y","DOIUrl":"https://doi.org/10.1007/s44007-023-00073-y","url":null,"abstract":"","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139213788","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}
Pub Date : 2023-11-20DOI: 10.1007/s44007-023-00078-7
Christina Frederick, Karamatou Yacoubou Djima
{"title":"Examples of Riesz Bases of Exponentials for Multi-tiling Domains and Their Duals","authors":"Christina Frederick, Karamatou Yacoubou Djima","doi":"10.1007/s44007-023-00078-7","DOIUrl":"https://doi.org/10.1007/s44007-023-00078-7","url":null,"abstract":"","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139255332","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}
Pub Date : 2023-11-10DOI: 10.1007/s44007-023-00067-w
Jürgen Klüners, Jiuya Wang
For odd primes $$ell $$ and number fields k, we study the asymptotic distribution of number fields L/k given as a tower of relative cyclic $$C_ell $$ -extensions L/F/k using the idélic approach of class field theory. This involves a classification for the Galois group of L/k based on local conditions on L/F and F/k, and an extension of the method of Wright in enumerating abelian extensions. We call the possible Galois groups for these extensions generalized and twisted Heisenberg groups. We then prove the strong Malle–conjecture for all these groups in their representation on $$ell ^2$$ points.
{"title":"Idélic Approach in Enumerating Heisenberg Extensions","authors":"Jürgen Klüners, Jiuya Wang","doi":"10.1007/s44007-023-00067-w","DOIUrl":"https://doi.org/10.1007/s44007-023-00067-w","url":null,"abstract":"For odd primes $$ell $$ and number fields k, we study the asymptotic distribution of number fields L/k given as a tower of relative cyclic $$C_ell $$ -extensions L/F/k using the idélic approach of class field theory. This involves a classification for the Galois group of L/k based on local conditions on L/F and F/k, and an extension of the method of Wright in enumerating abelian extensions. We call the possible Galois groups for these extensions generalized and twisted Heisenberg groups. We then prove the strong Malle–conjecture for all these groups in their representation on $$ell ^2$$ points.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136356","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}
Pub Date : 2023-11-10DOI: 10.1007/s44007-023-00075-w
Anwita Bhowmik, Rupam Barman
For a prime $$pequiv 3pmod 4$$ and a positive integer t, let $$q=p^{2t}$$ . The Peisert graph of order q is the graph with vertex set $$mathbb {F}_q$$ such that ab is an edge if $$a-bin langle g^4rangle cup glangle g^4rangle $$ , where g is a primitive element of $$mathbb {F}_q$$ . In this paper, we construct a similar graph with vertex set as the commutative ring $$mathbb {Z}_n$$ for suitable n, which we call Peisert-like graph and denote by $$G^*(n)$$ . Owing to the need for cyclicity of the group of units of $$mathbb {Z}_n$$ , we consider $$n=p^alpha $$ or $$2p^alpha $$ , where $$pequiv 1pmod 4$$ is a prime and $$alpha $$ is a positive integer. For primes $$pequiv 1pmod 8$$ , we compute the number of triangles in the graph $$G^*(p^{alpha })$$ by evaluating certain character sums. Next, we study cliques of order 4 in $$G^*(p^{alpha })$$ . To find the number of cliques of order 4 in $$G^*(p^{alpha })$$ , we first introduce hypergeometric functions containing Dirichlet characters as arguments and then express the number of cliques of order 4 in $$G^*(p^{alpha })$$ in terms of these hypergeometric functions.
{"title":"Hypergeometric Functions for Dirichlet Characters and Peisert-Like Graphs on $$mathbb {Z}_n$$","authors":"Anwita Bhowmik, Rupam Barman","doi":"10.1007/s44007-023-00075-w","DOIUrl":"https://doi.org/10.1007/s44007-023-00075-w","url":null,"abstract":"For a prime $$pequiv 3pmod 4$$ and a positive integer t, let $$q=p^{2t}$$ . The Peisert graph of order q is the graph with vertex set $$mathbb {F}_q$$ such that ab is an edge if $$a-bin langle g^4rangle cup glangle g^4rangle $$ , where g is a primitive element of $$mathbb {F}_q$$ . In this paper, we construct a similar graph with vertex set as the commutative ring $$mathbb {Z}_n$$ for suitable n, which we call Peisert-like graph and denote by $$G^*(n)$$ . Owing to the need for cyclicity of the group of units of $$mathbb {Z}_n$$ , we consider $$n=p^alpha $$ or $$2p^alpha $$ , where $$pequiv 1pmod 4$$ is a prime and $$alpha $$ is a positive integer. For primes $$pequiv 1pmod 8$$ , we compute the number of triangles in the graph $$G^*(p^{alpha })$$ by evaluating certain character sums. Next, we study cliques of order 4 in $$G^*(p^{alpha })$$ . To find the number of cliques of order 4 in $$G^*(p^{alpha })$$ , we first introduce hypergeometric functions containing Dirichlet characters as arguments and then express the number of cliques of order 4 in $$G^*(p^{alpha })$$ in terms of these hypergeometric functions.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141136","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}
Pub Date : 2023-10-31DOI: 10.1007/s44007-023-00071-0
Bérénice Grec, Srboljub Simić
{"title":"Higher-Order Maxwell–Stefan Model of Diffusion","authors":"Bérénice Grec, Srboljub Simić","doi":"10.1007/s44007-023-00071-0","DOIUrl":"https://doi.org/10.1007/s44007-023-00071-0","url":null,"abstract":"","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870274","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}
Pub Date : 2023-10-04DOI: 10.1007/s44007-023-00065-y
Marko Riedel, Hosam Mahmoud
Egorychev method is a potent technique for reducing combinatorial sums. In spite of the effectiveness of the method, it is not well known or widely disseminated. Our purpose in writing this manuscript is to bring light to this method. At the heart of this method is the representation of functions as series. The chief idea in Egorychev method is to reduce a combinatorial sum by recognizing some factors in it as coefficients in a series (possibly in the form of contour integrals), then identifying the parts that can be summed in closed form. Once the summation is gone, the rest can be evaluated via one of several techniques, which are namely: (I) Direct extraction of coefficients, after an inspection telling us it is the generating function (formal power series) of a known sequence, (II) Applying residue operators, and (III) Appealing to Cauchy’s residue theorem, when the coefficients alluded to appear as contour integrals. We present some background from the theory of complex variables and illustrate each technique with some examples. In concluding remarks, we compare Egorychev method to alternative methods, such as Wilf–Zeilberger theory, Zeilberger algorithm, and Almkvist–Zeilberger algorithm and to the performance of computer algebra systems.
