Pub Date : 2021-10-19DOI: 10.1017/s0963548322000311
Elvira Moreno, Mauricio Velasco
We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$ . We introduce the switched random walk determined by a finite set of probability distributions on $G$ , prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for the effective estimation of this quantity.
{"title":"On random walks and switched random walks on homogeneous spaces","authors":"Elvira Moreno, Mauricio Velasco","doi":"10.1017/s0963548322000311","DOIUrl":"https://doi.org/10.1017/s0963548322000311","url":null,"abstract":"We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$ . We introduce the switched random walk determined by a finite set of probability distributions on $G$ , prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for the effective estimation of this quantity.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"4 1","pages":"398-421"},"PeriodicalIF":0.9,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85933372","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 : 2021-10-11DOI: 10.1017/s0963548322000232
B. Nica
We study multivariate polynomials over ‘structured’ grids. Firstly, we propose an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results – notably, the Combinatorial Nullstellensatz and the Coefficient Theorem – to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.
{"title":"Polynomials over structured grids","authors":"B. Nica","doi":"10.1017/s0963548322000232","DOIUrl":"https://doi.org/10.1017/s0963548322000232","url":null,"abstract":"\u0000 We study multivariate polynomials over ‘structured’ grids. Firstly, we propose an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results – notably, the Combinatorial Nullstellensatz and the Coefficient Theorem – to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"1 1","pages":"284-298"},"PeriodicalIF":0.9,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72535674","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 : 2021-06-16DOI: 10.1017/s0963548322000013
N. Dragani'c, Stefan Glock, M. Krivelevich
We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies $hat{R}_{mathrm{ind}}(P_n)leq 5 cdot 10^7n$ , thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and Łuczak. We also provide a bound for the k-colour version, showing that $hat{R}_{mathrm{ind}}^k(P_n)=O(k^3log^4k)n$ . Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,frac{1+varepsilon}{n})$ , contains typically an induced path of length $Theta(varepsilon^2) n$ .
{"title":"Short proofs for long induced paths","authors":"N. Dragani'c, Stefan Glock, M. Krivelevich","doi":"10.1017/s0963548322000013","DOIUrl":"https://doi.org/10.1017/s0963548322000013","url":null,"abstract":"\u0000 We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies \u0000 \u0000 \u0000 \u0000$hat{R}_{mathrm{ind}}(P_n)leq 5 cdot 10^7n$\u0000\u0000 \u0000 , thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and Łuczak. We also provide a bound for the k-colour version, showing that \u0000 \u0000 \u0000 \u0000$hat{R}_{mathrm{ind}}^k(P_n)=O(k^3log^4k)n$\u0000\u0000 \u0000 . Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, \u0000 \u0000 \u0000 \u0000$G(n,frac{1+varepsilon}{n})$\u0000\u0000 \u0000 , contains typically an induced path of length \u0000 \u0000 \u0000 \u0000$Theta(varepsilon^2) n$\u0000\u0000 \u0000 .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"16 1","pages":"870-878"},"PeriodicalIF":0.9,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80770829","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 : 2021-06-15DOI: 10.1017/s0963548322000190
Catherine S. Greenhill, M. Isaev, Tamás Makai, B. McKay
We find an asymptotic enumeration formula for the number of simple $r$ -uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$ -uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random $r$ -uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum.
