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":"40 1","pages":"0"},"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":"18 1","pages":"0"},"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":"81 1","pages":"0"},"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}
Pub Date : 2023-09-27DOI: 10.1007/s44007-023-00068-9
Timothy Myers
In this paper we will present a new construction of any real geometric (Clifford) algebra $${mathbb {G}}^{(p,q)}$$ with signature (p, q) where $$p+q=n$$ by defining a product on the vector space $${mathbb {R}}^{(2^n)}$$ in a manner similar to Gauss’ ordered pair construction of the complex numbers ( $${mathbb {C}}$$ ) and Hamilton’s ordered quadruple construction of the quaternions ( $${mathbb {H}}$$ ). We will motivate the definition of a geometric product on $${mathbb {G}}^{(p,q)}$$ by generalizing the ordered tuple definition of multiplication on each of $${mathbb {C}}$$ and $${mathbb {H}}$$ . Similar to the way in which Gauss obtains the basis $${1, i}$$ from the ordered pair definition of multiplication on $${mathbb {C}}$$ , we will likewise derive a basis of monomials for $${mathbb {G}}^{(p,q)}$$ by multiplying those ordered $$2^n$$ tuples that generate $${mathbb {G}}^{(p,q)}$$ .
{"title":"An Ordered Tuple Construction of Geometric Algebras","authors":"Timothy Myers","doi":"10.1007/s44007-023-00068-9","DOIUrl":"https://doi.org/10.1007/s44007-023-00068-9","url":null,"abstract":"In this paper we will present a new construction of any real geometric (Clifford) algebra $${mathbb {G}}^{(p,q)}$$ with signature (p, q) where $$p+q=n$$ by defining a product on the vector space $${mathbb {R}}^{(2^n)}$$ in a manner similar to Gauss’ ordered pair construction of the complex numbers ( $${mathbb {C}}$$ ) and Hamilton’s ordered quadruple construction of the quaternions ( $${mathbb {H}}$$ ). We will motivate the definition of a geometric product on $${mathbb {G}}^{(p,q)}$$ by generalizing the ordered tuple definition of multiplication on each of $${mathbb {C}}$$ and $${mathbb {H}}$$ . Similar to the way in which Gauss obtains the basis $${1, i}$$ from the ordered pair definition of multiplication on $${mathbb {C}}$$ , we will likewise derive a basis of monomials for $${mathbb {G}}^{(p,q)}$$ by multiplying those ordered $$2^n$$ tuples that generate $${mathbb {G}}^{(p,q)}$$ .","PeriodicalId":74051,"journal":{"name":"La matematica","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135534943","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 : 2023-09-19DOI: 10.1007/s44007-023-00061-2
Werner Schachinger
Abstract Consider Young diagrams of n boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set $${1,ldots ,n}$$ {1,…,n} . Here we are interested in asymptotics, as $$nrightarrow infty $$ n→∞ , of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice’s integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.
{"title":"Asymptotics of Some Plancherel Averages Via Polynomiality Results","authors":"Werner Schachinger","doi":"10.1007/s44007-023-00061-2","DOIUrl":"https://doi.org/10.1007/s44007-023-00061-2","url":null,"abstract":"Abstract Consider Young diagrams of n boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set $${1,ldots ,n}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . Here we are interested in asymptotics, as $$nrightarrow infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice’s integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135060581","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 : 2023-09-19DOI: 10.1007/s44007-023-00069-8
Hanli Tang
In this paper, we establish sharp subcritical and critical second-order Adams inequalities in Lorentz–Sobolev spaces. We also prove the subcritical and critical Adams inequalities are actually equivalent and our results extend existing ones in Tang (Potential Anal 53(1):297–314, 2020) to second order.
{"title":"Sharp Second-Order Adams Inequalities in Lorentz–Sobolev Spaces","authors":"Hanli Tang","doi":"10.1007/s44007-023-00069-8","DOIUrl":"https://doi.org/10.1007/s44007-023-00069-8","url":null,"abstract":"In this paper, we establish sharp subcritical and critical second-order Adams inequalities in Lorentz–Sobolev spaces. We also prove the subcritical and critical Adams inequalities are actually equivalent and our results extend existing ones in Tang (Potential Anal 53(1):297–314, 2020) to second order.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135060580","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 : 2023-09-18DOI: 10.1007/s44007-023-00062-1
Zhicheng Gao
A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the numbers of irreducible monic polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field $${mathbb F}_{q}$$ . The new lower bounds are used to derive some existence results about irreducible monic polynomials of degree d and self-reciprocal irreducible monic polynomials of degree 2d with roughly d/2 coefficients prescribed at positions including the middle range $$d/2-log _q dle jle d/2+log _q d$$ .
如果一个多项式的系数序列是回文的，那么它就被称为自互反的(或回文的)。本文在有限域$${mathbb F}_{q}$$上得到了不可约一元多项式和具有规定系数的自互易不可约一元多项式数目的改进误差界。利用新的下界，导出了d次不可约一元多项式和2d次自倒不可约一元多项式的存在性结果，这些多项式的系数大致为d/2，在包括中间范围$$d/2-log _q dle jle d/2+log _q d$$的位置上。
{"title":"New Estimates and Existence Results About Irreducible Polynomials and Self-Reciprocal Irreducible Polynomials with Prescribed Coefficients Over a Finite Field","authors":"Zhicheng Gao","doi":"10.1007/s44007-023-00062-1","DOIUrl":"https://doi.org/10.1007/s44007-023-00062-1","url":null,"abstract":"A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the numbers of irreducible monic polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field $${mathbb F}_{q}$$ . The new lower bounds are used to derive some existence results about irreducible monic polynomials of degree d and self-reciprocal irreducible monic polynomials of degree 2d with roughly d/2 coefficients prescribed at positions including the middle range $$d/2-log _q dle jle d/2+log _q d$$ .","PeriodicalId":74051,"journal":{"name":"La matematica","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135151246","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 : 2023-09-13DOI: 10.1007/s44007-023-00063-0
Michael Drmota, Eva-Maria Hainzl
Functional equations with one catalytic variable appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions, the dominant singularity of the solution has a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we—usually—have a singularity of type 3/2.
{"title":"Universal Asymptotic Properties of Positive Functional Equations with One Catalytic Variable","authors":"Michael Drmota, Eva-Maria Hainzl","doi":"10.1007/s44007-023-00063-0","DOIUrl":"https://doi.org/10.1007/s44007-023-00063-0","url":null,"abstract":"Functional equations with one catalytic variable appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions, the dominant singularity of the solution has a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we—usually—have a singularity of type 3/2.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734181","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 : 2023-09-13DOI: 10.1007/s44007-023-00064-z
Christoffer Olsson, Stephan Wagner
Abstract We study the size of the automorphism group of two different types of random trees: Galton–Watson trees and rooted Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for the mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton–Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted Pólya trees. We also show how to extend the results to some classes of unrooted trees.
{"title":"The Distribution of the Number of Automorphisms of Random Trees","authors":"Christoffer Olsson, Stephan Wagner","doi":"10.1007/s44007-023-00064-z","DOIUrl":"https://doi.org/10.1007/s44007-023-00064-z","url":null,"abstract":"Abstract We study the size of the automorphism group of two different types of random trees: Galton–Watson trees and rooted Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for the mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton–Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted Pólya trees. We also show how to extend the results to some classes of unrooted trees.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734178","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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