Counting Solutions to Binomial Complete Intersections

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.Comment: Several minor improvements. Final version to appear in the J. of Complexit

A construction of Frobenius manifolds with logarithmic poles and applications

A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a generalization of the reconstruction theorem of Kontsevich and Manin for projective smooth varieties with convergent Gromov-Witten potential. A second application is a construction of Frobenius manifolds out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled.Comment: 46 page

Residues and Resultants

Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jacobians.Comment: Plain TeX, 22 page

Infinitesimal Variations of Hodge Structure at Infinity

By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure (IVHS) at finite points. As an illustration of the rich information encoded in this new structure, some instances of the maximal dimension problem for this type of infinitesimal variation are presented and contrasted with the "classical" case of IVHS at finite points

Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow

AbstractThe stability analysis, the accuracy and the efficiency of a semi-implicit finite difference scheme for the numerical solution of a three-dimensional shallow water model are presented and discussed. The governing equations are the three-dimensional Reynolds equations in which pressure is assumed to be hydrostatic. The pressure gradient in the momentum equations and the velocities in the vertically integrated continuity equation are discretized with the θ-method, with θ being an implicitness parameter. It is shown that the method is stable for 12 ≤ θ ≤ 1, unstable for θ < 12 and highest accuracy and efficiency is achieved when θ = 12. The resulting algorithm is mass conservative and naturally allows for the simulation of flooding and drying of tidal flats

Modelling the hyperboloid elastic shell

This paper deals with the momentless state equations of a hyperboloid elastic shell which is obtained as a surface of revolution. The compatibility conditions are investigated and used to derive a class of analytical solutions. A computer model of given initial-boundary problems show some nonlinear effects such as the bending of the wave propagation surface

Computing Multidimensional Residues

Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial. This is an algebraic, but generally not rational, function of the coefficients. On the other hand, the global residue, which is the sum of the local residues over all roots, depends rationally on the coefficients. This paper deals with symbolic algorithms for evaluating that rational function. Under the assumption that the deformation to the initial forms is flat, for some choice of weights on the variables, we express the global residue as a single residue integral with respect to the initial forms. When the input equations are a Groebner basis, this leads to an efficient series expansion algorithm for global residues, and to a vanishing theorem with respect to the corresponding cone in the Groebner fan. The global residue of a polynomial equals the highest coefficient of its (Groebner basis) normal form, and, conversely, the entire normal form is expressed in terms of global residues. This yields a method for evaluating traces over zero-dimensional complete intersections. Applications include the counting of real roots, the computation of the degree of a polynomial map, and the evaluation of multivariate symmetric functions. All algorithms are illustrated for an explicit system in three variables

Friends, Cliques and Gifts: Social Proximity and Recognition in Peer-Based Tournament Rituals

Two main accounts of the effect of proximity between candidates competing for recognition and members of the evaluating audience in the underlying social structure can be extrapolated from extant literature on peer-based tournament rituals and cultural fields. Following a Bourdieusian tradition, one account – which we label self-reproduction – insists on the catalyzing effect of social proximity in shaping recognition along relational lines. Drawing from recent scholarship on social evaluation, a second account – which we label intellectual distance – suggests that social proximity deters recognition. We probe the influence of different articulations of social proximity (i.e., direct ties, cliquishness and reciprocity) on recognition by studying awarding decisions within the context of the Norwegian advertising industry. Interviews with key informants and econometric results suggest that, while self-reproduction tends to prevail over intellectual distance, these effects co-exist and their relative influence varies across levels of recognition. We gauge the relative saliency of the two accounts by using a mix-method approach. Important implications for research on social evaluation and recognition in peer-based tournament rituals are drawn

Gravitational Waves from Wobbling Pulsars

The prospects for detection of gravitational waves from precessing pulsars have been considered by constructing fully relativistic rotating neutron star models and evaluating the expected wave amplitude $h$ from a galactic source. For a "typical" neutron matter equation of state and observed rotation rates, it is shown that moderate wobble angles may render an observable signal from a nearby source once the present generation of interferometric antennas becomes operative.Comment: PlainTex, 7 pp. , no figures, IAG/USP Rep. 6