{"title":"Degree sequences of sufficiently dense random uniform hypergraphs","authors":"Catherine S. Greenhill, M. Isaev, Tamás Makai, B. McKay","doi":"10.1017/s0963548322000190","DOIUrl":"https://doi.org/10.1017/s0963548322000190","url":null,"abstract":"\u0000 We find an asymptotic enumeration formula for the number of simple \u0000 \u0000 \u0000 \u0000$r$\u0000\u0000 \u0000 -uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random \u0000 \u0000 \u0000 \u0000$r$\u0000\u0000 \u0000 -uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random \u0000 \u0000 \u0000 \u0000$r$\u0000\u0000 \u0000 -uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"5 1","pages":"183-224"},"PeriodicalIF":0.9,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88826802","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}
The aim of this study is to show the anticorrosive properties of 1,3-dimethyl-7H-purine-2,6-dione on aluminium corrosion in 1M hydrochloric acid and to study the synergy effect between iodide ions and this molecule. Research of eco-friendly, low toxic and biodegradable corrosion inhibitors capable of protecting metals in order to support industrialists who spend large sums of money to replace their corroded equipment. The main objective is to study the anticorrosive properties of theophylline on aluminium corrosion in 1M HCl. The anticorrosive properties study of theophylline on aluminium corrosion in 1M HCl was evaluated using mass loss, Density Functional Theory at B3LYP/6-31G (d) and Quantitative Structure-Property Relationship methods. The results obtained show that theophylline inhibition efficiency increases with concentration but decreases with increasing temperature with a maximum value of 88% for 5.10-3M at T = 298 K. The result from absorption isotherms reveals that theophylline adsorbs to the aluminium surface according to the modified Langmuir isotherm. Adejo Ekwenshi's isotherm has shown that the molecule adsorption on aluminium is essentially of a physical nature. Thermodynamic adsorption and activation parameters were calculated and analyzed. A synergistic effect between the molecule studied and the iodide ions was found. Furthermore, global and local reactivity were analyzed through density functional theory calculations. Quantitative structure-property relationship methods model has been permitted to correlate experimental and theoretical inhibition efficiencies. Theophylline is an excellent aluminum inhibitor corrosion in the studied solution. Theoretical results were in agreement with experimental data. Finally to find the best set of parameters for modeling the inhibition efficiency.
{"title":"Anticorrosive properties of theophylline on aluminium corrosion in 1M HCl: Experimental, Computational assessment and synergistic effect iodide ions","authors":"Tigori Mougo André, Beda Reginald Henriette Blanche, Niamien Paulin Marius, Diabaté Donourou","doi":"10.2174/1877946811666210526152314","DOIUrl":"https://doi.org/10.2174/1877946811666210526152314","url":null,"abstract":"\u0000\u0000The aim of this study is to show the anticorrosive properties of 1,3-dimethyl-7H-purine-2,6-dione on aluminium corrosion in 1M hydrochloric acid and to study the synergy effect between iodide ions and this molecule. \u0000\u0000\u0000\u0000\u0000 Research of eco-friendly, low toxic and biodegradable corrosion inhibitors capable of protecting metals in order to support industrialists who spend large sums of money to replace their corroded equipment.\u0000\u0000\u0000\u0000\u0000 The main objective is to study the anticorrosive properties of theophylline on aluminium corrosion in 1M HCl.\u0000\u0000\u0000\u0000\u0000 The anticorrosive properties study of theophylline on aluminium corrosion in 1M HCl was evaluated using mass loss, Density Functional Theory at B3LYP/6-31G (d) and Quantitative Structure-Property Relationship methods.\u0000\u0000\u0000\u0000\u0000The results obtained show that theophylline inhibition efficiency increases with concentration but decreases with increasing temperature with a maximum value of 88% for 5.10-3M at T = 298 K. The result from absorption isotherms reveals that theophylline adsorbs to the aluminium surface according to the modified Langmuir isotherm. Adejo Ekwenshi's isotherm has shown that the molecule adsorption on aluminium is essentially of a physical nature. Thermodynamic adsorption and activation parameters were calculated and analyzed. A synergistic effect between the molecule studied and the iodide ions was found. Furthermore, global and local reactivity were analyzed through density functional theory calculations. Quantitative structure-property relationship methods model has been permitted to correlate experimental and theoretical inhibition efficiencies.\u0000\u0000\u0000\u0000\u0000Theophylline is an excellent aluminum inhibitor corrosion in the studied solution. Theoretical results were in agreement with experimental data.\u0000\u0000\u0000\u0000\u0000 Finally to find the best set of parameters for modeling the inhibition efficiency.\u0000\u0000","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48862710","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 : 2021-05-25DOI: 10.2174/1877946811666210525102725
