Leibniz's Principles and Topological Extensions

Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object", "everything can possibly exist, unless it yields contradiction", "the ideal elements correctly determine the real things". Here we give a precise formulation of these principles within the framework of the Topological Extensions of [8], structures that generalize at once compactifications, completions, and nonstandard extensions. In this topological context, the above Leibniz's principles appear as a property of separation, a property of compactness, and a property of analyticity, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnz's principles can be fulfilled in pairs, but not all three together.Comment: 16 page

A topological interpretation of three Leibnizian principles within the functional extensions

Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object" (Identity of indiscernibles), "everything can possibly exist, unless it yields contradiction" (Possibility as consistency), and "the ideal elements correctly determine the real things" (Transfer). Here we give a precise logico-mathematical formulation of these principles within the framework of the Functional Extensions, mathematical structures that generalize at once compactifications, completions, and elementary extensions of models. In this context, the above Leibnizian principles appear as topological or algebraic properties, namely: a property of separation, a property of compactness, and a property of directeness, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnizian principles may be fulfilled in pairs, but not all three together.Comment: arXiv admin note: substantial text overlap with arXiv:1012.434

Quasi-selective ultrafilters and asymptotic numerosities

We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of "asymptotic numerosities" for all sets of tuples of natural numbers. Such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptotic density to all sequences of tuples of natural numbers.Comment: 27 page

A Euclidean comparison theory for the size of sets

We discuss two main ways in comparing and evaluating the size of sets: the "Cantorian" way, grounded on the so called Hume principle (two sets have equal size if they are equipotent), and the "Euclidean" way, maintaining Euclid's principle "the whole is greater than the part". The former being deeply investigated since the very birth of set theory, we concentrate here on the "Euclidean" notion of size (numerosity), that maintains the Cantorain defiitions of order, addition and multiplication, while preserving the natural idea that a set is (strictly) larger than its proper subsets. These numerosities satisfy the five Euclid's common notions, and constitute a semiring of nonstandarda natural numbers, thus enjoying the best arithmetic. Most relevant is the natural set theoretic definition} of the set-preordering: X\prec Y\ \ \Iff\ \ \exists Z\ X\simeq Z\subset Y Extending this ``proper subset property" from countable to uncountable sets has been the main open question in this area from the beginning of the century.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:2207.0750

Dynamic factor models: improvements and applications

In the last two decades data collection, aided by an increased computational capability, has considerably increased both dimension and structure of the datasets; given this, statisticians and economists may today work with time series of remarkable dimension which may come from different sources. Dealing with such datasets may not be so easy and requires the development of ad hoc mathematical models. Dynamic Factor Models (DFM) represent one of the newest techniques in big data management. The adoption of those models allowed me to deepen the study of volatility while introducing Bayesian non-parametric techniques, and to do structural analysis improving the generated impulse response functions. The application of this all was made in the field of economics and finance

Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities

We introduce a "Euclidean" notion of size (numerosity) for "Punktmengen", i.e. sets of points of Euclidean (finitely dimensional) spaces over any "line" L, namely one that maintains the Cantorian defiitions of order, addition and multiplication, while preserving the ancient principle that "the whole is greater than the part" (a set is (strictly) larger than its proper subsets). These numerosities satisfy the five Euclid's common notions, thus enjoying a very good arithmetic, since they constitute the nonnegative part of the ordered ring of the Euclidean integers, here introduced by suitably assigning a transfinite sum to (ordinally indexed) kappa-sequences of integers (so generating a semiring of nonstandard natural numbers). Most relevant is the natural set theoretic definition of the set-preordering <: given any two sets X, Y of any cardinality, one has X<Y if and only if there exists a proper superset of X that is equinumerous to Y . Extending this "superset property" from countable to uncountable sets has been one of the main open question in this area from the beginning of the century

Prevalence of Chronic Cancer and No-Cancer Pain in Elderly Hospitalized Patients: Elements for the Early Assessment of Palliative Care Needs

