Search

In a sequence $S=\{(X,f_n)\}_{n\in\mathbb{N}}$ of dynamical systems sharing a common ambient space, the point $f^k_n(x)$ visited by a certain $x\in X$ depends on the iteration order $k$ and on the index $n$ specifying the system in $S$. If the sequence of maps $\{f_n\}_{n\in\mathbb{N}}$ is eventually constant at every point, the $f_n$-orbits show an emergent poset structure. A maximal initial segment of this poset is isomorphic to a certain countable ordinal $\ge\omega$. We study this transfinite emergent structure from the point of view of topological dynamics, investigating orbits, recurrence, limit sets, attractors and conjugacy.

Among solar system objects, comets coming from the Oort Cloud are an elusive population, intrinsically rare and difficult to detect. Nonetheless, as the more pristine objects we can observe, they encapsulate critical cues on the formation of planetary systems and are the focus of many scientific investigations and science missions. The Legacy Survey of Space and Time (LSST), which will start to operate from the Vera C. Rubin Observatory in 2025, is expected to dramatically improve our detection ability of these comets by performing regular monitoring of the Southern sky deep down to magnitude 24.5 with excellent astrometry. However, making straightforward predictions on future LSST detection rates is challenging due to our biased knowledge of the underlying population. This is because identifications to date have been conducted by various surveys or individual observers, often without detailed information on their respective selection functions. Recent efforts to predict incoming flux of Long Period Comets still suffer of the lack of systematic, well-characterized, homogeneous cometary surveys. Here, we adopt a different point of view by asking how much earlier~on known comets on long-period or hyperbolic orbits would have been discovered by a LSST-like survey if it was already in place 10 years prior to their perihelion epoch. In this case, we are not simulating a real flux of incoming comet, as all comets in our sample reach the perihelion simultaneously, but we can analyze the impact of a LSST-like survey on individual objects. We find that LSST would have found about 40% of comets in our sample at least 5 years prior to their perihelion epoch, and at double (at least) the distance at which they were actually discovered. Based on this approach, we find that LSST has the potentiality to at least twofold the current discovery rate of long-period and hyperbolic comets., Comment: 14 pages, 7 figures. Accepted for publication on Icarus

This paper is concerned with a comparison principle for viscosity solutions to Hamilton-Jacobi (HJ), -Bellman (HJB), and -Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with L\'evy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to differential, difference, and integral operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous L\'evy processes.

We give a new perspective on the existence of viscosity solutions for a stationary and a time-dependent first order Hamilton-Jacobi equation. Following recent comparison principles, we work in a framework in which we consider a subsolution and a supersolution for two equations in terms of two Hamiltonians that can be seen as an upper and lower bound of our original Hamiltonian respectively. The upper and lower bound are formulated in terms of a relaxed Lyapunov function which allows us to restrict part of the analysis to compact sets and to work with almost optimizers of the considered control problems. For this reason, we can relax assumptions on the control problem: most notably, we do not need completeness of set of controlled paths. Moreover, our strategy avoids a-priori analysis on the regularity of the candidate solutions. We then explore the context of Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations to verify our assumptions in a convex and non-convex setting respectively.

In the last years the attention towards topological dynamical properties of highly discontinuous maps has increased significantly. In [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4), 315-332, 2022], a class of densely discontinuous interval maps, called "critical exponent maps", was introduced. These maps are defined through the word-combinatorics concept of critical exponent applied to the binary expansion of reals and show highly chaotically properties as well as some challenging problems. In this paper we identify an error in the proof of Theorem 7 in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4), 315-332, 2022], a purely combinatorial result which in fact does not hold. We show that most of the results in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4), 315-332, 2022], obtained there through Theorem 7, can be recovered. Moreover, we propose as a conjecture a weaker form of Theorem 7.

Despite considerable progress in developing artificial intelligence (AI) algorithms for prostate cancer detection from whole slide images, the clinical applicability of these models remains limited due to variability in pathological annotations and existing dataset limitations. This article proposes a novel approach to overcome these challenges by leveraging a Bayesian framework to seamlessly integrate new data, and present results as a panel of annotations. The framework is demonstrated by integrating a Bayesian prior with one trained AI model to generate a distribution of Gleason patterns for each pixel of an image. It is shown that using this distribution of Gleason patterns rather than a ground-truth label can improve model applicability, mitigate errors, and highlight areas of interest for pathologists. Additionally, we present a high-quality, hand-curated dataset of prostate histopathological images annotated at the gland level by trained pre-medical students and verified by an expert pathologist. We highlight the potential of this adaptive and uncertainty-aware framework for developing clinically deployable AI tools that can support pathologists in accurate prostate cancer grading, improve diagnostic accuracy, and create positive patient outcomes., Comment: 5 pages, 3 figures, one table. Accepted for presentation at the 22nd International Conference on Artificial Intelligence in Medicine (AIME 2024)

We prove comparison principle for viscosity solutions of a Hamilton-Jacobi-Bellman equation in a strong coupling regime considering a stationary and a time-dependent version of the equation. We consider a Hamiltonian that has a representation as the supremum of a difference of two functions: an internal Hamiltonian depending on a control variable and a function interpreted as a cost of applying the controls. Our major innovation lies in the use of a cost function that can be discontinuous, unbounded and depending on momenta, enabling us to address previously unexplored scenarios such as cases arising from the theory of large deviations and homogenisation. For completeness, we also state the existence of viscosity solutions and we verify the assumptions for an example arising from biochemistry.

