Algebraic Resolution Research Articles

Algebraic resolution of the duality ansatz for poincaré gauge fields

It has been known for sime time that exact torsion solutions of the Poincaré gauge theory with duality properties “live” on Einstein spaces. Via a (3+1)-decomposition we are now able to resolve the modified double duality ansatz for the Riemann-Cartan curvature in the static case. Thereby the independent linear connection can be expressed explicitly in terms of a given einsteinian coframe.

Eutrophication model of the Venice lagoon: Statistical treatment of ‘in situ’ measurements of phytoplankton growth parameters

A mathematical eutrophication model is proposed, which was obtained by combining a pre-existing three-dimensional model based on eddy-diffusive transport, embodying tidal agitation, and a ‘trophic dynamic reactor’ interconnecting eight intensive physico-chemical and biological variables. Parameters included in the phytoplankton growth equation, K N, K P, μ max, were obtained through several ‘in situ’ investigations performed under varying lagoon conditions. The usual statistical treatments for analyzing collected data, combined with nonlinear multiple variability, were found unsuitable. Therefore, three statistical methods were elaborated: algebraic resolution, including analyses of normal and log-normal distributions of the results; multiple regression analyses with constrained minima through a secular equation of the sixth degree; and analysis of the hypersurfaces of variance and reconstruction, from computer maps, of three-dimensional profiles with paraboloid-like base. Comparisons of results obtained with these methods permit delimitation of variability ranges of constants. Further, the values adopted in the model to simulate trophic equilibria, reproduced real lagoon conditions.

Multiplicative Structure on Resolutions of Algebras Defined by Herzog Ideals

There is only a small collection of algebras whose finite free resolutions are known to admit the structure of an associative, differential, graded, commutative algebra. Avramov and Hinič have shown that there are algebras (even Gorenstein ones) whose resolutions admit no such structure. We enlarge the affirmative list by showing that Herzog's algebras k0(g) defined by ‘a sequence and a matrix’ have resolutions that do support associative DGC structure. In fact, we do this for the versal deformations k(g). The proof, which is valid in arbitrary codimension g ⩾ 4, rests on our earlier ‘big from small’ construction and involves combining two Koszul resolutions (exterior algebras). As corollaries we prove that the k(g) are factorial, with normal, rigid, generic completions. We use the DGC structure to identify the singular locus; it follows that perfect specializations of k(g) in affine space of dimension at most g + 6 have smooth deformations.

On projective resolutions of Frobenius algebras and Gorenstein rings

On projective resolutions of Frobenius algebras and Gorenstein rings

Projective resolutions of Cohen-Macaulay algebras

The problem of explicitly finding a free resolution, minimal in some suitable sense, of a module over a polynomial ring is solved in principle by the algorithm of Hilbert [H]. However, this algorithm is of enormous computational difficulty. If the module happens to be finite dimensional over the ground field, and if the module structure is given by specifying the commuting linear transformations induced by the indeterminates, then a little-known result of Scheja and Storch, resumed in Sect. 1, allows one to write down an explicit free resolution of the right length without computation. The same idea can be used for many CohenMacaulay modules (Example 1.1). But although the Scheja-Storch resolution is minimal in some cases, it is not minimal in the main case of interest where the module is a factor-ring of the polynomial ring and not the ground field itself.

Formal ring of a cubic solid angle

A rational cubic form with real plane factors determines a solid angle for whose lattice points a formal ring can be constructed and can be made invariant under units. As in the first half of this work (in Volume 8 of this Journal), there is a finite basis for the invariant ring but peculiar effects occur in higher dimension. There might be, for instance, no minimal basis subdivision for the support polyhedron, and no cyclic basis, (simulating algebraic resolution difficulties in C 8). Furthermore, the support polyhedron is “flat” in some affine sense. Cases are computed for small discriminant cubics.

Hall and Knight's “Higher Algebra”

PERMIT me to remark that far from “ignoring” Messrs. Hall and Knight's use of the terms “Elementary” and “Higher” for the two parts of their Algebra, I have called special attention to it. It seems to me that the term “Higher Algebra” (algèbre supérieure) having been employed by Serret as embracing “the algebraic resolution of equations” in general, and that of “Modern Higher Algebra” by Salmon for what “with greater precision might be called the Algebra of Linear Transformations,” it is hardly open to the writers of what is really a text-book of elementary algebra in two parts to apply the same term to the second part of their work, and to object to a gentle protest from a reviewer. I am at a loss to see where in my subsequent remarks I have practically ignored their own use of the terms.