Nitin Sharma, T. Alam, Ajay Kumar
This study aims at the Kinetic Study of Iodination of Propanone in Different Acidic Medium by using Colorimeter. The kinetic experimentation of iodination of propanone has been performed in presence of different acids such as sulphuric acid, hydrochloric acid and acetic acid. The rate law of the iodination of propanone in acidic medium is determined by observing the disappearance of the brownish yellow colour of iodine in aqueous solution. The objective of this study is the preparation of standard solution, determination of absorbance of iodine solution at λmax= 480 nm, Kinetic study of iodination of propanone with different acidic medium. The kinetic analysis of this reaction has been carried out by colorimetry. The extent of the reaction has been monitored by measuring the absorbance of the reaction mixture after a suitable time interval. The rate law expression is determined to be rate = k [〖"propanone]" 〗_0^0.728 [acid]0. The rates of iodination of propanone in presence of different acidic medium are H2SO4 > HCl > CH3COOH. The rate of the reaction is found to be independent of the concentration of iodine i.e. it is zero order reaction with respect to iodine. However, it depends on the concentration of propanone and acid catalyst. By using this experimental technique, the consumption of chemicals is very less.
{"title":"Kinetic Study of Iodination of Propanone in Different Acidic Medium by using Colorimeter","authors":"Nitin Sharma, T. Alam, Ajay Kumar","doi":"10.2174/1877946811666210525102725","DOIUrl":"https://doi.org/10.2174/1877946811666210525102725","url":null,"abstract":"\u0000\u0000This study aims at the Kinetic Study of Iodination of Propanone in Different Acidic Medium by using Colorimeter.\u0000\u0000\u0000\u0000The kinetic experimentation of iodination of propanone has been performed in presence of different acids such as sulphuric acid, hydrochloric acid and acetic acid. The rate law of the iodination of propanone in acidic medium is determined by observing the disappearance of the brownish yellow colour of iodine in aqueous solution.\u0000\u0000\u0000\u0000The objective of this study is the preparation of standard solution, determination of absorbance of iodine solution at λmax= 480 nm, Kinetic study of iodination of propanone with different acidic medium.\u0000\u0000\u0000\u0000The kinetic analysis of this reaction has been carried out by colorimetry. The extent of the reaction has been monitored by measuring the absorbance of the reaction mixture after a suitable time interval.\u0000\u0000\u0000\u0000The rate law expression is determined to be rate = k [〖\"propanone]\" 〗_0^0.728 [acid]0. The rates of iodination of propanone in presence of different acidic medium are H2SO4 > HCl > CH3COOH.\u0000\u0000\u0000\u0000The rate of the reaction is found to be independent of the concentration of iodine i.e. it is zero order reaction with respect to iodine. However, it depends on the concentration of propanone and acid catalyst.\u0000\u0000\u0000\u0000By using this experimental technique, the consumption of chemicals is very less.\u0000","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49586416","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 : 2021-05-09DOI: 10.2174/1877946811666210509032002
Laboni Das, Shashi P. Shukla, S. Chatterjee, A. Satpati, S. Adhikari
{"title":"From Reduction to Oxidation: pH Controlled Reaction of 1 Hydroxyethyl Radical with Caffeic Acid Analogues.","authors":"Laboni Das, Shashi P. Shukla, S. Chatterjee, A. Satpati, S. Adhikari","doi":"10.2174/1877946811666210509032002","DOIUrl":"https://doi.org/10.2174/1877946811666210509032002","url":null,"abstract":"","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41793767","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 : 2021-04-29DOI: 10.1017/S0963548321000109
Bodan Arsovski
Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - mln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ in ^l}}}$$ for certain c=c(m) and $$ in = in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.
