Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.
设 K 是一个代数闭域的特征零，K[x,y] 是变量 x, y 的多项式环，W2(K) 是 K[x,y] 上所有 K 派生的李代数。如果存在 f∈K[x,y]，使得对于任意 h∈K[x,y]，D(h)=det J(f,h)（此处 J(f,h) 是 f 和 h 的雅各布矩阵），则导数 D∈W2(K) 称为雅各布导数。这样的推导用 Df 表示。Df 在 K[x,y] 中的内核是子代数 K[p]，其中 p=p(x,y) 是最小度的多项式，使得 f(x,y)=φ(p(x,y) 对于某个 φ(t)∈K[t] 。设 C=CW2(K)(Df) 是 Df 在 W2(K) 中的中心子。我们证明 C 是 K[p] 上阶 1 或阶 2 的自由 K[p] 模块，并指出了成为阶 2 模块的标准。我们利用这些结果得到了一类可积分自洽微分方程系统。
{"title":"Centralizers of Jacobian derivations","authors":"D. Efimov, A. Petravchuk, M. Sydorov","doi":"10.12958/adm2186","DOIUrl":"https://doi.org/10.12958/adm2186","url":null,"abstract":"Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139282635","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}
It is examined finite state automorphisms of regular rooted trees constructed in [6] to represent groups GL(n,Z). The number of states of automorphisms that correspond to elementary matrices i computed. Using the representation of GL(2,Z) over an alphabet of size 4 a finite state representation of the freegroup of rank 2 over binary alphabet is constructed.
{"title":"On a finite state representation of GL(n,Z)","authors":"A. Oliynyk, V. Prokhorchuk","doi":"10.12958/adm2158","DOIUrl":"https://doi.org/10.12958/adm2158","url":null,"abstract":"It is examined finite state automorphisms of regular rooted trees constructed in [6] to represent groups GL(n,Z). The number of states of automorphisms that correspond to elementary matrices i computed. Using the representation of GL(2,Z) over an alphabet of size 4 a finite state representation of the freegroup of rank 2 over binary alphabet is constructed.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139343019","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}
Recent development in the classification of p-groups often concentrate on the coclass graph G(p,r) associated with the finitep-groups coclassr, specially on periodicity results on these graphs. In particular, the structure of the subgraph inducedby ‘skeleton groups’ is of notable interest. Given their importance, in this paper, we investigate periodicity results of skeleton groups. Our results concentrate on the skeleton groups in G(p,1). We find a family of skeleton groups in G(7,1) whose 6-step parent is not aperiodic parent. This shows that the periodicity results available inthe current literature for primes p≡5 mod 6 do not hold for the primes p≡1 mod 6. We also improve a known periodicity result in a special case of skeleton groups.
最近在p群分类方面的进展通常集中在与有限群共类相关的协类图G(p,r)上，特别是在这些图的周期性结果上。特别是，由“骨架群”引起的子图结构值得注意。鉴于它们的重要性，本文研究了骨架群的周期性结果。我们的结果集中在G(p,1)中的骨架群上。我们在G(7,1)中发现了一个六步亲本不是非周期亲本的骨架群家族。这表明目前文献中关于素数p≡5 mod 6的周期性结果并不适用于素数p≡1 mod 6。我们还改进了一个已知的骨架群的特殊情况下的周期性结果。
{"title":"Orbit isomorphic skeleton groups","authors":"Subhrajyoti Saha","doi":"10.12958/adm1886","DOIUrl":"https://doi.org/10.12958/adm1886","url":null,"abstract":"Recent development in the classification of p-groups often concentrate on the coclass graph G(p,r) associated with the finitep-groups coclassr, specially on periodicity results on these graphs. In particular, the structure of the subgraph inducedby ‘skeleton groups’ is of notable interest. Given their importance, in this paper, we investigate periodicity results of skeleton groups. Our results concentrate on the skeleton groups in G(p,1). We find a family of skeleton groups in G(7,1) whose 6-step parent is not aperiodic parent. This shows that the periodicity results available inthe current literature for primes p≡5 mod 6 do not hold for the primes p≡1 mod 6. We also improve a known periodicity result in a special case of skeleton groups.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136302937","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}
For a group G, FG denotes the set of all non-empty finite subsets of G. We extend the finitary coarse structure of G from G×G to FG×FG and say that a macro-uniform mapping f:FG→FG (resp. f:[G]2→G) is a finitary selector (resp. 2-selector) of G if f(A)∈A for each A ∈ FG (resp. A∈[G]2). Weprove that a group G admits a finitary selector if and only if G admits a 2-selector and if and only if G is a finite extension of an infinite cyclic subgroup or G is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures.
{"title":"Coarse selectors of groups","authors":"Igor Protasov","doi":"10.12958/adm2127","DOIUrl":"https://doi.org/10.12958/adm2127","url":null,"abstract":"For a group G, FG denotes the set of all non-empty finite subsets of G. We extend the finitary coarse structure of G from G×G to FG×FG and say that a macro-uniform mapping f:FG→FG (resp. f:[G]2→G) is a finitary selector (resp. 2-selector) of G if f(A)∈A for each A ∈ FG (resp. A∈[G]2). Weprove that a group G admits a finitary selector if and only if G admits a 2-selector and if and only if G is a finite extension of an infinite cyclic subgroup or G is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783919","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}
Noncommutative cryptography is based on applications of algebraic structures like noncommutative groups, semigroups, and noncommutative rings. Its intersection with Multivariate cryptography contains studies of cryptographic applications of subsemigroups and subgroups of affine Cremona semigroups defined over finite commutative rings. Efficiently computed homomorphisms between stable subsemigroups of affine Cremona semigroups can be used in tame homomorphisms protocols schemes and their inverse versions. The implementation scheme with the sequence of subgroups of affine Cremona group that defines the projective limit was already suggested. We present the implementation of another scheme that uses two projective limits which define two different infinite groups and the homomorphism between them. The security of the corresponding algorithm is based on complexity of the decomposition problem for an element of affine Cremona semigroup into a product of given generators. These algorithms may be used in postquantum technologies.