Summary: Background: We studied prevalence of chronic pain, related or not to cancer, in elderly patients, its correlation with socio-clinical factors, and its effects on daily living, to estimate feasibility of an early assessment of palliative care needs in a non-specialist hospital setting. Methods: In this prospective study, a questionnaire concerning pain and multidimensional assessment tools were administered to patients consecutively admitted to a Department of Internal Medicine comprising a Stroke Unit. Results: One hundred patients were recruited, 38 of whom experiencing pain, chronic in 26 patients (68%). A total of 34.3% of patients with pain and 12.5% of patients without pain suffered from depression (P = 0.013). Depressed patients showed significantly higher median values in all Brief Pain Inventory (BPI) scores and items. Depressed patients also obtained less pain relief from therapies. Patients with mild dementia showed, significantly or as a trend, a higher median least, average and "pain right now" pain values. Worst pain values in the previous 24 h increased with age. Only 42% of patients reported to be on pain therapy upon admission to hospital, whereas 62% were undergoing treatment at the time of discharge. A correlation was found between the pain value and the level of interference with daily activities. Pain was mentioned in the discharge letter in 36% of cases. Conclusion: Pain is a critical underestimated problem in elderly patients. A timely systematic evaluation of the pain would call attention to palliative care needs and reduce the negative effects of uncontrolled pain on the quality of life. Keywords: Pain assessment, Pain prevalence, Elderly patients, Pain and depression, Pain and activities of daily livin

Cardinal invariants and independence results in the poset of precompact group topologies

AbstractWe study the poset B(G) of all precompact Hausdorff group topologies on an infinite group G and its subposet Bσ(G) of topologies of weight σ, extending earlier results of Berhanu, Comfort, Reid, Remus, Ross, Dikranjan, and others. We show that if Bσ(G) ≠ ∅ and 2¦GG′¦ = 2¦G¦ (in particular, if G is abelian) then the poset [2¦G¦]σ of all subsets of 2¦G¦ of size σ can be embedded into Bσ(G) (and vice versa). So the study of many features (depth, height, width, size of chains, etc.) of the poset Bσ(G) is reduced to purely set-theoretical problems. We introduce a cardinal function Dede(σ) to measure the length of chains in [X]σ for ¦X¦> σ generalizing the well-known cardinal function Ded(σ). We prove that Dede(σ) = Ded(σ) iff cf Ded(σ) ≠ σ+ and we use earlier results of Mitchell and Baumgartner to show that Dede(N1) = Ded(N1) is independent of Zermelo-Fraenkel set theory (ZFC). We apply this result to show that it cannot be established in ZFC whether BN1(Z) has chains of bigger size than those of the bounded chains.We prove that the poset HN0(G) of all Hausdorff metrizable group topologies on the group G = ⊕N0 Z2 has uncountable depth, hence cannot be embedded into BN0(G). This is to be contrasted with the fact that for every infinite abelian group G the poset H(G) of all Hausdorff group topologies on G can be embedded into B(G). We also prove that it is independent of ZFC whether the poset HN0(G) has the same height as the poset BN0(G)

Convergence of Discrete-Time Cellular Neural Networks with Application to Image Processing

The paper considers a class of discrete-time cellular neural networks (DT-CNNs) obtained by applying Euler's discretization scheme to standard CNNs. Let T be the DT-CNN interconnection matrix which is defined by the feedback cloning template. The paper shows that a DT-CNN is convergent, i.e. each solution tends to an equilibrium point, when T is symmetric and, in the case where T + En is not positive-semidefinite, the step size of Euler's discretization scheme does not exceed a given bound (En is the n × n unit matrix). It is shown that two relevant properties hold as a consequence of the local and space-invariant interconnecting structure of a DT-CNN, namely: (1) the bound on the step size can be easily estimated via the elements of the DT-CNN feedback cloning template only; (2) the bound is independent of the DT-CNN dimension. These two properties make DT-CNNs very effective in view of computer simulations and for the practical applications to high-dimensional processing tasks. The obtained results are proved via Lyapunov approach and LaSalle's Invariance Principle in combination with some fundamental inequalities enjoyed by the projection operator on a convex set. The results are compared with previous ones in the literature on the convergence of DT-CNNs and also with those obtained for different neural network models as the Brain-State-in-a-Box model. Finally, the results on convergence are illustrated via the application to some relevant 2D and 1D DT-CNNs for image processing tasks