Entropy on normed semigroups (towards a unifying approach to entropy)

We present a unifying approach to the study of entropies in mathematics, such as measure entropy, various forms of topological entropy, several notions of algebraic entropy, and two forms of set-theoretic entropy. We take into account only discrete dynamical systems, that is, pairs (X,\u3c6), where X is the underlying space (e.g., a probability space, a compact topological space, a group, a set) and \u3c6:X\u2192X is a transformation of X (e.g., a measure preserving transformation, a continuous selfmap, a group homomorphism, a selfmap). We see entropies as functions h:X\u2192R+, associating to each flow (X,\u3c6) of a category C either a non-negative real number or 1e. First, we introduce the notion of semigroup entropy hS:S\u2192R+, which is a numerical invariant attached to endomorphisms of the category S of normed semigroups. Then, for a functor F:C\u2192S from any specific category C to S, we define the functorial entropy hF:C\u2192R+ as the composition hS\u25e6F, that is, hF(\u3c6) = hS(F\u3c6) for any endomorphism \u3c6:X\u2192X in C. Clearly, hF inherits many of the properties of hS, depending also on the functor F. Motivated by this aspect, we study in detail the properties of hS. Such a general scheme, using elementary category theory, permits one to obtain many relevant known entropies as functorial entropies hF, for appropriately chosen categories C and functors F:C\u2192S. All of the above mentioned entropies are functorial. Furthermore, we exploit our scheme to elaborate a common approach to establishing the properties shared by those entropies that we find as functorial entropies, pointing out their common nature. We give also a detailed description of the limits of our approach, namely entropies which cannot be covered. Finally, we discuss and deeply analyze the relations between pairs of entropies through the looking glass of our unifying approach. To this end we first formalize the notion of Bridge Theorem between two entropies h1:C1\u2192R+ and h2:C2\u2192R+ with respect to a functor \u3b5:C1\u2192C2, taking inspiration from the known relation between the topological and the algebraic entropy via the Pontryagin duality functor. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem. It allows us to shelter various relations between pairs of entropies under the same umbrella (e.g., the above mentioned connection of the topological and the algebraic entropy, as well as their relation to the set-theoretic entropy)

Metric Versus Topological Receptive Entropy of Semigroup Actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54\u201398, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin\u2013Katok Formula and the local Variational Principle

TOPOLOGICAL ENTROPY, UPPER CARATHEODORY CAPACITY AND FRACTAL DIMENSIONS OF SEMIGROUP ACTIONS

We study dynamical systems given by the action T : G x X -> X of a finitely generated semigroup G with identity 1 on a compact metric space X by continuous selfmaps and with T(1, -) = id(X).For any finite generating set G(1) of G containing 1, the receptive topological entropy of G(1) (in the sense of Ghys et al. (1988) and Hofmann and Stoyanov (1995)) is shown to coincide with the limit of upper capacities of dynamically defined Caratheodory structures on X depending on G(1), and a similar result holds true for the classical topological entropy when G is amenable. Moreover, the receptive topological entropy and the topological entropy of G(1) are lower bounded by respective generalizations of Katok's delta-measure entropy, for delta is an element of (0, 1).In the case when T(g, -) is a locally expanding selfmap of X for every g is an element of G {1}, we show that the receptive topological entropy of G(1) dominates the Hausdorff dimension of X modulo a factor log lambda determined by the expanding coefficients of the elements of {T(g, -) : g is an element of G(1) {1}}

Finiteness of topological entropy for locally compact abelian groups

We study the locally compact abelian groups in the class E_infty, that is, having only continuous endomorphisms of finite topological entropy, and in its subclass E_0, that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to E_infty, and that those of them that belong to E_0 are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem

Algebraic entropy of shift endomorphisms on abelian groups

For every finite-to-one map \u3bb:\u393\u2192\u393 and for every abelian group K, the generalized shift \u3c3\u3bb of the direct sum 95_\u393 K is the endomorphism defined by (x_ i)\u21a6(x_\u3bb(i)). In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of K, but mainly on the function \u3bb. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples

Algebraic entropy in locally linearly compact vector spaces

We introduce algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as a natural extension of the algebraic entropy for endomorphisms of discrete vector spaces studied in Giordano Bruno and Salce (Arab J Math 1:69\u201387, 2012). We show that the main properties continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem

PLANT EXTRACTS AS GREEN POTENTIAL STRATEGIES TO CONTROL THE BIODETERIORATION OF CULTURAL HERITAGE

The biodeterioration of historic-artistic manufacts is related to several biological systems, including fungi and bacteria, whose metabolic activities and vegetative development have a direct consequence on the conservation of cultural assets. Generally, different chemical compounds are utilized as biocides in order to control biodeteriogens growth, but recently the attention has been focused on potential risks of their use towards human health (operators, visitors) and the environment. In order to develop alternative methods, various natural products have been tested, particularly to control the colonization by fungi and bacteria. In this study, antimicrobial activity of three different plant products, Tea tree essential oil, Calamintha nepeta and Allium sativum L. extracts, has been evaluated against Bacillus subtilis, Micrococcus luteus, Penicillium chrysogenum and Aspergillus spp. (previously isolated from colonized artworks) through three different in vitro antimicrobial assays (micro-dilution in microtiter plates, well plates diffusion and agar disc diffusion method). The bioassays show a different microbial susceptibility to plant extracts, establishing for each bacteria and fungi the Minimum Inhibitory Concentration (MIC) and defining the diameter of the growth inhibition area. This result supports the data reported in literature and shows an important potential suggestion for the possible use in the control of biodeterioration of cultural heritage, safe both for human health and environment

