Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2))...Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2)).Suppose x≥(q_(1)q_(2))^(1+ε),1≤r_i≤q_i,and(r_i,q_i)=1(i=1,2).This paper studies the power moments and sign changes ofΔ(x;r_(1),q_(1),r_(2),q_(2)).We prove that for sufficiently large constant C,Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))changes sign in the interval[T,T+C√T]for any large T.Meanwhile,we show that for small constants c and c,there exist infinitely many subintervals of length c√log^(-7)T in[T,2T]where±Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))>cx^(1/4)always holds.展开更多
We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left ...We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.展开更多
In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this ...In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F<sub>0</sub>) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−1<s , witch is absurd. To reach such a result, we suppose first that (F<sub>0</sub>) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F<sub>0</sub>): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=0 ( b 2 , b ′ 2 , b ″ 2 ) if λ=1;From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article. .展开更多
In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defin...In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if xRy = 0. We show that the following three conditions on an FIC ring R are equivalent： （1） χ（R） is finite; （2） ω（R） is finite; （3） Nil＊ R is finite where Nil.R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Г＊ （R）.展开更多
Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible...Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let<sub></sub><sub></sub> (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .展开更多
Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph correspond...Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).展开更多
In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those gra...In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.展开更多
This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study ...This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.展开更多
Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ...Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .展开更多
Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a an...Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.展开更多
We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that...We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.展开更多
Abstract This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polynomials which are subject to cer...Abstract This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polynomials which are subject to certain constrains at the endpoints of a prescribed parametric interval and minimized in a certain sense. This result can be combined with subdivision technique to obtain a continuous piecewise approximation to a rational curve.展开更多
We propose the following conjecture on(σ)n the sum-of-divisors function:log(e'nlog logn-σ(n))/log(e'nlog log n)will increase strictly and converge to1 when n runs from the colossally abundant numbers to infi...We propose the following conjecture on(σ)n the sum-of-divisors function:log(e'nlog logn-σ(n))/log(e'nlog log n)will increase strictly and converge to1 when n runs from the colossally abundant numbers to infinity This conjecture is a sufficient condition for the and converge to1 when n nuns from the colossally abundant numbers to infinity.This conjecture is a sufficient condition for the Ricemann hypothesis by Robin's theorem,and it is confirmed for n from10^(4 )up to 10^(103078) Further,we present two additional Riemann hypothesis by Robin's theorem,and it is confirmed forn from 10^(4) up to 10^(103078) Further,we present two additional conjectures that are related to Robin's theorem.展开更多
Let D be an increasing sequence of positive integers, and consider the divisor functions：d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1,where [d,δ]=1.c.m.（d,δ）. A probabilistic argumen...Let D be an increasing sequence of positive integers, and consider the divisor functions：d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1,where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.展开更多
Let X be a compact K?hler manifold and D be a simple normal crossing divisor. If D is the support of some effective q-ample divisor, we show H^i(X, ?_X^j (log D)) = 0, for i + j > n + q.
In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompo...In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices,and prove that the complete lower continuous,principal divisor lattices have irredundant join irreducible decompositions.In the end,we show the descriptions of lattices that have unique(resp.replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.展开更多
Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line.This article is a survey of recent development...Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line.This article is a survey of recent developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.展开更多
In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry.In particular,we generalizeMaulik-Pandharipande...In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry.In particular,we generalizeMaulik-Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence.Then,we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.展开更多
A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgra...A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgraph of G induced on the vertex set V（G） ／ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G=Г（R）, which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G＊ has at least two connected components. We prove that the diameter of the induced graph G＊ is two if Z（R）2 ≠{0}, Z（R）3 = {0} and Gc is connected. We determine the structure of R which has two distinct nonadjacent vertices a, fl C Z（R）＊／{c} such that the ideal [N（a）N（β）]{0} is generated by only one element of Z（R）＊／{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kn with some end vertices adjacent to a single vertex of Kn.展开更多
Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos^2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the seque...Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos^2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y^3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.展开更多
基金supported by the Talent Fund of Beijing Jiaotong University(No.2020RC012)NSFC(No.11871295),supported by NSFC(No.11971476),supported by NSFC(No.12071421)。
文摘Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2)).Suppose x≥(q_(1)q_(2))^(1+ε),1≤r_i≤q_i,and(r_i,q_i)=1(i=1,2).This paper studies the power moments and sign changes ofΔ(x;r_(1),q_(1),r_(2),q_(2)).We prove that for sufficiently large constant C,Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))changes sign in the interval[T,T+C√T]for any large T.Meanwhile,we show that for small constants c and c,there exist infinitely many subintervals of length c√log^(-7)T in[T,2T]where±Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))>cx^(1/4)always holds.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1107109711101217)
文摘We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.
文摘In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F<sub>0</sub>) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−1<s , witch is absurd. To reach such a result, we suppose first that (F<sub>0</sub>) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F<sub>0</sub>): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=0 ( b 2 , b ′ 2 , b ″ 2 ) if λ=1;From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article. .
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137134311161006+4 种基金1166101411171142)the Guangxi Science Research and Technology Development Project(Grant No.1599005-2-13)the Scientic Research Fund of Guangxi Education Department(Grant No.KY2015ZD075)the Natural Science Foundation of Guangxi(Grant No.2016GXSFDA380017)
文摘In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if xRy = 0. We show that the following three conditions on an FIC ring R are equivalent： （1） χ（R） is finite; （2） ω（R） is finite; （3） Nil＊ R is finite where Nil.R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Г＊ （R）.
文摘Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let<sub></sub><sub></sub> (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .
文摘Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).
基金Supported by the Natural Sciences Foundation of Guangxi Province(0575052, 0640070)Supported by the Innovation Project of Guangxi Graduate Education(2006106030701M05)Supported by the Scientific Research Foundation of Guangxi Educational Committee(200707LX233
文摘In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.
文摘This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.
文摘Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .
文摘Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.
基金Partially supported by the NSF (10071035) of China.
文摘We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.
文摘Abstract This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polynomials which are subject to certain constrains at the endpoints of a prescribed parametric interval and minimized in a certain sense. This result can be combined with subdivision technique to obtain a continuous piecewise approximation to a rational curve.
文摘We propose the following conjecture on(σ)n the sum-of-divisors function:log(e'nlog logn-σ(n))/log(e'nlog log n)will increase strictly and converge to1 when n runs from the colossally abundant numbers to infinity This conjecture is a sufficient condition for the and converge to1 when n nuns from the colossally abundant numbers to infinity.This conjecture is a sufficient condition for the Ricemann hypothesis by Robin's theorem,and it is confirmed for n from10^(4 )up to 10^(103078) Further,we present two additional Riemann hypothesis by Robin's theorem,and it is confirmed forn from 10^(4) up to 10^(103078) Further,we present two additional conjectures that are related to Robin's theorem.
文摘Let D be an increasing sequence of positive integers, and consider the divisor functions：d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1,where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.
文摘Let X be a compact K?hler manifold and D be a simple normal crossing divisor. If D is the support of some effective q-ample divisor, we show H^i(X, ?_X^j (log D)) = 0, for i + j > n + q.
基金supported by National Natural Science Foundation of China (Grant No.10671138)
文摘In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices,and prove that the complete lower continuous,principal divisor lattices have irredundant join irreducible decompositions.In the end,we show the descriptions of lattices that have unique(resp.replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.
文摘Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line.This article is a survey of recent developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.
文摘In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry.In particular,we generalizeMaulik-Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence.Then,we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.
基金Supported by National Natural Science Foundation of China （Grant No. 10671122） the first author is supported by Youth Foundation of Shanghai （Grant No. sdl10017） and also partly supported by Natural Science Foundation of Shanghai （Grant No. 10ZR1412500） the second author is partly supported by STCSM （Grant No. 09XD1402500）
文摘A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgraph of G induced on the vertex set V（G） ／ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G=Г（R）, which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G＊ has at least two connected components. We prove that the diameter of the induced graph G＊ is two if Z（R）2 ≠{0}, Z（R）3 = {0} and Gc is connected. We determine the structure of R which has two distinct nonadjacent vertices a, fl C Z（R）＊／{c} such that the ideal [N（a）N（β）]{0} is generated by only one element of Z（R）＊／{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kn with some end vertices adjacent to a single vertex of Kn.
基金supported by the Government of the Russian Federation (Grant No. 14.Z50.31.0030)
文摘Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos^2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y^3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.
