The Hilbert metric between two points 𝑥, 𝑦 in a bounded convex domain 𝐺 is defined as the logarithm of the cross-ratio 𝑥, 𝑦 and the intersection points of the Euclidean line passing through the points 𝑥, 𝑦 and the boundary of the domain. Here, we study this metric in the case of the unit ball 𝔹 𝑛 . We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal and quasiregular mappings.
{"title":"Hilbert Metric in the Unit Ball","authors":"Oona Rainio, Matti Vuorinen","doi":"10.1556/012.2023.01544","DOIUrl":"https://doi.org/10.1556/012.2023.01544","url":null,"abstract":"The Hilbert metric between two points 𝑥, 𝑦 in a bounded convex domain 𝐺 is defined as the logarithm of the cross-ratio 𝑥, 𝑦 and the intersection points of the Euclidean line passing through the points 𝑥, 𝑦 and the boundary of the domain. Here, we study this metric in the case of the unit ball 𝔹 𝑛 . We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal and quasiregular mappings.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"38 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273536","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}
We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4.
证明了最大惠特尼数不超过4的梯度序集的自同态猜想。
{"title":"The Endomorphism Conjecture for Graded Posets with Whitney Numbers at most 4","authors":"Miklós Bóna, Ryan R. Martin","doi":"10.1556/012.2023.01540","DOIUrl":"https://doi.org/10.1556/012.2023.01540","url":null,"abstract":"We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"12 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267976","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}
A proper edge coloring of a graph 𝐺 is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings ). The strong chromatic index 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛 2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞 𝐴 (𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring . The second is 𝑞 𝐵 (𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a B-coloring . These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞 𝐴 (𝐺), 𝑞 𝐵 (𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special (7, 4)-conjecture. In order to understand the behavior of 𝑞 𝐴(𝐺) and 𝑞 𝐵(𝐺), we study these parameters for some graphs. We note that 𝑞 𝐴 (𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞 𝐵 (𝐺) is studied here for the first time.
{"title":"“Less” Strong Chromatic Indices and the (7, 4)-Conjecture","authors":"András Gyárfás, Gábor N. Sárközy","doi":"10.1556/012.2023.01539","DOIUrl":"https://doi.org/10.1556/012.2023.01539","url":null,"abstract":"A proper edge coloring of a graph 𝐺 is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings ). The strong chromatic index 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛 2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞 𝐴 (𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring . The second is 𝑞 𝐵 (𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a B-coloring . These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞 𝐴 (𝐺), 𝑞 𝐵 (𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special (7, 4)-conjecture. In order to understand the behavior of 𝑞 𝐴(𝐺) and 𝑞 𝐵(𝐺), we study these parameters for some graphs. We note that 𝑞 𝐴 (𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞 𝐵 (𝐺) is studied here for the first time.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"51 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267756","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}
We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension 𝑑, a cross-polytope of dimension 𝑑, respectively a hypercube of dimension 𝑑 ≤ 3, using commutative algebra. We deduce summation formulae for products of 𝑞-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.
{"title":"Refined Ehrhart Series and Bigraded Rings","authors":"Praise Adeyemo, Balázs Szendrői","doi":"10.1556/012.2023.01541","DOIUrl":"https://doi.org/10.1556/012.2023.01541","url":null,"abstract":"We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension 𝑑, a cross-polytope of dimension 𝑑, respectively a hypercube of dimension 𝑑 ≤ 3, using commutative algebra. We deduce summation formulae for products of 𝑞-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"17 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135268439","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}
This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼 𝑔 (𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼 𝑔 (𝐺), we completely characterize the graph 𝐺 for which 𝐼 𝑔 (𝐺) is integrally closed, and show that this is equivalent to 𝐼 𝑔 (𝐺) being normal i.e., all integral powers of 𝐼 𝑔 (𝐺) are integrally clased. We also give a necessary and sufficient condition for when 𝐺 is the star-shaped graph. Finally, we give a necessary and sufficient condition when the generalized edge ideal of a complete graph is integrally closed.