Modeling vaccination rollouts, SARS-CoV-2 variants and the requirement for non-pharmaceutical interventions in Italy

Despite progress in clinical care for patients with coronavirus disease 2019 (COVID-19)1, population-wide interventions are still crucial to manage the pandemic, which has been aggravated by the emergence of new, highly transmissible variants. In this study, we combined the SIDARTHE model2, which predicts the spread of SARS-CoV-2 infections, with a new data-based model that projects new cases onto casualties and healthcare system costs. Based on the Italian case study, we outline several scenarios: mass vaccination campaigns with different paces, different transmission rates due to new variants and different enforced countermeasures, including the alternation of opening and closure phases. Our results demonstrate that non-pharmaceutical interventions (NPIs) have a higher effect on the epidemic evolution than vaccination alone, advocating for the need to keep NPIs in place during the first phase of the vaccination campaign. Our model predicts that, from April 2021 to January 2022, in a scenario with no vaccine rollout and weak NPIs (R = 1.27), as many as 298,000 deaths associated with COVID-19 could occur. However, fast vaccination rollouts could reduce mortality to as few as 51,000 deaths. Implementation of restrictive NPIs (R = 0.9) could reduce COVID-19 deaths to 30,000 without vaccinating the population and to 18,000 with a fast rollout of vaccines. We also show that, if intermittent open\u2013close strategies are adopted, implementing a closing phase first could reduce deaths (from 47,000 to 27,000 with slow vaccine rollout) and healthcare system costs, without substantive aggravation of socioeconomic losses

Interventional radiology of the thyroid gland : critical review and state of the art

Thyroid nodules are a common incidental finding during a routinely ultrasound (US) exam unrelated to the thyroid gland in the healthy adult population with a prevalence of 20-76%. As treated before with surgery, in the last years new minimally invasive techniques have been developed as an alternative to surgery. The aim of this review, based on newly revised guidelines, is to provide some information regarding the basic principles, indications, materials, techniques, and results of mini-invasive procedures or treatments for thyroid nodules. We performed a narrative review including both newest and representative papers and guidelines based on the different procedures of ablation techniques developed in the last years for the diagnosis and the treatment of thyroid nodules. All examined papers referred very good results in term of volume nodule reduction, improvement in related symptoms and cosmetic problems, with a very low rate of complications and side effects for all the minimally invasive technique analyzed. Obviously, some differents between technique based on different kind of thyroid nodules and different indication were found. In conclusion, many thyroid nodules nowadays could be treated thanks to the advent of new mini-invasive technique that are less expensive and present a lower risk of major complications and side effects compared to surgery

FoxO3a as a positive prognostic marker and a therapeutic target in Tamoxifen-resistant breast cancer

Background: Resistance to endocrine treatments is a major clinical challenge in the management of estrogen receptor positive breast cancers. Although multiple mechanisms leading to endocrine resistance have been proposed, the poor outcome of this subgroup of patients demands additional studies. Methods: FoxO3a involvement in the acquisition and reversion of tamoxifen resistance was assessed in vitro in three parental ER+ breast cancer cells, MCF-7, T47D and ZR-75-1, in the deriving Tamoxifen resistant models (TamR) and in Tet-inducible TamR/FoxO3a stable cell lines, by growth curves, PLA, siRNA, RT-PCR, Western blot, Immunofluorescence, Transmission Electron Microscopy, TUNEL, cell cycle, proteomics analyses and animal models. FoxO3a clinical relevance was validated in silico by Kaplan−Meier survival curves. Results: Here, we show that tamoxifen resistant breast cancer cells (TamR) express low FoxO3a levels. The hyperactive growth factors signaling, characterizing these cells, leads to FoxO3a hyper-phosphorylation and subsequent proteasomal degradation. FoxO3a re-expression by using TamR tetracycline inducible cells or by treating TamR with the anticonvulsant lamotrigine (LTG), restored the sensitivity to the antiestrogen and strongly reduced tumor mass in TamR-derived mouse xenografts. Proteomics data unveiled novel potential mediators of FoxO3a anti-proliferative and pro-apoptotic activity, while the Kaplan−Meier analysis showed that FoxO3a is predictive of a positive response to tamoxifen therapy in Luminal A breast cancer patients. Conclusions: Altogether, our data indicate that FoxO3a is a key target to be exploited in endocrine-resistant tumors. In this context, LTG, being able to induce FoxO3a, might represent a valid candidate in combination therapy to prevent resistance to tamoxifen in patients at risk