The quantitative analysis of climate in relation to insect abundance

(1). All natural complexes are related, directly or indirectly, through climate. The importance of quantitative methods of evaluating the role of various climatic factors is, therefore, apparent.(2). In addition to a pronounced seasonal rhythm in abundance, plagues and dearths of insects come and go with considerable regularity, when viewed over a series of years. The most important period is one of approximately eleven years, with a tendency to mass periodicity during the years closely following the epoch of sunspot-minima.(3). Insect -populations are controlled by a complex of continually varying environmental factors, of which the climatic ones assume the dominant r6le in the causation (but not always the termination) of insect - outbreaks.(4). There are great differences in the relative susceptibility of the different phases of the life-cycle to environmental conditions, so that critical periods can be recognised. As the latter are usually restricted to a few days or weeks associated with reproduction or the early stages of existence, and the damage is caused by a more mature stage, the weather during the critical periods has a high predictive value.(5). Insect-outbreaks result from an unusual combination of circumstances during a particular period. These 'optimum' conditions are widely separated in space and time, so that the environment normally imposes control, and outbreaks mean escape from control - an abnormal phenomenon.(6). In addition to the academic interest of the results, particularly in relation to the equilibrium of Nature, it is believed that they may have considerable economic value, by providing a scientific basis for the prediction of outbreaks

Torsion functors with monomial support

The dependence of torsion functors on their supporting ideals is investigated, especially in the case of monomial ideals of certain subrings of polynomial algebras over not necessarily Noetherian rings. As an application it is shown how flatness of quasicoherent sheaves on toric schemes is related to graded local cohomology.Comment: updated reference

Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring

We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a formula for computing the Euler characteristic of a Weil divisor D on a complete simplicial toric variety in terms of graded pieces of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of the Euler characteristic of D, to the Stanley-Reisner ideal of the fan, which is used in defining the Chow ring. The formula also follows from work of Maclagan-Smith.Comment: 9 pages 1 figur

Multigraded Castelnuovo-Mumford Regularity

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.Comment: 30 pages, 5 figure

Segre Classes on Smooth Projective Toric Varieties

We provide a generalization of the algorithm of Eklund-Jost-Peterson for computing Segre classes of closed subschemes of projective k-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.Comment: 19 pages, 1 figure, added references, corrected typos, minor text replacement

Pruning Algorithms for Pretropisms of Newton Polytopes

Pretropisms are candidates for the leading exponents of Puiseux series that represent solutions of polynomial systems. To find pretropisms, we propose an exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton polytopes. We prefer exact arithmetic not only because of the exact input and the degrees of the output, but because of the often unpredictable growth of the coordinates in the face normals, even for polytopes in generic position. We provide experimental results with our preliminary implementation in Sage that compare favorably with the pruning method that relies only on cone intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning, accepted for presentation at Computer Algebra in Scientific Computing, CASC 201