Bayesian inference of a non-local proliferation model

From a systems biology perspective, the majority of cancer models, although interesting and providing a qualitative explanation of some problems, have a major disadvantage in that they usually miss a genuine connection with experimental data. Having this in mind, in this paper, we aim at contributing to the improvement of many cancer models which contain a proliferation term. To this end, we propose a new non-local model of cell proliferation. We select data that are suitable to perform Bayesian inference for unknown parameters and we provide a discussion on the range of applicability of the model. Furthermore, we provide proof of the stability of posterior distributions in total variation norm which exploits the theory of spaces of measures equipped with the weighted flat norm. In a companion paper, we provide detailed proof of the well-posedness of the problem and we investigate the convergence of the escalator boxcar train (EBT) algorithm applied to solve the equation.

Low energy configurations of topological singularities in two dimensions: A Γ-convergence analysis of dipoles

This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg–Landau energy. Denoting by [Formula: see text] the length scale parameter in such models, we focus on the [Formula: see text] energy regime. It is well known that, for configurations whose energy is bounded by [Formula: see text], the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying [Formula: see text] energy, plus a measure supported on small zero-average sets. Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses. Here, we perform a compactness and [Formula: see text]-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale [Formula: see text], for [Formula: see text]), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical [Formula: see text]-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order [Formula: see text] with [Formula: see text].

Strong approximation in h-mass of rectifiable currents under homological constraint

Let {h:\mathbb{R}\to\mathbb{R}_{+}} be a lower semicontinuous subbadditive and even function such that {h(0)=0} and {h(\theta)\geq\alpha|\theta|} for some {\alpha>0} . If {T=\tau(M,\theta,\xi)} is a k-rectifiable chain, its h-mass is defined as \mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}. Given such a rectifiable flat chain T with {\mathbb{M}_{h}(T)<\infty} and {\partial T} polyhedral, we prove that for every {\eta>0} , it decomposes as {T=P+\partial V} with P polyhedral, V rectifiable, {\mathbb{M}_{h}(V)<\eta} and {\mathbb{M}_{h}(P)<\mathbb{M}_{h}(T)+\eta} . In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint {\partial P=\partial T} . When {h^{\prime}(0^{+})} is well defined and finite, the definition of the h-mass extends as a finite functional on the space of finite mass k-chains (not necessarily rectifiable). We prove in this case a similar approximation result for finite mass k-chains with polyhedral boundary. These results are motivated by the study of approximations of {\mathbb{M}_{h}} by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of {T\mapsto\mathbb{M}_{h}(T)+\mathbb{I}_{\partial S}(\partial T)} with respect to the topology of the flat norm.

On the structure of flat chains modulo p

In this paper, we prove that every equivalence class in the quotient group of integral 1-currents modulo p in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for m-dimensional integral currents modulo p implies that the family of {(m-1)}-dimensional flat chains of the form pT, with T a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for 0-dimensional flat chains, and, using a proposition from The structure of minimizing hypersurfaces mod 4 by Brian White, also for flat chains of codimension 1.