Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some nece...Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.展开更多
we have discussed structures of Abelian group G by order |A(G) |of automoorphism group and have obtained all types of finite Abelian grooup G when the order of A(G) equals 27pq(p,q are odd primmes).
Canonical differential calculi are defined for finitely generated Abelian groups with involutions existing consistently. Two such the canonical calculi are presented. Fermionic representations for canonical calculi ar...Canonical differential calculi are defined for finitely generated Abelian groups with involutions existing consistently. Two such the canonical calculi are presented. Fermionic representations for canonical calculi are defined based on quantized calculi. Fermionic representations for aforementioned two canonical calculi are searched out.展开更多
In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain...In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain that the topological entropy of a transitive,almost Banach-mean equicontinuous dynamical system of Abelian group action is zero.As an application of our main result,we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.展开更多
Paper considers the calculation of the values of Gibbs derivatives on finite Abelian groups. The calculation procedure is based upon the decision diagram representation of functions defined on finite Abelian groups. A...Paper considers the calculation of the values of Gibbs derivatives on finite Abelian groups. The calculation procedure is based upon the decision diagram representation of functions defined on finite Abelian groups. Approach permits processing of large functions.展开更多
Tilings of p-groups are closely associated with error-correcting codes. In [1], M. Dinitz, attempting to generalize full-rank tilings of ?Zn2??to arbitrary finite abelian groups, was able to show that if p ≥5, ...Tilings of p-groups are closely associated with error-correcting codes. In [1], M. Dinitz, attempting to generalize full-rank tilings of ?Zn2??to arbitrary finite abelian groups, was able to show that if p ≥5, then?Znp? admits full-rank tiling and left the case p=3, as an open question. The result proved in this paper the settles of the question for the case p=3.展开更多
If a finite abelian group G is a direct product of its subsets such that G = A1···Ai···An, G is said to have the Hajos-n-proprty if it follows that one of these subsets, say Ai is periodi...If a finite abelian group G is a direct product of its subsets such that G = A1···Ai···An, G is said to have the Hajos-n-proprty if it follows that one of these subsets, say Ai is periodic, meaning that there exists a nonidentity element g in G such that gAi = Ai . Using some properties of cyclotomic polynomials, we will show that the cyclic groups of orders pα and groups of type (p2,q2) and (pα,pβ) where p and q are distinct primes and α, β integers ≥ 1 have this property.展开更多
The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth.Finite groups where the quotient space are Enriques surfaces are known.In this paper,by analyzing effec...The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth.Finite groups where the quotient space are Enriques surfaces are known.In this paper,by analyzing effective divisors on smooth rational surfaces,the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth.In particular,he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor.Furthermore,he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface.Subsequently,he studies the same theme for Enriques surfaces.展开更多
Let t（G） be the number of unitary factors of finite abelian group G. In this paper we prove T（x）=∑<sub>(</sub>G≤（）t（G） =main terms+O(x<sup>(</sup>（1+2k）/（3+4k）for any exponent pa...Let t（G） be the number of unitary factors of finite abelian group G. In this paper we prove T（x）=∑<sub>(</sub>G≤（）t（G） =main terms+O(x<sup>(</sup>（1+2k）/（3+4k）for any exponent pair （k, 1/2+2K）. which improves on the exponent 9/25 obtained by Xiaodong Cao and the author.展开更多
A finite group is said to be weakly separable if every algebraic isomorphism between two 5-rings over this group is induced by a combinatorial isomorphism.We prove that every abelian weakly separable group only belong...A finite group is said to be weakly separable if every algebraic isomorphism between two 5-rings over this group is induced by a combinatorial isomorphism.We prove that every abelian weakly separable group only belongs to one of several explicitly given families.展开更多
Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The followi...Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained： if G∈F, f（ ∞ ） include f（p）, f（p） ≠φ for each p∈π, and A has no nonzero F central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F central ZG-factors.展开更多
Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first invest...Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first investigated the distribution of the values of finite non-isomorphicAbelian groups in short intervals. E. Kratzel reduced the problem to estimate theerror term △(1, 2, 3;x) in the three-dimensional multiplicative problem, and furtherimproved Ivice’s result.展开更多
Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA<sub>k</sub>（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x<sup>581/1744</sup>logx...Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA<sub>k</sub>（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x<sup>581/1744</sup>logx=x<sup>0.33314…</sup>logx as x→∞,it was proved by A,Ivic thatA<sub>k</sub>（x, h）=（d<sub>k</sub>+o（1））h, （1）whered<sub>k</sub>=sum from n=1 to ∞ (1/2πn integral from n=-π to π(e<sup>ikt g<sub>t</sub>（n）dt≥0</sup>)),g<sub>t</sub>（n）=sum from n=d/n to (μ（n/d）e<sup>ita</sup>（d）).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x<sup>877/2653</sup>（logx）<sup>c</sup>=x<sup>0.3305…</sup>（logx）<sup>c</sup>,then Eq.（1）is true, where C is a computable constant. Based on the estimate for △（1, 2, 2;x） in Ref.[2] and elementary discussion, thisnote proves the following theorem, which gives an improvement to the problem.展开更多
It is well known that K<sub>0</sub>R（?）Z（?）（?）<sub>0</sub>R, where R is a commutative ring. So the Grothendieck group of R can be given by the reduced group （?）<sub>0</sub>R...It is well known that K<sub>0</sub>R（?）Z（?）（?）<sub>0</sub>R, where R is a commutative ring. So the Grothendieck group of R can be given by the reduced group （?）<sub>0</sub>R. On the other hand, linear representations of groups can be seen as the finitely generated projective modules over group rings. Thus, it is very useful to study the properties of reduced groups of group rings.展开更多
基金National Natural Science Foundation of China(No.11671258)。
文摘Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.
