Francesco Morandin

Uniqueness for a Stochastic Inviscid Dyadic Model

AbstractFor the deterministic dyadic model of turbulence, there are exam-ples of initial conditions in l 2 which have more than one solution. Theaim of this paper is to prove that uniqueness, for all l 2 -initial condi-tions, is restored when a suitable multiplicative noise is introduced.The noise is formally energy preserving. Uniqueness is understood inthe weak probabilistic sense. 1 Introduction The infinite system of nonlinear differential equationsdX n (t)dt= k n−1 X 2n−1 (t) −k n X n (t)X n+1 (t), t ≥ 0 (1.1)X n (0) = x n for n ≥ 1, with coefficients k n > 0 for each n ≥ 1, X 0 (t) = 0 and k 0 = 0, isone of the simplest models which presumablyreflect some of the propertiesof3D Euler equations. At least, it is infinite dimensional, formally conservative(the energyP ∞n=1 X 2n (t) is formally constant), and quadratic. One of its‘pathologies’ is the lack of uniqueness of solutions, in the space l 2 of squaresummable sequences: when, for instance, k n = λ n with λ > 1, there areexamples of initial conditions x = (x

Energy dissipation and self-similar solutions for an unforced inviscid dyadic model

A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved, and decay of energy like t ―2 is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions.

Smooth solutions for the dyadic model

We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier-Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier-Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the non-linearity.

Anomalous dissipation in a stochastic inviscid dyadic model

A stochastic version of an inviscid dyadic model of turbulence, with multiplicative noise, is proved to exhibit energy dissipation in spite of the formal energy conservation. The proof is based on the reduction to a linear stochastic equation with multiplicative noise, by Girsanov transform, and the interpretation of the second moment equation as the master equation of a birth and death process.

Global regularity for a slightly supercritical hyperdissipative Navier–Stokes system

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier--Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao2009].

TAMGeS: a Three-array Method for Genotyping of SNPs by a dual-colour approach

BackgroundMany of the most effective high-throughput protocols for SNP genotyping employ microarrays. Genotypes are assessed by comparing the signal intensities that derive from the hybridization of different allele-specific probes labelled either by using four fluorescent dyes, one for each base, or by using only two dyes and investigating the polymorphic alleles two by two on separate arrays. The employment of only two dyes makes it possible to use a dual-laser scanner, which has the advantage of being present in every microarray laboratory. However, this protocol may present some drawbacks. To infer all the six possible genotypes it is necessary to compare signals from two arrays, but this comparison not always is successful. A number of systematic errors in the experimental protocol, in fact, may differently affect signal intensities on separate arrays. Here we present TAMGeS (Three-Array Method for Genotyping of SNPs), an exhaustive method for SNP genotyping through SBE (Single Base Extension) and dual-colour microarrays, which makes the comparison of signals on distinct arrays reliable by using a third array and a data handling method for signal normalization based on bilinear regression theory.ResultsWe tested the effectiveness of the proposed method by evaluating the results obtained from the direct comparison of the two arrays or by applying TAMGeS, both on experimental and synthetic data. With synthetic data, TAMGeS reduced the frequency of errors by an order of magnitude, when the incidence of systematic errors was not negligible. With the experimental data, produced by genotyping 25 SNPs in 437 subjects, TAMGeS reduced the percentage of missing genotypes from 54% (Two-Array Method) to 14.5%. Allelic and genotypic call rates were 99.3% and 99.5%, respectively. The normalization procedure takes into account also systematic errors, which can be generated by a time-delayed assay, thus making the protocol more flexible.ConclusionTAMGeS represents an innovative method, which proved to be very effective in producing reliable SNP genotyping data by dual-colour microarrays. The requirement of a third array is well balanced by the strong enhancement in data quality and by the greater flexibility of the experimental protocol.

A dyadic model on a tree

We study an infinite system of nonlinear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and Navier-Stokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.

Stochastic inviscid shell models: well-posedness and anomalous dissipation

In this paper we study a stochastic version of an inviscid shell model of turbulence with multiplicative noise. The deterministic counterpart of this model is quite general and includes inviscid GOY and Sabra shell models of turbulence. We prove global weak existence and uniqueness of solutions for any finite energy initial condition. Moreover energy dissipation of the system is proved in spite of its formal energy conservation.

Recurrence of prostate cancer after HIFU. Proposal of a novel predictive index

Background and aim of the work: Prostate cancer is one of the most common cancers in men over 50 years of age. Surgery, radiotherapy and hormonal manipulation represent its typical treatment. High-Intensity Focused Ultrasound (HIFU) is an alternative choice in localized prostate cancer. To date, an index for prediction of recurrence in patients treated with HIFU is not availabe. Our study proposes a novel index for the predition of recurrence able to determine if a candidate is fit for this tratment. methods: 107 patients underwent HIFU fram 2010 to 2015. A total of 12 variables were considered for the analysis. The final predictive model was obtained through a stepwise forward selection method. Results: The final model used a total of 6 variables, all correlated to the response variable. The Index is able to predict the recurrence after HIFU tratment in the most majority of candidates to treatment. The index may be used to make a more scientific decision with regard to choosing optimal candidates for HIFU. (www.actabiomedica.it)

Structure Function and Fractal Dissipation for an Intermittent Inviscid Dyadic Model

We study a generalization of the original tree-indexed dyadic model by Katz and Pavlović for the turbulent energy cascade of the three-dimensional Euler equation. We allow the coefficients to vary with some restrictions, thus giving the model a realistic spatial intermittency. By introducing a forcing term on the first component, the fixed point of the dynamics is well defined and some explicit computations allow us to prove the rich multifractal structure of the solution. In particular the exponent of the structure function is concave in accordance with other theoretical and experimental models. Moreover, anomalous energy dissipation happens in a fractal set of dimension strictly less than 3.