将Alon, Linial, Meshulam的结果推广到阿贝尔群，证明了如果G是指数为m的有限阿贝尔群，且S是G的元素序列，使得S的任何子序列至少含有$$|S| - mln |G|$$个元素生成G，则S是G的可加基。我们还证明了对于c=c(m)和$$ in = in (m) < 1$$，任意l个G的生成集的加性张成包含一个至少为$$|G{|^{1 - c{ in ^l}}}$$的子群的余集;当G是向量空间时，我们使用概率方法给出更清晰的c(m)和$$ in (m)$$值;并对相关已知结果给出了新的证明。
{"title":"Additive bases via Fourier analysis","authors":"Bodan Arsovski","doi":"10.1017/S0963548321000109","DOIUrl":"https://doi.org/10.1017/S0963548321000109","url":null,"abstract":"\u0000 Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least \u0000 \u0000 \u0000 \u0000$$|S| - mln |G|$$\u0000\u0000 \u0000 elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least \u0000 \u0000 \u0000 \u0000$$|G{|^{1 - c{ in ^l}}}$$\u0000\u0000 \u0000 for certain c=c(m) and \u0000 \u0000 \u0000 \u0000$$ in = in (m) < 1$$\u0000\u0000 \u0000 ; we use the probabilistic method to give sharper values of c(m) and \u0000 \u0000 \u0000 \u0000$$ in (m)$$\u0000\u0000 \u0000 in the case when G is a vector space; and we give new proofs of related known results.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"102 1","pages":"930-941"},"PeriodicalIF":0.9,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74713713","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 : 2021-04-24DOI: 10.17074/CPC.V1I40.36287
Carlos Augusto de Oliveira Carvalhar
Anito é mais conhecido por ser um dos acusadores de Sócrates, mas este político teve outra proeza: sua sagacidade o fez criar um método obscuro de identificar os jurados e assim saber exatamente a quem subornar em um processo legal, ou seja, ele conseguia identificar quem eram os dikastaí que estariam agrupados no dikastḗrion específico que julgaria determinado caso em um tribunal. Com essa metodologia ele conseguiu se safar de uma condenação em 409 a.e.c., quando por sua falha Atenas perdeu Pilos (Aristóteles, Constituição de Atenas, 27.5). A façanha desse cidadão possivelmente foi a origem de dois novos verbos (dekázein e syndekázein) e o motivo de uma nova lei contra esse método (reportada no segundo Contra Estéfano de Demóstenes, 46.26). Além disso, duas décadas depois, o sistema por sorteio para a escolha de jurados foi totalmente reformulado, justamente para evitar esse esquema corrupto que ele empregou. Este artigo fará a exposição desses aspectos legais relacionados à escolha de jurados no século V, bem como abordará a corrupção e suborno que foi feita a partir desse método de Anito, discorrendo também sobre uma alteração proposta no século IV (que aparece na Assembleia das Mulheres de Aristófanes) e a terceira reformulação, apresentada em Aristóteles, Constituição de Atenas, 63-66. Também se explorará essa temática com a Apologia de Platão, uma vez que Anito, aquele que subornou jurados, é o mesmo agente que está processando Sócrates por corromper (diaphtheírein) os jovens, aproveitando-se assim a ambiguidade deste verbo, pois pode significar tanto a corrupção moral em sentido largo, como também aquela motivada por suborno.
