Collection Of Integers Research Articles

On the Quantum K-Theory of the Quintic

Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series $J(Q,q,t)$ that satisfies a system of linear differential equations with respect to $t$ and $q$-difference equations with respect to $Q$. With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small $J$-function $J(Q,q,0)$ which, in the case of Fano manifolds, is a vector-valued $q$-hypergeometric function. On the other hand, for the quintic 3-fold we formulate an explicit conjecture for the small $J$-function and its small linear $q$-difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants, and the case of Fano manifolds, the coefficients of the small linear $q$-difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small $J$-function agrees with a proposal of Jockers-Mayr.

Sufficient conditions for the existence of a graph with a given variety of balls

It is proved that for each positive integer d and each collection of integers $$\bar \tau $$ = (τ 0, τ 1 …, τ d ) such that τ 0 ⩾ τ 1 ⩾ … ⩾ τ d = 1 and τ d−1 ⩾ d 2 + 1, there exists a graph of diameter d whose variety vector of the balls is equal to $$\bar \tau $$ ; if d ⩾ 3 then there is no graph of diameter d whose variety vector of balls (τ 0, τ 1, …, τ d ) satisfies the condition τ 0 = τ 1 = … = τ d−1 ⩽ 2d − 1.

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: [Formula: see text] for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form [Formula: see text] for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

On the Existence of Certain Families of Curves

Introduction. Given a proper surjective map f: XY from a non-singular surface X to a non-singular curve Y over an algebraically closed field k, one can write a closed fiber as a divisor Z = mlCl + + m C, We call the type of Z the integers n, mi, p(Ci) = dimkHl(Ci, sc), and (Ci, Cj) for i #j. We also say a type is any collection of integers n > 1, mi > 1, pi > 0, ci4 = cii > 0 (i#j; i,j = 1,...,n) such that mi divides E 1=,m1ci,. This divisibility is automatic for the type of Z, since

Equipartition of Energy for Nonlinear Systems

A system of harmonic oscillators weakly coupled by nonlinear forces will not achieve equipartition of energy as long as the uncoupled frequencies ωk are linearly independent on the integers, i.e., as long as there is no collection of integers {nk} for which Σnkωk=0 other than all nk=0. This result is shown to follow from the general form of the Kryloff and Bogoliuboff series solution to the equations of motion. Physically, the linear independence of the uncoupled frequencies means that none of the interacting oscillators drives another at its resonant frequency, and this lack of internal resonance precludes appreciable energy sharing in the limit as the coupling tends to zero. It is shown that the lack of equipartition of energy observed by Ulam, Fermi, and Pasta for certain nonlinear systems may be explained in terms of the preceding remarks. Moreover, a Kryloff and Bogoliuboff series solution to the appropriate equations of motion is shown to yield qualitative agreement with the Ulam, Fermi, and Pasta computer solution. Finally, a particular system of linear differential equations is solved which illustrates a mechanism whereby oscillator systems may achieve equipartition of energy.