文摘we have discussed structures of Abelian group G by order |A(G) |of automoorphism group and have obtained all types of finite Abelian grooup G when the order of A(G) equals 27pq(p,q are odd primmes).
基金Climb-Up (Pan Deng) Project of Department of Science and Technology of China，国家自然科学基金，Doctoral Programme Foundation of Institution of Higher Education of China
文摘Canonical differential calculi are defined for finitely generated Abelian groups with involutions existing consistently. Two such the canonical calculi are presented. Fermionic representations for canonical calculi are defined based on quantized calculi. Fermionic representations for aforementioned two canonical calculi are searched out.
基金supported by NSF of China(11671057)NSF of Chongqing(cstc2020jcyj-msxmX0694)the Fundamental Research Funds for the Central Universities(2018CDQYST0023).
文摘In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain that the topological entropy of a transitive,almost Banach-mean equicontinuous dynamical system of Abelian group action is zero.As an application of our main result,we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.
文摘Paper considers the calculation of the values of Gibbs derivatives on finite Abelian groups. The calculation procedure is based upon the decision diagram representation of functions defined on finite Abelian groups. Approach permits processing of large functions.
文摘Tilings of p-groups are closely associated with error-correcting codes. In [1], M. Dinitz, attempting to generalize full-rank tilings of ?Zn2??to arbitrary finite abelian groups, was able to show that if p ≥5, then?Znp? admits full-rank tiling and left the case p=3, as an open question. The result proved in this paper the settles of the question for the case p=3.
文摘If a finite abelian group G is a direct product of its subsets such that G = A1···Ai···An, G is said to have the Hajos-n-proprty if it follows that one of these subsets, say Ai is periodic, meaning that there exists a nonidentity element g in G such that gAi = Ai . Using some properties of cyclotomic polynomials, we will show that the cyclic groups of orders pα and groups of type (p2,q2) and (pα,pβ) where p and q are distinct primes and α, β integers ≥ 1 have this property.
文摘The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth.Finite groups where the quotient space are Enriques surfaces are known.In this paper,by analyzing effective divisors on smooth rational surfaces,the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth.In particular,he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor.Furthermore,he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface.Subsequently,he studies the same theme for Enriques surfaces.
基金Supported by MCME and Natural Science Foundation of Shandong Province(Grant No. Q98A02110)
文摘Let t（G） be the number of unitary factors of finite abelian group G. In this paper we prove T（x）=∑<sub>(</sub>G≤（）t（G） =main terms+O(x<sup>(</sup>（1+2k）/（3+4k）for any exponent pair （k, 1/2+2K）. which improves on the exponent 9/25 obtained by Xiaodong Cao and the author.
基金Supported by the Russian Foundation for Basic Research(project 18-01-00752).
文摘A finite group is said to be weakly separable if every algebraic isomorphism between two 5-rings over this group is induced by a combinatorial isomorphism.We prove that every abelian weakly separable group only belongs to one of several explicitly given families.
文摘Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained： if G∈F, f（ ∞ ） include f（p）, f（p） ≠φ for each p∈π, and A has no nonzero F central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F central ZG-factors.
文摘Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first investigated the distribution of the values of finite non-isomorphicAbelian groups in short intervals. E. Kratzel reduced the problem to estimate theerror term △(1, 2, 3;x) in the three-dimensional multiplicative problem, and furtherimproved Ivice’s result.
文摘Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA<sub>k</sub>（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x<sup>581/1744</sup>logx=x<sup>0.33314…</sup>logx as x→∞,it was proved by A,Ivic thatA<sub>k</sub>（x, h）=（d<sub>k</sub>+o（1））h, （1）whered<sub>k</sub>=sum from n=1 to ∞ (1/2πn integral from n=-π to π(e<sup>ikt g<sub>t</sub>（n）dt≥0</sup>)),g<sub>t</sub>（n）=sum from n=d/n to (μ（n/d）e<sup>ita</sup>（d）).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x<sup>877/2653</sup>（logx）<sup>c</sup>=x<sup>0.3305…</sup>（logx）<sup>c</sup>,then Eq.（1）is true, where C is a computable constant. Based on the estimate for △（1, 2, 2;x） in Ref.[2] and elementary discussion, thisnote proves the following theorem, which gives an improvement to the problem.
文摘It is well known that K<sub>0</sub>R（?）Z（?）（?）<sub>0</sub>R, where R is a commutative ring. So the Grothendieck group of R can be given by the reduced group （?）<sub>0</sub>R. On the other hand, linear representations of groups can be seen as the finitely generated projective modules over group rings. Thus, it is very useful to study the properties of reduced groups of group rings.