{"title":"Integral Closure of Powers of Generalized Edge Ideals","authors":"Sirajul Haque","doi":"10.1556/012.2023.01543","DOIUrl":"https://doi.org/10.1556/012.2023.01543","url":null,"abstract":"This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼 𝑔 (𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼 𝑔 (𝐺), we completely characterize the graph 𝐺 for which 𝐼 𝑔 (𝐺) is integrally closed, and show that this is equivalent to 𝐼 𝑔 (𝐺) being normal i.e., all integral powers of 𝐼 𝑔 (𝐺) are integrally clased. We also give a necessary and sufficient condition for when 𝐺 is the star-shaped graph. Finally, we give a necessary and sufficient condition when the generalized edge ideal of a complete graph is integrally closed.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273885","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}
Lovejoy introduced the partition function as the number of 𝑙-regular overpartitions of 𝑛. Andrews defined (𝑘, 𝑖)-singular overpartitions counted by the partition function , and pointed out that . Meanwhile, Andrews derived an interesting divisibility property that (mod 3). Recently, we constructed the partition statistic 𝑟 𝑙 -crank of 𝑙-regular overpartitions and give combinatorial interpretations for some congruences of as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the 𝑟 3 -crank of 3-regular overpartitions.
{"title":"Equalities for the 𝑟3-Crank of 3-Regular Overpartitions","authors":"Robert X. J. Hao, Erin Y. Y. Shen","doi":"10.1556/012.2023.01542","DOIUrl":"https://doi.org/10.1556/012.2023.01542","url":null,"abstract":"Lovejoy introduced the partition function as the number of 𝑙-regular overpartitions of 𝑛. Andrews defined (𝑘, 𝑖)-singular overpartitions counted by the partition function , and pointed out that . Meanwhile, Andrews derived an interesting divisibility property that (mod 3). Recently, we constructed the partition statistic 𝑟 𝑙 -crank of 𝑙-regular overpartitions and give combinatorial interpretations for some congruences of as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the 𝑟 3 -crank of 3-regular overpartitions.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"22 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267112","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}
Jesús A. De Loera, Christopher O’Neill, Chengyang Wang
In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups , where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.
{"title":"Convexity in (Colored) Affine Semigroups","authors":"Jesús A. De Loera, Christopher O’Neill, Chengyang Wang","doi":"10.1556/012.2023.01545","DOIUrl":"https://doi.org/10.1556/012.2023.01545","url":null,"abstract":"In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups , where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"39 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267577","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}
John Horton Conway stood out from many famous mathematicians for his love of games and puzzles. Among others, he is known for inventing the two-player topological games called Sprouts and Brussels Sprouts. These games start with n spots (n crosses resp.), have simple rules, last for finitely many moves, and the player who makes the last move wins. In the misère versions, the player who makes the last move loses. In this paper, we make Brussels Sprouts colored, preserving the aesthetic interest and balance of the game. In contrast to the original Sprouts, Colored Brussels Sprouts allows mathematical analysis without computer programming and has winning strategies for a large family of the number of spots.
{"title":"On Conway’s Brussels Sprouts","authors":"Andras Bezdek, Haile Gilroy, Owen Henderschedt, Alason Lakhani","doi":"10.1556/012.2023.01535","DOIUrl":"https://doi.org/10.1556/012.2023.01535","url":null,"abstract":"John Horton Conway stood out from many famous mathematicians for his love of games and puzzles. Among others, he is known for inventing the two-player topological games called Sprouts and Brussels Sprouts. These games start with n spots (n crosses resp.), have simple rules, last for finitely many moves, and the player who makes the last move wins. In the misère versions, the player who makes the last move loses. In this paper, we make Brussels Sprouts colored, preserving the aesthetic interest and balance of the game. In contrast to the original Sprouts, Colored Brussels Sprouts allows mathematical analysis without computer programming and has winning strategies for a large family of the number of spots.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"31 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80886503","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}
We give a full, correct proof of the following result, earlier claimed in [1]. If the Continuum Hypothesis holds then there is a coloring of the plane with countably many colors, with no monocolored right triangle.
{"title":"A Coloring of the Plane Without Monochromatic Right Triangles","authors":"Bal'azs Bursics, P'eter Komj'ath","doi":"10.1556/012.2023.01537","DOIUrl":"https://doi.org/10.1556/012.2023.01537","url":null,"abstract":"We give a full, correct proof of the following result, earlier claimed in [1]. If the Continuum Hypothesis holds then there is a coloring of the plane with countably many colors, with no monocolored right triangle.","PeriodicalId":51187,"journal":{"name":"Studia Scientiarum Mathematicarum Hungarica","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87848992","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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