{"title":"Egorychev Method: A Hidden Treasure","authors":"Marko Riedel, Hosam Mahmoud","doi":"10.1007/s44007-023-00065-y","DOIUrl":"https://doi.org/10.1007/s44007-023-00065-y","url":null,"abstract":"Egorychev method is a potent technique for reducing combinatorial sums. In spite of the effectiveness of the method, it is not well known or widely disseminated. Our purpose in writing this manuscript is to bring light to this method. At the heart of this method is the representation of functions as series. The chief idea in Egorychev method is to reduce a combinatorial sum by recognizing some factors in it as coefficients in a series (possibly in the form of contour integrals), then identifying the parts that can be summed in closed form. Once the summation is gone, the rest can be evaluated via one of several techniques, which are namely: (I) Direct extraction of coefficients, after an inspection telling us it is the generating function (formal power series) of a known sequence, (II) Applying residue operators, and (III) Appealing to Cauchy’s residue theorem, when the coefficients alluded to appear as contour integrals. We present some background from the theory of complex variables and illustrate each technique with some examples. In concluding remarks, we compare Egorychev method to alternative methods, such as Wilf–Zeilberger theory, Zeilberger algorithm, and Almkvist–Zeilberger algorithm and to the performance of computer algebra systems.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135591395","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}
Pub Date : 2023-10-02DOI: 10.1007/s44007-023-00060-3
Benjamin Hackl, Alois Panholzer, Stephan Wagner
Abstract We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with n vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this work: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling; second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase; and lastly, the largest connected component, for which we also observe a phase transition.
{"title":"The Uncover Process for Random Labeled Trees","authors":"Benjamin Hackl, Alois Panholzer, Stephan Wagner","doi":"10.1007/s44007-023-00060-3","DOIUrl":"https://doi.org/10.1007/s44007-023-00060-3","url":null,"abstract":"Abstract We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with n vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this work: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling; second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase; and lastly, the largest connected component, for which we also observe a phase transition.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135898619","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}
Pub Date : 2023-10-02DOI: 10.1007/s44007-023-00070-1
Niclas Bernhoff
Abstract A semi-classical approach to the study of the evolution of bosonic or fermionic excitations is through the Nordheim—Boltzmann- or, Uehling—Uhlenbeck—equation, also known as the quantum Boltzmann equation. In some low ranges of temperatures—e.g., in the presence of a Bose condensate—also other types of collision operators may render in essential contributions. Therefore, extended— or, even other—collision operators are to be considered as well. This work concerns a discretized version—a system of partial differential equations—of such a quantum equation with an extended collision operator. Trend to equilibrium is studied for a planar stationary system, as well as the spatially homogeneous system. Some essential properties of the linearized operator are proven, implying that results for general half-space problems for the discrete Boltzmann equation can be applied. A more general collision operator is also introduced, and similar results are obtained also for this general equation.
{"title":"Discrete Quantum Kinetic Equation","authors":"Niclas Bernhoff","doi":"10.1007/s44007-023-00070-1","DOIUrl":"https://doi.org/10.1007/s44007-023-00070-1","url":null,"abstract":"Abstract A semi-classical approach to the study of the evolution of bosonic or fermionic excitations is through the Nordheim—Boltzmann- or, Uehling—Uhlenbeck—equation, also known as the quantum Boltzmann equation. In some low ranges of temperatures—e.g., in the presence of a Bose condensate—also other types of collision operators may render in essential contributions. Therefore, extended— or, even other—collision operators are to be considered as well. This work concerns a discretized version—a system of partial differential equations—of such a quantum equation with an extended collision operator. Trend to equilibrium is studied for a planar stationary system, as well as the spatially homogeneous system. Some essential properties of the linearized operator are proven, implying that results for general half-space problems for the discrete Boltzmann equation can be applied. A more general collision operator is also introduced, and similar results are obtained also for this general equation.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135895324","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}
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