{"title":"ANITO E O SUBORNO DE JURADOS (DEKÁZEIN) NOS PROCESSOS ATENIENSES","authors":"Carlos Augusto de Oliveira Carvalhar","doi":"10.17074/CPC.V1I40.36287","DOIUrl":"https://doi.org/10.17074/CPC.V1I40.36287","url":null,"abstract":"Anito é mais conhecido por ser um dos acusadores de Sócrates, mas este político teve outra proeza: sua sagacidade o fez criar um método obscuro de identificar os jurados e assim saber exatamente a quem subornar em um processo legal, ou seja, ele conseguia identificar quem eram os dikastaí que estariam agrupados no dikastḗrion específico que julgaria determinado caso em um tribunal. Com essa metodologia ele conseguiu se safar de uma condenação em 409 a.e.c., quando por sua falha Atenas perdeu Pilos (Aristóteles, Constituição de Atenas, 27.5). A façanha desse cidadão possivelmente foi a origem de dois novos verbos (dekázein e syndekázein) e o motivo de uma nova lei contra esse método (reportada no segundo Contra Estéfano de Demóstenes, 46.26). Além disso, duas décadas depois, o sistema por sorteio para a escolha de jurados foi totalmente reformulado, justamente para evitar esse esquema corrupto que ele empregou. Este artigo fará a exposição desses aspectos legais relacionados à escolha de jurados no século V, bem como abordará a corrupção e suborno que foi feita a partir desse método de Anito, discorrendo também sobre uma alteração proposta no século IV (que aparece na Assembleia das Mulheres de Aristófanes) e a terceira reformulação, apresentada em Aristóteles, Constituição de Atenas, 63-66. Também se explorará essa temática com a Apologia de Platão, uma vez que Anito, aquele que subornou jurados, é o mesmo agente que está processando Sócrates por corromper (diaphtheírein) os jovens, aproveitando-se assim a ambiguidade deste verbo, pois pode significar tanto a corrupção moral em sentido largo, como também aquela motivada por suborno.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46488987","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 : 2021-04-23DOI: 10.1017/s0963548321000572
J. Fox, Yufei Zhao
We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $epsilon>0$ there exists M such that every triangle-free graph G has an $epsilon$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $epsilon$ -approximate homomorphism is a map $V(G) to V(F)$ where all but at most $epsilon leftlvert{V(G)}rightrvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $epsilon^{-1}$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.
我们研究了三角形去除引理和它的几个变体之间的定量关系。其中一种变体，我们称之为无三角引理，说明对于每个$epsilon>0$存在M，使得每个无三角形图G在最多M个顶点上与无三角形图F具有$epsilon$ -近似同态(这里的$epsilon$ -近似同态是一个映射$V(G) to V(F)$，其中G的除最多$epsilon leftlvert{V(G)}rightrvert^2$边外的所有边都映射到F的边)。我们的结果的一个结果是，在无三角形引论中，最小可能的M比任何多项式中的指数增长更快在$epsilon^{-1}$。我们还证明了任意图的更一般的结果，以及有限域上的算术类似，其中边界接近最优。
{"title":"Removal lemmas and approximate homomorphisms","authors":"J. Fox, Yufei Zhao","doi":"10.1017/s0963548321000572","DOIUrl":"https://doi.org/10.1017/s0963548321000572","url":null,"abstract":"\u0000 We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each \u0000 \u0000 \u0000 \u0000$epsilon>0$\u0000\u0000 \u0000 there exists M such that every triangle-free graph G has an \u0000 \u0000 \u0000 \u0000$epsilon$\u0000\u0000 \u0000 -approximate homomorphism to a triangle-free graph F on at most M vertices (here an \u0000 \u0000 \u0000 \u0000$epsilon$\u0000\u0000 \u0000 -approximate homomorphism is a map \u0000 \u0000 \u0000 \u0000$V(G) to V(F)$\u0000\u0000 \u0000 where all but at most \u0000 \u0000 \u0000 \u0000$epsilon leftlvert{V(G)}rightrvert^2$\u0000\u0000 \u0000 edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in \u0000 \u0000 \u0000 \u0000$epsilon^{-1}$\u0000\u0000 \u0000 . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"69 1","pages":"721-736"},"PeriodicalIF":0.9,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80025932","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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