Pedro A. García-Sánchez

CSpace: an integrated workplace for the graphical and algebraic analysis of phase assemblages on 32-bit wintel platforms

Abstract CSpace is a program for the graphical and algebraic analysis of composition relations within chemical systems. The program is particularly suited to the needs of petrologists, but could also prove useful for mineralogists, geochemists and other environmental scientists. A few examples of what can be accomplished with CSpace are the mapping of compositions into some desired set of system/phase components, the estimation of reaction/mixing coefficients and assessment of phase-rule compatibility relations within or between complex mineral assemblages. The program also allows dynamic inspection of compositional relations by means of barycentric plots. CSpace provides an integrated workplace for data management, manipulation and plotting. Data management is done through a built-in spreadsheet-like editor, which also acts as a data repository for the graphical and algebraic procedures. Algebraic capabilities are provided by a mapping engine and a matrix analysis tool, both of which are based on singular-value decomposition. The mapping engine uses a general approach to linear mapping, capable of handling determined, underdetermined and overdetermined problems. The matrix analysis tool is implemented as a task “wizard” that guides the user through a number of steps to perform matrix approximation (finding nearest rank-deficient models of an input composition matrix), and inspection of null-reaction space relationships (i.e. of implicit linear relations among the elements of the composition matrix). Graphical capabilities are provided by a graph engine that directly links with the contents of the data editor. The graph engine can generate sophisticated 2-D ternary (triangular) and 3D quaternary (tetrahedral) barycentric plots and includes features such as interactive re-sizing and rotation, on-the-fly coordinate scaling and support for automated drawing of tie lines.

The catenary and tame degree of numerical monoids

Abstract We construct an algorithm which computes the catenary and tame degree of a numerical monoid. As an example we explicitly calculate the catenary and tame degree of numerical monoids generated by arithmetical sequences in terms of their first element, the number of elements in the sequence and the difference between two consecutive elements of the sequence.

Systems of Inequalities and Numerical Semigroups

A one-to-one correspondence is described between the setS(m) of numerical semigroups with multiplicity m and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in S(m) a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of Nm−1. Finally, this result is particularized to the symmetric case.

Uniquely presented finitely generated commutative monoids

A finitely generated commutative monoid is uniquely presented if it has a unique minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally, for some relevant families of numerical semigroups we describe the elements that are uniquely presented.

numericalsgps, a GAP package for numerical semigroups

The package numericalsgps performs computations with and for numerical and affine semigroups. This manuscript is a survey of what the package does, and at the same time intends to gather the trending topics on numerical semigroups.

On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators

The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behavior of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases.

ON PRESENTATIONS OF COMMUTATIVE MONOIDS

In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid.

NUMERICAL SEMIGROUPS AND APPLICATIONS

The aim of this manuscript is to give some basic notions related to numerical semigroups, and from these on the one hand describe a classical application to the study of singularities of plane algebraic curves, and on the other, show how numerical semigroups can be used to obtain handy examples of nonunique factorization invariants.

Every Numerical Semigroup is One Half of Infinitely Many Symmetric Numerical Semigroups

Let S be a numerical semigroup. Then there exist infinitely many symmetric numerical semigroups such that . We give an explicit description of them.

Cyclotomic numerical semigroups

Given a numerical semigroup