This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy.We first analyze the strength of the open and (...) clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility.We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain $\mathbb {N}$ or is single-valued.These notions proved to be extremely useful when exploring the Weihrauch degree of the problem $\mathsf {DS}$ of computing descending sequences in ill-founded linear orders. They allow us to show that $\mathsf {DS}$, and the Weihrauch equivalent problem $\mathsf {BS}$ of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$, $\boldsymbol {\Sigma }^1_1$, $\boldsymbol {\Pi }^1_1$. We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the Baire hierarchy and show that they do not collapse at any finite level.Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace $\mathbf {K}$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol {\Pi }^0_3$ -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf {K}$. We also generalize the results by relaxing the compactness of the ambient space and show that the closed Salem sets are still $\boldsymbol {\Pi }^0_3$ -complete when we endow $\mathbf {F}$ with the Fell topology. A similar result holds also for the Vietoris topology.We conclude by analyzing the same notions from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity, and show that the complexities remain the same also in the lightface case. In particular, we show that the family of all the closed Salem sets is $\Pi ^0_3$ -complete. We furthermore characterize the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.prepared by Manlio Valenti.E-mail:[email protected]. (shrink)

The aim of this article is to identify the main factors of the current crisis of the nation-state and to demonstrate how many of the voids left by this crisis are filled by religions. The main characteristic of the nation-state is the principle of sovereignty. The apogee of the nation-state is the political form of industrialization. National identity is possible only when the state proves to its citizens that the fact of being a member of it carries benefits and privileges (...) and will always bring more. Today, the majority of nation-states, in particular the oldest great powers, no longer have this capability. The weakening of the nation-state began at the end of the 19th century. The first wave of globalization multiplied the cases of reciprocal interferences and trespassed on the theoretical impermeability of the sovereign states. The outcome of the First World War, with the creation of the first supranational body, and much more the outcome of the Second World War, were two important steps of this crisis. The birth of the United Nations, and of other supranational bodies, as well as the creation of the first court called to judge an entire political class, were an assault on the principle of sovereignty. The second wave of globalization, characterized by the free circulation of goods, money, people and cultures, did the rest. Moreover, the countries that ‘invented’ the principle of sovereignty are today in relative decline as new powers are emerging. The nation-state is no longer able to keep its promises. The less effective states become at offering their citizens both meaning and social services, the more do religions tend to reoccupy the public stage. The less national and political legitimacy they have, the more powers use the religious tool against one another. (shrink)

Inquieta anche il fatto che i nuovi giardini tendano ad assomigliare in modo impressionante a quei “rendering”, anch’essi uno standard, che si trovano esposti nelle vetrine delle sempre più numerose agenzie immobiliari che impestano la periferia diffusa. Ne avevo una vicino a casa qualche trasloco fa, quando abitavo in un appartamento di due stanze con bagno nel quartiere di Santa Lucia.Vitaliano Trevisan, Tristissimi giardini, pp. 46-47 Dopo la mela biblica consumata in un giardino terrestre...

Inquieta anche il fatto che i nuovi giardini tendano ad assomigliare in modo impressionante a quei “rendering”, anch’essi uno standard, che si trovano esposti nelle vetrine delle sempre più numerose agenzie immobiliari che impestano la periferia diffusa. Ne avevo una vicino a casa qualche trasloco fa, quando abitavo in un appartamento di due stanze con bagno nel quartiere di Santa Lucia.Vitaliano Trevisan, Tristissimi giardini, pp. 46-47 Dopo la mela biblica consumata in un giardino terrestre...

Contrary to the position sustained in recent German- and English-language studies, the Author of this note rejects the suitability of using the modern and generic term “Homoianism” to describe the different subordinationist doctrinal positions circulating during the second half of the fourth century. In the case of Eudoxius, Valens, Urascius and Ulfila, among others, it is more appropriate to continue to speak in terms of “Arianism”, as their Nicene opponents had already realized, not without reason.

Consider any subvariety of BL-algebras generated by a single BL-chain which is the ordinal sum of the standard MV-algebra on [0, 1] and a basic hoop H. We present a geometrical characterization of elements in the finitely generated free algebra of each of these subvarieties. In this characterization there is a clear insight of the role of the regular and dense elements of the generating chain. As an application, we analyze maximal and prime filters in the free algebra.

