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"To which is now added a tabular series of complete decimal quotients ... " Errata at end of text. Book.
0000 - Gale Group | The Making of Modern World
Leonard Gillman, Melvin Henriksen,
... Following Kaplansky [4], we call such rings elementary divisor rings.A necessary condition is that 5 satisfy ... chain condition on ideals, and has no zero-divisors, then F is also sufficient.Helmer [3] showed ... of any two elements, one has a "largest" divisor that is relatively prime to the other(2)).Kaplansky[4] generalized this further by permitting zero-divisors, provided that they are all in the (Perlis- ... argument, we find that the condition on zero-divisors can be replaced by the hypothesis that 5 ... and sufficient condition that S be an elementary divisor ring.Along the way, we obtain a necessary ...
Tópico(s): Commutative Algebra and Its Applications
1956 - American Mathematical Society | Transactions of the American Mathematical Society
Without title page, being printed as a specimen of a work in course of publication. Book.
0000 - Gale Group | The Making of Modern World
Leonard Gillman, Melvin Henriksen,
... Following Kaplansky [4], we call such rings elementary divisor rings.A necessary condition is that 5 satisfy ... chain condition on ideals, and has no zero-divisors, then F is also sufficient.Helmer [3] showed ... of any two elements, one has a "largest" divisor that is relatively prime to the other(2)).Kaplansky[4] generalized this further by permitting zero-divisors, provided that they are all in the (Perlis- ... argument, we find that the condition on zero-divisors can be replaced by the hypothesis that 5 ... and sufficient condition that S be an elementary divisor ring.Along the way, we obtain a necessary ...
Tópico(s): Algebraic structures and combinatorial models
1956 - American Mathematical Society | Transactions of the American Mathematical Society
I.Introduction.The notion of non-degenerate divisors on an abelian variety was introduced first by Morikawa [2]Il, then after Weil proved that if Xis a non-digenerate divisor on an abelian variety, then the complete linear ... sufficiently large m [6].The latter property of divisors can be transferred to any variety and we shall call, in this paper, a divisor X on an algebraic variety "non-degenerate", if ... one can distinguish the class of non-degenerate divisors among the set of divisors.For this purpose ...
Tópico(s): Advanced Topics in Algebra
1960 - Hiroshima University - Department of Mathematics | Hiroshima Mathematical Journal
Michel Granger, David Mond, Mathias Schulze,
We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give ... the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric- and ... can only give rise to a linear free divisor if it has no (oriented or unoriented) cycles. We also deduce that the linear free divisors which appear in Sato and Kimura's list ... spaces are the only irreducible reductive linear free divisors. Furthermore, we show that all quiver linear free ...
Tópico(s): Advanced Algebra and Geometry
2010 - Wiley | Proceedings of the London Mathematical Society
Lisa Ann Vermeire DeMeyer, Larisa Greve, Arman Sabbaghi, Jonathan Wang,
Abstract The zero-divisor graph of a commutative semigroup with zero is the graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices adjacent ... elements is zero. New criteria to identify zero-divisor graphs are derived using both graph-theoretic and ... necessary to guarantee a graph is a zero-divisor graph. In addition, the removal or addition of vertices to a zero-divisor graph is investigated by using equivalence relations and ... with more than two triangles attached are zero-divisor graphs. Lastly, we classify several graph structures that ...
Tópico(s): Algebraic structures and combinatorial models
2010 - Taylor & Francis | Communications in Algebra
... smooth element, is called a large prime. A divisor, written by the sum of several smooth elements ... one large prime, is called an almost smooth divisor. By the use of the almost smooth divisor, Thériault improved this index calculus. In this paper, a divisor, written by the sum of several smooth elements ... large primes, is called a 2-almost smooth divisor. By use of the 2-almost smooth divisor, we are able to give more improvements. The ... following seven parts: 1) Preparing, 2) Collecting reduced divisors, 3) Making sufficiently large sets of almost smooth ...