We investigate the quantification of demographic risk in a framework consistent with the market-consistent valuation imposed by Solvency II. We provide compact formulas for evaluating inflows and outflows of a portfolio of insurance policies based on a cohort approach. In this context, we maintain the highest level of generality in order to consider both traditional policies and equity-linked policies: therefore, we propose a market-consistent valuation of the liabilities. In the second step we evaluate the Solvency Capital Requirement of the idiosyncratic risk, linked to accidental mortality, and the systematic risk one, also known as trend risk, proposing a formal closed formula for the former and an algorithm for the latter. We show that accidental volatility depends on the intrinsic characteristics of the policies of the cohort (Sums-at-Risk), on the age of the policyholders and on the variability of the sums insured; trend risk depends both on accidental volatility and on the longevity forecasting model used.

In this note, our aim is to show that families of smooth hypersurfaces of $\mathbb R^{n+1}$ which are all $C^1$--close enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo-Nirenberg and ``geometric'' Calder\'on-Zygmund inequalities. This technical result is quite useful, in particular, in the study of the geometric flows of hypersurfaces.

The NASA Double Asteroid Redirection Test (DART) mission performed a kinetic impact on asteroid Dimorphos, the satellite of the binary asteroid (65803) Didymos, at 23:14 UTC on September 26, 2022 as a planetary defense test. DART was the first hypervelocity impact experiment on an asteroid at size and velocity scales relevant to planetary defense, intended to validate kinetic impact as a means of asteroid deflection. Here we report the first determination of the momentum transferred to an asteroid by kinetic impact. Based on the change in the binary orbit period, we find an instantaneous reduction in Dimorphos's along-track orbital velocity component of 2.70 +/- 0.10 mm/s, indicating enhanced momentum transfer due to recoil from ejecta streams produced by the impact. For a Dimorphos bulk density range of 1,500 to 3,300 kg/m$^3$, we find that the expected value of the momentum enhancement factor, $\beta$, ranges between 2.2 and 4.9, depending on the mass of Dimorphos. If Dimorphos and Didymos are assumed to have equal densities of 2,400 kg/m$^3$, $\beta$= 3.61 +0.19/-0.25 (1 $\sigma$). These $\beta$ values indicate that significantly more momentum was transferred to Dimorphos from the escaping impact ejecta than was incident with DART. Therefore, the DART kinetic impact was highly effective in deflecting the asteroid Dimorphos., Comment: accepted by Nature

The European Space Agency's Rosetta mission to comet 67P/Churyumov-Gerasimenko has offered scientists the opportunity to study a comet in unprecedented detail. Four instruments of the Rosetta orbiter, namely, the Micro-Imaging Dust Analysis System (MIDAS), the Grain Impact Analyzer and Dust Accumulator (GIADA), the COmetary Secondary Ion Mass Analyser (COSIMA), and the Rosetta Orbiter Spectrometer for Ion and Neutral Analysis (ROSINA) have provided information on cometary dust particles. Cross-instrument comparisons are crucial to characterize cometary dust particles beyond the capabilities of individual sensors, as they are sensitive to different dust components. We present the first comparison between detections of the ROSINA COmet Pressure Sensor (COPS) and GIADA. These two instruments are complementary as the former is sensitive solely to volatiles of icy particles, while the latter measured the dust particle as a whole, including refractories and condensed (semi)volatiles. Our goal is to correlate the particles detected by COPS and GIADA and to assess whether they belong to a common group. We statistically analyzed the in situ data of COPS and GIADA by calculating Pearson correlation coefficients. Among the several types of particles detected by GIADA, we find that COPS particles are significantly correlated solely with GIADA fluffy agglomerates (Pearson correlation coefficient of 0.55 and p-value of $4.6\cdot 10^{-3}$). This suggests that fluffy particles are composed of both refractories and volatiles. COPS volatile volumes, which may be represented by equivalent spheres with a diameter in the range between 0.06 $\mu$m and 0.8 $\mu$m, are similar to the sizes of the fractal particle's subunits identified by MIDAS (i.e., 0.05-0.18 $\mu$m)., Comment: 6 pages, 3 figures, accepted for publication in A&A

We consider the context of molecular motors modelled by a diffusion process driven by the gradient of a weakly periodic potential that depends on an internal degree of freedom. The switch of the internal state, that can freely be interpreted as a molecular switch, is modelled as a Markov jump process that depends on the location of the motor. Rescaling space and time, the limit of the trajectory of the diffusion process homogenizes over the periodic potential as well as over the internal degree of freedom. Around the homogenized limit, we prove the large deviation principle of trajectories with a method developed by Feng and Kurtz based on the analysis of an associated Hamilton--Jacobi--Bellman equation with an Hamiltonian that here, as an innovative fact, depends on both position and momenta.