Abstract:
We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{\alpha_1,...,\alpha_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus, (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in $\G$ and which generate the \emph{toric algebra of $\G$} (type II generators) and a set of binomials of degree $q-1$ obtained combinatorially from $\G$ (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo--Mumford regularity is equal to $\max\set{\alpha_1(q-2),...,\alpha_r(q-2), \lceil (n-1)(q-2)/2\rceil}$, where $n=\alpha_1+... + \alpha_r$.

Abstract:
Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.

Abstract:
We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with respect to a minimal reduction J of I.

Abstract:
Let X be a subset of a projective space, over a finite field K, which is parameterized by the monomials arising from the edges of a clutter. Let I(X) be the vanishing ideal of X. It is shown that I(X) is a complete intersection if and only if X is a projective torus. In this case we determine the minimum distance of any parameterized linear code arising from X.

Abstract:
Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).

Abstract:
We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields, with prescribed regularity and degree of a certain type. Let X be a subset of a projective space over a field K. It is shown that the vanishing ideal of X is a lattice ideal of dimension 1 if and only if X is a finite subgroup of a projective torus. For finite fields, it is shown that X is a subgroup of a projective torus if and only if X is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartie graph in terms of the regularities of the vanishing ideals of the blocks of the graph.

Abstract:
Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise disjoint even cycles. In this case, a fomula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph is connected. We are able to show a formula for the length of the parameterized linear code associated with any graph, in terms of the number of bipartite and non-bipartite components.

Abstract:
Let K=Fq be a finite field. We introduce a family of projective Reed-Muller-type codes called projective Segre codes. Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct products of projective Reed-Muller-type codes. As a consequence we recover some results on projective Reed-Muller-type codes over the Segre variety and over projective tori.

Abstract:
three hundred and sixty-three pregnant women enrolled in the pregnancy medical care program of s. paulo health department in the district of butantan, s. paulo city, brazil, were studied at their first routine consultation between april and october, 1988. their average age was 25 and 65,9% of them belonged to families with a monthly income below us$ 50.00 per capita. only 3.1% presented an income above us$ 150.00 per capita. taking the minimum transferrin saturation threshold of 15% as determining iron deficiency, a 4.6% prevalence of iron deficiency was observed in the first trimester, 17.3% in the second trimester and 42.8% in the third trimester, resulting in an overall prevalence of 12.4%. there was no significant difference between prevalences of iron deficiency according to the number of pregnancies. the prevalence of iron deficiency was higher in women presenting incomes below us$ 50.00 per capita.

Abstract:
No período compreendido entre abril e outubro de 1988, foram estudadas 363 gestantes de primeira consulta , que estavam inscritas no Programa de Atendimento à Gestante em oito Centros de Saúde da Secretaria da Saúde do Estado de S o Paulo (Brasil). Na ocasi o da coleta de material estas gestantes n o faziam uso de medicamentos que continham ferro, ácido fólico, vitamina B12 ou associa es destes. A idade média das gestantes foi de 25 anos; 65,9% delas pertenciam a famílias com renda de até um SMPC (salário mínimo per capita) e apenas 3,1% pertenciam a famílias com renda superior a 3 SMPC. Tomando-se a satura o da transferrina inferior a 15% como índice mínimo para definir a deficiência de ferro, a prevalência de deficiência de ferro no primeiro trimestre (4,6%) foi significativamente menor do que a observada no segundo (17,3%), e esta foi menor do que no terceiro trimestre (42,8%). A prevalência de deficiência de ferro total agrupada nos três trimestres foi de 12,4%. N o houve diferen a significativa entre as prevalências de deficiência de ferro segundo o número de partos. Esta prevalência foi maior no grupo das gestantes que pertenciam a famílias com renda de até 0,5 SMPC. Nas gestantes anêmicas, 46,7% eram deficientes de ferro, 44,4% de ácido fólico, 20,0% de ferro e ácido fólico e nenhuma delas eram deficientes de vitamina B12.