Virtual prototyping of medieval weapons for historical reconstruction of siege scenarios starting from topography and archaeological investigations

Chronicles of sieges to castles or fortresses, using “machinae”, can often be found in historical sources. Moreover, archaeological excavations of castles or fortresses has brought to light rocks or projectiles whose carving suggests a military usage. Nevertheless, chronicles and discoveries alone, are seldom enough to propose a faithful reconstruction of these machines. Therefore, the aim of this research is the development of methodologies for reconstructing virtual scenarios of sieges, starting from the scarce information available. In order to achieve it, a procedure for the virtual reconstruction of the siege machine has been set up, focusing on typology and dimensions of the machines, also investigating possible fire positions according to topography. The entire procedure has been developed using the siege of Cervara di Roma’s Rocca as a case study. Late medieval chronicles (end of 13th Century) report the siege brought by the papal army in order to restore the jurisdiction on the Cervara’s stronghold, following the insurrection of a group of vassals headed by a monk named Pelagio. The discovery, in the area of the Rocca, of a stone that could have been used as a projectile confirms what reported. The proposed methodology is composed of two parts. The first one is connected to the study of the “internal ballistics”, to understand the performances and to build virtual models of siege machines. The second part is the study of the “external ballistics”, then to the positioning and shooting ability of possible machines, analysing the topography of the area. In this paper, we present the feasibility of this methodology through the preliminary results achieved correlating internal and external ballistics

Dissipative continuous Euler flows

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy

SBV regularity for Hamilton-Jacobi equations in Rn\mathbb R^n

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$\partial_t u + H(D_{x} u)=0 \qquad \textrm{in} \Omega\subset \mathbb R\times \mathbb R^{n} .$$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.Comment: 15 page

Lack of uniqueness for weak solutions of the incompressible porous media equation

In this work we consider weak solutions of the incompressible 2-D porous media equation. By using the approach of De Lellis-Sz\'ekelyhidi we prove non-uniqueness for solutions in $L^\infty$ in space and time.Comment: 23 pages, 2 fugure

Regularity of higher codimension area minimizing integral currents

This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher codimension, due originally to F. Almgren, which is contained in a series of papers in collaboration with C. De Lellis (University of Zurich).Comment: This text will appear in "Geometric Measure Theory and Real Analysis", pp. 131--192, Proceedings of the ERC school in Pisa (2013), L. Ambrosio Ed., Edizioni SNS (CRM Series

Weak Solutions to the Stationary Incompressible Euler Equations

We consider weak stationary solutions to the incompressible Euler equations and show that the analogue of the h-principle obtained in [5, 7] for time-dependent weak solutions continues to hold. The key difference arises in dimension d = 2, where it turns out that the relaxation is strictly smaller than what one obtains in the time-dependent case.Comment: 16 pages, 2 figures. Corrected a mistake in the proof of Theorem 17. Results unchanged. Corrected a typographical erro

On the concentration of entropy for scalar conservation laws

We prove that the entropy for an $L^\infty$-solution to a scalar conservation laws with continuous initial data is concentrated on a countably $1$-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution

Analysis of extended genomic rearrangements in oncological research.

Screening for genomic rearrangements is a fundamental task in the genetic diagnosis of many inherited disorders including cancer-predisposing syndromes. Several methods were developed for analysis of structural genomic abnormalities, some are targeted to the analysis of one or few specific loci, others are designed to scan the whole genome. Locus-specific methods are used when the candidate loci responsible for the specific pathological condition are known. Whole-genome methods are used to discover loci bearing structural abnormalities when the disease-associated locus is unknown. Three main approaches have been employed for the analysis of locus-specific structural changes. The first two are based on probe hybridization and include cytogenetics and DNA blotting. The third approach is based on PCR amplification and includes microsatellite or single nucleotide polymorphism (SNP) genotyping, relative allele quantitation, real-time quantitative PCR, long PCR and multiplex PCR-based methods such as multiplex ligation-dependent probe amplification and the recently developed nonfluorescent multiplex PCR coupled to high-performance liquid chromatography analysis. Whole-genome methods include cytogenetic methods, array-comparative genomic hybridization, SNP array and other sequence-based methods. The goal of the present review is to provide an overview of the main features and advantages and limitations of methods for the screening of structural genomic abnormalities relevant to oncological research

Weak solutions to problems involving inviscid fluids

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method of convex integration we show the existence of infinitely many weak solutions for prescribed initial data and kinetic energy