Tópico(s): Polynomial and algebraic computation
2007 - Springer Science+Business Media | Japan Journal of Industrial and Applied Mathematics
Qianrong Tan, Zongbing Lin, Liu Liu,
... positive integers. The set S is called a divisor chain if there is a permutation σ of { ... that the set S consists of two coprime divisor chains if we can partition S as S = ... 2, where S 1 and S 2 are divisor chains and each element of S 1 is ... i , x j ) a of the greatest common divisor (GCD) of x i and x j as ... and among power LCM matrices on two coprime divisor chains, Linear Multilinear Algebra 58 (2010), pp. 659-- ... showed that if S consists of two coprime divisor chains and 1 ∈ S and a|b, then ( ... we assume that S consists of two coprime divisor chains and 1 ∉ S. We show the following ...
Tópico(s): graph theory and CDMA systems
2011 - Taylor & Francis | Linear and Multilinear Algebra
In this paper we study zero-divisor graphs of rings and semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less ... equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than ... to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one ... characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative ...
Tópico(s): Fuzzy and Soft Set Theory
2012 - World Scientific | International Journal of Algebra and Computation
... onto a smooth curve with the three given divisors as fibers, possibly multiple fibers (Theorem 2.1). ... fails completely if we have only two disjoint divisors in a homology class, as we will explain. ... projective variety that has a holomorphic section whose divisor of zeros is smooth.Can we compute the Betti numbers of this divisor in terms of the given variety and the ... we compute the Betti numbers of any smooth divisor in a smooth projective variety X in terms ... Betti numbers (and Hodge numbers) of a smooth divisor are determined by its cohomology class if the ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2000 - Institute of Mathematical Statistics | The Michigan Mathematical Journal
Given a divisor D D on a tropical curve Γ \Gamma , we show that reduced divisors define an integral affine map from the tropical ... an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into ... the tropical curves with a very ample canonical divisor. As an application of the reduced-divisor map, we show the existence of Weierstrass points ... discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve C C ...
Tópico(s): Cryptography and Residue Arithmetic
2013 - American Mathematical Society | Transactions of the American Mathematical Society
... a toric variety with enough invariant effective Cartier divisors in detail.Here a toric variety is said to have enough invariant effective Cartier divisors if, for each nonempty affine open subset stable ... of the torus, there exists an effective Cartier divisor whose support equals its complement.Both quasi-projective ... simplicial toric varieties have enough invariant effective Cartier divisors.In terms of the homogeneous coordinate ring, we ... of linear equivalence classes of invariant effective Cartier divisors.A toric variety X Δ is said to ...
Tópico(s): Nonlinear Waves and Solitons
1998 - Tohoku University | Tohoku Mathematical Journal
... splitting comes from a compatible splitting of a divisor on some birational model E ⊆ X ′ -→ X (ie, ... log canonical center), then we show that the divisor corresponding to the splitting on W is bounded ... others.We also show that difference between the divisor associated to the splitting and the divisorial part ... splittings., and hence φ corresponds to an effective divisor linearly equivalent to (1 -p e )K X .• We use D φ to denote this effective divisor corresponding to φ.• We write ∆ φ := 1 p ... this process can be reversed.Given an effective divisor ∆ such that∆ with ∆ φ = ∆.The choice of φ ...
Tópico(s): Advanced Algebra and Geometry
2017 - | ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
... power (xi, xj)a of the greatest common divisor of xi and xj as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (Sa). ... set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = ... integer and all Si (1 ⩽ i ⩽ k) are divisor chains such that (max(Si), max(Sj)) = gcd( ... set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, ... and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations ...
Tópico(s): Finite Group Theory Research
2010 - Elsevier BV | Applied Mathematics and Computation
... problem is by induction on k .If the divisor D on the curve C is nonspecial, then ... special, then it is contained in a canonical divisor.The canonical divisors are cut out on C by curves of ... reason we have developed a theory of generalized divisors on Gorenstein curves, which appears in § 1 .We ... of Fujita (1.6) telling when the canonical divisor on a Gorenstein curve is very ample.This result should simplify the beginning theory of divisors on K3 surfaces as given in the paper ... formulate and prove Noether's theorem for generalized divisors on irreducible plane curves.This is done in § ...
