The Steiner tree problem revisited through rectifiable G-currents

The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization problem for $1$-dimensional currents with coefficients in a suitable normed group. The representation used for these currents allows to state a calibration principle for this problem. We also exhibit calibrations in some examples

A multi-material transport problem and its convex relaxation via rectifiable GG-currents

In this paper we study a variant of the branched transportation problem, that we call multi-material transport problem. This is a transportation problem, where distinct commodities are transported simultaneously along a network. The cost of the transportation depends on the network used to move the masses, as it is common in models studied in branched transportation. The main novelty is that in our model the cost per unit length of the network does not depend only on the total flow, but on the actual quantity of each commodity. This allows to take into account different interactions between the transported goods. We propose an Eulerian formulation of the discrete problem, describing the flow of each commodity through every point of the network. We provide minimal assumptions on the cost, under which existence of solutions can be proved. Moreover, we prove that, under mild additional assumptions, the problem can be rephrased as a mass minimization problem in a class of rectifiable currents with coefficients in a group, allowing to introduce a notion of calibration. The latter result is new even in the well studied framework of the "single-material" branched transportation.Comment: Accepted: SIAM J. Math. Ana

A multi-material transport problem with arbitrary marginals

In this paper we study general transportation problems in $\mathbb{R}^n$, in which $m$ different goods are moved simultaneously. The initial and final positions of the goods are prescribed by measures $\mu^-$, $\mu^+$ on $\mathbb{R}^n$ with values in $\mathbb{R}^m$. When the measures are finite atomic, a discrete transportation network is a measure $T$ on $\mathbb{R}^n$ with values in $\mathbb{R}^{n\times m}$ represented by an oriented graph $\mathcal{G}$ in $\mathbb{R}^n$ whose edges carry multiplicities in $\mathbb{R}^m$. The constraint is encoded in the relation ${\rm div}(T)=\mu^--\mu^+$. The cost of the discrete transportation $T$ is obtained integrating on $\mathcal{G}$ a general function $\mathcal{C}:\mathbb{R}^m\to\mathbb{R}$ of the multiplicity. When the initial data $\left(\mu^-,\mu^+\right)$ are arbitrary (possibly diffuse) measures, the cost of a transportation network between them is computed by relaxation of the functional on graphs mentioned above. Our main result establishes the existence of cost-minimizing transportation networks for arbitrary data $\left(\mu^-,\mu^+\right)$. Furthermore, under additional assumptions on the cost integrand $\mathcal{C}$, we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data. Finally, we provide an explicit integral representation formula for the cost of rectifiable transportation networks, and we characterize the costs such that every transportation network with finite cost is rectifiable.Comment: In V3 we have added an explicit integral representation formula for the cost of rectifiable transportation networks and characterized the cost functionals such that every transportation network with finite energy is rectifiable. The representation formula is proved in the general framework of $k$-currents with coefficients in group

The Steiner tree problem revisited through rectifiable G-currents

The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization problem for 1-dimensional currents with coefficients in a suitable normed group. The representation used for these currents allows to state a calibration principle for this problem. We also exhibit calibrations in some examples

Diversity Assessment and DNA-Based Fingerprinting of Sicilian Hazelnut (Corylus avellana L.) Germplasm

The characterization of plant genetic resources is a precondition for genetic improvement and germplasm management. The increasing use of molecular markers for DNA-based genotype signature is crucial for variety identification and traceability in the food supply chain. We collected 75 Sicilian hazelnut accessions from private and public field collections, including widely grown varieties from the Nebrodi Mountains in north east Sicily (Italy). The germplasm was fingerprinted through nine standardized microsatellites (SSR) for hazelnut identification to evaluate the genetic diversity of the collected accessions, validating SSR discrimination power. We identified cases of homonymy and synonymy among acquisitions and the unique profiles. The genetic relationships illustrated by hierarchical clustering, structure, and discriminant analyses revealed a clear distinction between local and commercial varieties. The comparative genetic analysis also showed that the Nebrodi genotypes are significantly different from the Northern Italian, Iberian, and Turkish genotypes. These results highlight the need and urgency to preserve Nebrodi germplasm as a useful and valuable source for traits of interest employable for breeding. Our study demonstrates the usefulness of molecular marker analysis to select a reference germplasm collection of Sicilian hazelnut varieties and to implement certified plants’ production in the supply chain

An investigation of the self- and inter-incompatibility of the olive cultivars 'Arbequina' and 'Koroneiki' in the Mediterranean climate of Sicily

In this investigation, the self-(in)compatibility of the Spanish cultivar Arbequina and the Greek cultivar Koroneiki was studied for the first time in Sicily, where these low vigour cultivars were recently introduced in super-intensive olive groves. Self- (S.P.) and openpollination (O.P.) tests, observation of fruit set and paternity test of seeds with microsatellite (SSR) markers, were performed to ascertain whether these cultivars were self-fertile and/or inter-compatible. For S.P. tests, branches with flowers at the balloon stage were bagged. For the O.P. tests, flowers were left to pollinate under natural conditions. Fruits from S.P. and O.P. were collected in November and fruit set was calculated. Genomic DNA was extracted from seeds. None of the 'Arbequina' seeds studied in either the S.P. or O.P. tests originated from self-fertilization. In addition, none of these seeds had 'Koroneiki' as the pollen parent. In contrast, 'Koroneiki' was found to be predominantly self-compatible in self-bagged branches, with 70% of the seeds originating from selffertilization. However, the incidence of self-fertilization was low (11%) in seeds from the O.P. test. Low levels of inter-compatibility were found between 'Arbequina' and 'Koroneiki', while many local cultivars were found to be good pollinators. The information presented here will be useful to growers for planning their orchards with suitable pollinators and for our breeding program aiming at obtaining new low vigour olive genotypes. In addition, our results suggested that the recent model of attribution of S-alleles and the prediction of suitable pollinizers for a given variety should be more cautious and always based on controlled crosses and paternity testing of seed from those crosses

The first high-density sequence characterized SNP-based linkage map of olive (Olea europaea L. subsp. europaea) developed using genotyping by sequencing

A number of linkage maps have been previously developed in olive; however, these are mostly composed of markers that have not been characterized at the sequence level, supplemented with smaller numbers of microsatellite markers. In this investigation, we sought to develop a saturated linkage mapping resource for olive composed entirely of sequence characterized markers. We employed genotyping by sequencing to develop a map of a F2 population derived from the selfing of the cultivar Koroneiki. The linkage map contained a total of 23 linkage groups comprised of 1,597 tagged SNP markers in 636 mapping bins spanning a genetic distance of 1189.7 cM. An additional 6,658 segregating SNPs were associated with the 23 linkage groups identified but their marker order was not determined in this investigation. The SNP markers sequences were submitted to NCBI database. The linkage map produced will be an invaluable resource for the study of tree habit and vigour traits segregating in the progeny, and will assist to anchor and orientate sequencing scaffolds from future genome sequencing efforts

Numerical Calibration of Steiner trees

International audienceIn this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided