Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

On the existence of non-special divisors of degree $g$ and $g-1$ in algebraic function fields over $\F_q$

Abstract: We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $g\geq 1$ defined over a finite field $\F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $g\geq 2$ if $q\geq 3$ and that there always exists a non-special divisor of degree $g-1\geq 1$ if $q\geq 4$. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension $\F_{q^n}$ of $\F_q$, when $q=2^r\geq 16$.

The group ring of Q/Z and an application of a divisor problem

Abstract: First we prove some elementary but useful identities in the group ring of Q/Z. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest.

Matrix factorizations over elementary divisor domains

Abstract: We study the homotopy category $\mathrm{hmf}(R,W)$ of matrix factorizations of non-zero elements $W\in R^\times$, where $R$ is an elementary divisor domain. When $R$ has prime elements and $W$ factors into a square-free element $W_0$ and a finite product of primes of multiplicity greater than one and which do not divide $W_0$, we show that $\mathrm{hmf}(R,W)$ is triangle-equivalent with an orthogonal sum of the triangulated categories of singularities $\mathrm{D}_{\mathrm sing}(A_n(p))$ of the local Artinian rings $A_n(p)=R/\langle p^n\rangle$, where $p$ runs over the prime divisors of $W$ of order $n\geq 2$. This result holds even when $R$ is not Noetherian. The triangulated categories $\mathrm{D}_{\mathrm sing}(A_n(p))$ are Krull-Schmidt and we describe them explicitly. We also study the cocycle category $\mathrm{zmf}(R,W)$, showing that it is additively generated by elementary matrix factorizations. Finally, we discuss a few classes of examples.

Determination of Supersymmetric Particle Masses and Attributes with Genetic Divisors

Abstract: Arithmetic conditions relating particle masses can be defined on the basis of (1) the supersymmetric conservation of congruence and (2) the observed characteristics of particle reactions and stabilities. Stated in the form of common divisors, these relations can be interpreted as expressions of genetic elements that represent specific particle characteristics. In order to illustrate this concept, it is shown that the pion triplet ({pi}{sup {+-}}, {pi}{sup 0}) can be associated with the existence of a greatest common divisor d{sub 0{+-}} in a way that can account for both the highly similar physical properties of these particles and the observed {pi}{sup {+-}}/{pi}{sup 0} mass splitting. These results support the conclusion that a corresponding statement holds generally for all particle multiplets. Classification of the respective physical states is achieved by assignment of the common divisors to residue classes in a finite field F{sub P{sub {alpha}}} and the existence of the multiplicative group of units F{sub P{sub {alpha}}} enables the corresponding mass parameters to be associated with a rich subgroup structure. The existence of inverse states in F{sub P{sub {alpha}}} allows relationships connecting particle mass values to be conveniently expressed in a form in which the genetic divisor structure is prominent. An examplemore » is given in which the masses of two neutral mesons (K{degree} {r_arrow} {pi}{degree}) are related to the properties of the electron (e), a charged lepton. Physically, since this relationship reflects the cascade decay K{degree} {r_arrow} {pi}{degree} + {pi}{degree}/{pi}{degree} {r_arrow} e{sup +} + e{sup {minus}}, in which a neutral kaon is converted into four charged leptons, it enables the genetic divisor concept, through the intrinsic algebraic structure of the field, to provide a theoretical basis for the conservation of both electric charge and lepton number. It is further shown that the fundamental source of supersymmetry can be expressed in terms of hierarchical relationships between odd and even order subgroups of F{sub P{sub {alpha}}}, an outcome that automatically reflects itself in the phenomenon of fermion/boson pairing of individual particle systems. Accordingly, supersymmetry is best represented as a group rather than a particle property. The status of the Higgs subgroup of order 4 is singular; it is isolated from the hierarchical pattern and communicates globally to the mass scale through the seesaw congruence by (1) fusing the concepts of mass and space and (2) specifying the generators of the physical masses.« less

Divisor problem in arithmetic progressions modulo a prime power

Abstract: We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In particular, we break the classical barrier $q \le x^{2/3}$ for such formulas, and generalise a recent result of R.~Khan (2015), making it uniform in $k$.