An l-hemi-implicative semilattice is an algebra $$\mathbf {A} = $$ A= such that $$$$ is a semilattice with a greatest element 1 and satisfies: for every $$a,b,c\in A$$ a,b,c∈A, $$a\le b\rightarrow c$$ a≤b→c implies $$a\wedge b \le c$$ a∧b≤c and $$a\rightarrow a = 1$$ a→a=1. An l-hemi-implicative semilattice is commutative if if it satisfies that $$a\rightarrow b = b\rightarrow a$$ a→b=b→a for every $$a,b\in A$$ a,b∈A. It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide (...) a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an derived operation by $$a \sim b := \wedge $$ a∼b:=∧. Endowing $$$$ with the binary operation $$\sim $$ ∼ the algebra $$$$ results an l-hemi-implicative semilattice, which also satisfies the identity $$a \sim b = b \sim a$$ a∼b=b∼a. In this article, we characterize the commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element. Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices. (shrink)

Una prima parte del saggio sintetizza le linee principali di un recente libro dell’autore: Per un paradigma del corpo: una rifondazione filosofica dell’ecologia, Quodlibet, Macerata, 2019. Viene sostenuto un paradigma in cui corpo e biologia mantengano la loro irriducibilità nei confronti della cultura e in cui il cosiddetto “linguistic turn”, che ha dominato la filosofia del XX secolo, viene fortemente criticato; in coerenza con questo, la riflessione di Merleau-Ponty sul concetto di natura viene considerata lo strumento più adeguato e più (...) attuale per ripensare la questione ecologica; ne vengono esaminate le conseguenze nel campo della filosofia della tecnologia e del problema del soggetto. Una seconda parte del saggio, che si concentra sull’incontro che ebbe luogo nel 1961 fra Adorno e Merleau-Ponty, cerca di abbozzare un paradigma materialistico per l’ecologia, correggendo l’avvicinamento che alla fine della sua vita vi fu fra Merleau-Ponty e Heidegger. (shrink)

The emergence of ecology as a fundamental horizon not only of politics, but also of contemporary philosophy, pushes us to rethink the relationship that currents of thought such as Kuturkritik and phenomenology, especially the phenomenology of Maurice Merleau-Ponty, can maintain with it. After a preliminary consideration of post-structuralist and postmodern positions from this perspective, the essay focuses on the French philosopher, and in particular on his courses on Nature and on his elaboration of the Husserlian concept of Stiftung. This results (...) in Merleau-Ponty’s original position on the Kultur – Civilization alternative: critique of unlimited productivism, respect for the environment, and, at the same time, defense of the variety of cultures and traditions from a homogenizing globalization are valuable tools that his philosophy offers to the debates of our time. (shrink)

Classification learning is a preeminent human ability within the animal kingdom but the key mechanisms of brain networks regulating learning remain mostly elusive. Recent neuroimaging advancements have depicted human brain as a complex graph machinery where brain regions are nodes and coherent activities among them represent the functional connections. While long-term motor memories have been found to alter functional connectivity in the resting human brain, a graph topological investigation of the short-time effects of learning are still not widely investigated. For (...) instance, classification learning is known to orchestrate rapid modulation of diverse memory systems like short-term and visual working memories but how the brain functional connectome accommodates such modulations is unclear. We used publicly available repositories (openfmri.org) selecting three experiments, two focused on short-term classification learning along two consecutive runs where learning was promoted by trial-by-trial feedback errors, while a further experiment was used as supplementary control. We analyzed the functional connectivity extracted from BOLD fMRI signals, and estimated the graph information processing in the cerebral networks. The information processing capability, characterized by complex network statistics, significantly improved over runs, together with the subject classification accuracy. Instead, null-learning experiments, where feedbacks came with poor consistency, did not provoke any significant change in the functional connectivity over runs. We propose that learning induces fast modifications in the overall brain network dynamics, definitely ameliorating the short-term potential of the brain to process and integrate information, a dynamic consistently orchestrated by modulations of the functional connections among specific brain regions. (shrink)

The COVID-19 outbreak has seen people in many countries asked to radically modify their way of life in compliance with sweeping safety measures. During the current crisis, technology is turning out to be key, in that it allows practitioners to deliver psychological services to people who would otherwise be unreachable. However, professionals cannot solely rely on their traditional modes of practice, in that different methods are required to bring to light the needs of those affected by the emergency. People are (...) being overwhelmed by a cascade of unusual and unexpected events that are putting a strain on their everyday routines and usual meaning-making systems; ongoing challenges to their employment and financial status will likely divert personal resources away from psychological well-being. We therefore argue that psychologists should also consider the needs of the general population. Among those who may require help – aside from the main targets of psychological intervention, such as healthcare personnel and COVID-19 patients and their relatives – specific attention should be paid to those who are not at the center of the crisis. We suggest that this large segment of potential users may benefit from a non-medical approach focused on the promotion of meaning-making processes. Indeed, the disruptive nature of the current situation hinders sense-making and threatens to undermine psychological balance and well-being, at an individual as well as at a societal level. The present article proposes a methodological perspective based on the reconstruction of meaning-making processes (sense of coherence, predictability, metaphors, narratives). Specifically, psychological interventions should promote personal and collective resources with a view to: “normalizing” current distressful experiences (i.e., acknowledging that such reactions are normal in light of the present situation); widening the observational field, taking relational contexts into account, and promoting an understanding of distressful experiences as coping strategies; fostering meaning-making/reconstruction processes through the use of appropriate metaphors and narratives; promoting a sense of coherence. We present two clinical vignettes to illustrate how these principles might be applied in practice. In conclusion, the exceptional psychological challenges posed by the COVID-19 pandemic require practitioners to adopt a broad and flexible perspective on clinical intervention. (shrink)