Tópico(s): Algebraic structures and combinatorial models
1986 - Duke University Press | Kyoto journal of mathematics
The Neron-Severi group of divisor classes modulo algebraic equivalence on a smooth algebraic surface is often not difficult to calculate, and has classically been studied as ... more difficult problem is the determination of those divisor classes which can be represented by effective divisors; these divisor classes form a monoid contained in the Neron- ... whole Neron-Severi group, the monoid of effective divisor classes may or may not be finitely generated, ... bundle techniques to describe the monoid of effective divisor classes modulo algebraic equivalence on a complex ruled ...
Tópico(s): Polynomial and algebraic computation
2002 - Mathematical Sciences Publishers | Pacific Journal of Mathematics
The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and the k -zero-divisor hypergraph of a commutative ring is a nice ... long and detailed, any reader familiar with zero-divisors will be able to read through the exposition ... R , k ) , the set of all k -zero-divisors in R , is associated to R , where each ... Z ( R , k ) that satisfies the k -zero-divisor condition is an edge in H k ( R ) . ... where Z ′ is the set of all zero divisors of R which are not k -zero-divisors ...
Tópico(s): Algebraic structures and combinatorial models
2007 - Hindawi Publishing Corporation | International Journal of Mathematics and Mathematical Sciences
Gavril Farkas, Rahul Pandharipande,
The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in $\overline{{\mathcal{M}}}_{g,n}$ which includes the space of canonical divisors as an open subset. The theory leads to ... the closures of the moduli spaces of canonical divisors. In case the differentials have at least one ... meromorphic case), the moduli spaces of twisted canonical divisors on genus $g$ curves are of pure codimension $ ...
Tópico(s): Analytic Number Theory Research
2016 - Cambridge University Press | Journal of the Institute of Mathematics of Jussieu
Hiren Maharaj, Gretchen L. Matthews,
Given a divisor A of a function field, there is a unique divisor of minimum degree that defines the same vector ... functions as A and there is a unique divisor of maximum degree that defines the same vector space of rational differentials as A. These divisors are called the floor and the ceiling of ... both the floor and the ceiling of a divisor. The floor and the ceiling of a divisor give new bounds for the minimum distance of ... codes. The floor and the ceiling of a divisor supported by collinear places of the Hermitian function ...
Tópico(s): Finite Group Theory Research
2005 - Elsevier BV | Finite Fields and Their Applications
Stephen J. Sangwine, Daniel Alfsmann,
... nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in ... formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in ... present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares ... give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement ...
Tópico(s): Advanced Topics in Algebra
2010 - Birkhäuser | Advances in Applied Clifford Algebras
... positive integers. The set S is called a divisor chain if there is a permutation σ on { ... the set S consists of finitely many coprime divisor chains if there is a positive integer k ... S=S1∪...∪Sk, where all the Si are divisor chains and each element of Si is coprime ... power (xi,xj)a of the greatest common divisor of xi and xj as its (i,j)-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (Sa). ... b and S consists of finitely many coprime divisor chains with 1∈S, then in the ring ...
Tópico(s): Algebraic structures and combinatorial models
2012 - Elsevier BV | Linear Algebra and its Applications
... is a rational integer, the of 0. A divisor to the indices of every integer of the ... the field. We shall use the term common divisor.t The existence of divisors was first established in 1871 by Dedekindt who ... that a rational prime p can be a divisor of a field K if and only if ... r(f) is the number of prime ideal divisors of p of degree f, and g(f) ... called index form for p to be a divisor. In 1907, Bauer? showed that if p <n ... of nth degree in which p is a divisor. Von Zylinsky** has
Tópico(s): Advanced Topics in Algebra
1930 - American Mathematical Society | Transactions of the American Mathematical Society
... positive integers. The set S is called a divisor chain if there is a permutation σ of { ... that the set S consists of two coprime divisor chains if we can partition S as S = ... 2, where S 1 and S 2 are divisor chains and each element of S 1 is ... xj ) a of the GCD (GCD: greatest common divisor) of xi and xj as its i, j- ... results: assume that S consists of two coprime divisor chains and 1∈S. We first show that ...
Tópico(s): Advanced Topics in Algebra
2009 - Taylor & Francis | Linear and Multilinear Algebra