{"title":"On new protocols of Noncommutative Cryptography in terms of homomorphism of stable multivariate transformation groups","authors":"Vasyl Ustimenko, Michał Klisowski","doi":"10.12958/adm1523","DOIUrl":"https://doi.org/10.12958/adm1523","url":null,"abstract":"Noncommutative cryptography is based on applications of algebraic structures like noncommutative groups, semigroups, and noncommutative rings. Its intersection with Multivariate cryptography contains studies of cryptographic applications of subsemigroups and subgroups of affine Cremona semigroups defined over finite commutative rings. Efficiently computed homomorphisms between stable subsemigroups of affine Cremona semigroups can be used in tame homomorphisms protocols schemes and their inverse versions. The implementation scheme with the sequence of subgroups of affine Cremona group that defines the projective limit was already suggested. We present the implementation of another scheme that uses two projective limits which define two different infinite groups and the homomorphism between them. The security of the corresponding algorithm is based on complexity of the decomposition problem for an element of affine Cremona semigroup into a product of given generators. These algorithms may be used in postquantum technologies.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136304805","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}
We investigate circular planar nearrings con-structed from finite fields as well the complex number field using amultiplicative subgroup of order k, and characterize the overlaps of the basic graphs which arise in the associated 2-designs.
{"title":"Overlaps in field generated circular planar nearrings","authors":"Wen-Fong Ke, Hubert Kiechle","doi":"10.12958/adm2130","DOIUrl":"https://doi.org/10.12958/adm2130","url":null,"abstract":"We investigate circular planar nearrings con-structed from finite fields as well the complex number field using amultiplicative subgroup of order k, and characterize the overlaps of the basic graphs which arise in the associated 2-designs.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136305693","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}
The main point of our research is to obtain the estimates for Kloosterman sums K(α, β; h, q; k) considered on the ellipse bound for the case of the integer rational moduleq and forsome natural number k with conditions (α, q)=(β, q)=1 on the integer numbers of imaginary quadratic field. These estimates can be used to construct the asymptotic formulas for the sum of divisors function τℓ(α)forℓ= 2,3, . . . over the ring of integer elements of imaginary quadratic field in arithmetic progression.
{"title":"The Kloosterman sums on the ellipse","authors":"Sergey Varbanets, Yakov Vorobyov","doi":"10.12958/adm2048","DOIUrl":"https://doi.org/10.12958/adm2048","url":null,"abstract":"The main point of our research is to obtain the estimates for Kloosterman sums K(α, β; h, q; k) considered on the ellipse bound for the case of the integer rational moduleq and forsome natural number k with conditions (α, q)=(β, q)=1 on the integer numbers of imaginary quadratic field. These estimates can be used to construct the asymptotic formulas for the sum of divisors function τℓ(α)forℓ= 2,3, . . . over the ring of integer elements of imaginary quadratic field in arithmetic progression.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136304818","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}
Let R be a commutative ring with unity. The R-algebra G=G(A,M,N,B) is a generalized matrix algebra defined by the Morita context (A,B,M,N, ξMN,ΩNM). In this article, westudy generalized Lie derivation and show that every generalized Lie derivation on a generalized matrix algebra has the standard form under certain assumptions.
{"title":"A study on generalized matrix algebras having generalized Lie derivations","authors":"Aisha Jabeen, Musheer Ahmad, Adnan Abbasi","doi":"10.12958/adm1722","DOIUrl":"https://doi.org/10.12958/adm1722","url":null,"abstract":"Let R be a commutative ring with unity. The R-algebra G=G(A,M,N,B) is a generalized matrix algebra defined by the Morita context (A,B,M,N, ξMN,ΩNM). In this article, westudy generalized Lie derivation and show that every generalized Lie derivation on a generalized matrix algebra has the standard form under certain assumptions.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136305463","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}
We study sets of triple points of Böröczky’s arrangements of lines in the context of the containment problem proposed by Harbourne and Huneke. We show that in the class of those arrangements, the smallest counterexample to the containment I(3) ⊂ I2 is obtained when the number of lines is equal to 12.
{"title":"On the containment I(3) ⊂ I2 and configurationsof triple points in Böröczky line arrangements","authors":"Jakub Kabat","doi":"10.12958/adm1959","DOIUrl":"https://doi.org/10.12958/adm1959","url":null,"abstract":"We study sets of triple points of Böröczky’s arrangements of lines in the context of the containment problem proposed by Harbourne and Huneke. We show that in the class of those arrangements, the smallest counterexample to the containment I(3) ⊂ I2 is obtained when the number of lines is equal to 12.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136305711","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}
For every prime p it is shown that a wide class of HNN extensions of free abelian groups admits faithful representation by finite p-automata.
对于每一个素数p，证明了自由阿贝尔群的广泛的HNN扩展允许用有限p自动机忠实地表示。
{"title":"On exponentiation, p-automata and HNN extensions of free abelian groups","authors":"Andriy Oliynyk, Veronika Prokhorchuk","doi":"10.12958/adm2132","DOIUrl":"https://doi.org/10.12958/adm2132","url":null,"abstract":"For every prime p it is shown that a wide class of HNN extensions of free abelian groups admits faithful representation by finite p-automata.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136305469","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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