Codimension 3 Arithmetically Gorenstein Subschemes of projective NN-space

We study the lowest dimensional open case of the question whether every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^N$ is glicci, that is, whether every zero-scheme in $\mathbb{P}^3$ is glicci. We show that a set of $n \geq 56$ points in general position in \PP^3 admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in $\mathbb{P}^3$.Comment: to appear in Annales de l'Institut Fourie

Racks and blocked braids

In the paper Blocked-braid Groups, submitted to Applied Categorical Structures, the present authors together with Davide Maglia introduced the blocked-braid groups BB_n on n strands, and proved that a blocked torsion has order either 2 or 4. We conjectured that the order was actually 4 but our methods in that paper, which involved introducing for any group G a braided monoidal category of tangled relations, were inadequate to demonstrate this fact. Subsequently Davide Maglia in unpublished work investigated exactly what part of the structure and properties of a group G are needed to permit the construction of a braided monoidal category with a tangle algebra and was able to distinguish blocked two-torsions from the identity. In this paper we present a simplification of his answer, which turns out to be related to the notion of rack. We show that if G is a rack then there is a braided monoidal category TRel_G generalizing that of the above paper. Further we introduce a variation of the notion of rack which we call irack which yields a tangle algebra in TRel_G. Iracks are in particular racks but have in addition to the operations abstracting group conjugation also a unary operation abstracting group inverse. Using iracks we obtain new invariants for tangles and blocked braids permitting us to present a proof of Maglia's result that a blocked double torsion is not the identity. This work was presented at the Conference in Memory of Aurelio Carboni, Milan, 24-26 June 2013

Tangled Circuits

The theme of the paper is the use of commutative Frobenius algebras in braided strict monoidal categories in the study of varieties of circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras

Blocked-braid Groups

We introduce and study a family of groups $\mathbf{BB}_n$, called the blocked-braid groups, which are quotients of Artin's braid groups $\mathbf{B}_n$, and have the corresponding symmetric groups $\Sigma_n$ as quotients. They are defined by adding a certain class of geometrical modifications to braids. They arise in the study of commutative Frobenius algebras and tangle algebras in braided strict monoidal categories. A fundamental equation true in $\mathbf{BB}_n$ is Dirac's Belt Trick; that torsion through $4\pi$ is equal to the identity. We show that $\mathbf{BB}_n$ is finite for $n=1,2$ and 3 but infinite for $n>3$

The compositional construction of Markov processes II

In an earlier paper we introduced a notion of Markov automaton, together with parallel operations which permit the compositional description of Markov processes. We illustrated by showing how to describe a system of n dining philosophers, and we observed that Perron-Frobenius theory yields a proof that the probability of reaching deadlock tends to one as the number of steps goes to infinity. In this paper we add sequential operations to the algebra (and the necessary structure to support them). The extra operations permit the description of hierarchical systems, and ones with evolving geometry

Ground motion and stress accumulation driven by density anomalies in a viscoelastic lithosphere. Some results for the Apennines

SUMMARY We provide the analytical formulation for calculating the displacement and stress field forced by internal sources in a stratified, self-gravitating, viscoelastic earth. This model is specialized to study the rate of vertical motion and shear stress accumulation produced by lithospheric density anomalies. These sources are allowed to vary in the lateral direction. We show that sphericity plays a crucial role for elongated lithospheric anomalies while self-gravitation produces minor deviations from a gravitating Earth. When the model is applied to the Apennines we get, for lithospheric viscosity ranging between 10" and loz3 Pas, the subsidence of the plate underlying the active front of the overthrusting load to be around 0.5-1.0 mm yr-'. This is consistent with the amount of sedimentation in the Adriatic foredeep. The deformation pattern is very peculiar, with the largest subsidence localized beneath the active front of the topography. Our model enlightens the impact of discontinuities of tectonic phases on vertical motions in collision zones. If lithospheric viscosity is around 10'' Pa s, vertical motions decay drastically on time scales of 105yr if lateral migration of density anomalies comes to an end. For higher viscosities, deformation rates are maintained longer. This correlation between horizontal and vertical motions suggests that altimetric geodetic surveying along levelling lines of a few hundred kilometers can be an important tool to constrain the tectonics of the studied region. Results are also shown for vertical motions along a transect perpendicular to the Apennines, when the crustal structure inverted from Bouguer gravity data is considered. Analysis of the stress field induced by an overthrusting load shows that principal stress differences of a few bar (or a few tenths of MPa) can be accumulated on time scales of 102-103yr. These low values agree with the average stress drop of earthquakes in the Appalachians and northern Apennines where our modelling can be applied. We find that lateral density variations certainly contribute to intraplate stresses, but they are less efficient in triggering earthquakes than other mechanisms, such as transcurrent motions along active plate margins. Seismicity induced by lateral variations of crustal and lithospheric density must be moderate, characterized by long return times. These results are in agreement with the recorded seismicity in the northern Apennines where the largest earthquakes have return times of lo2 yr. If shear stress is forced by an overthrusting load, we find that the largest rate of stress accumulation in the crust is concentrated beneath the active front, close to the boundary with the ductile lithosphere. Discontinuities of tectonic phases play an important role in controlling the amount of shear stress due to density anomalies