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We face with the question of a suitable measure theory in Euclidean point-free geometry and we sketch out some possible solutions. The proposed measures, which are positive and invariant with respect to movements, are based on the notion of infinitesimal masses, i.e. masses whose associated supports form a sequence of finer and finer partitions.

We introduce a concept of modularity for a $$\hbox {d}_{0}$$ -algebraA in such a way that, in case of A being a lattice, it coincides with the usual one. We also show that if $$\mu $$ is a $$\hbox {d}_{0}$$ -measure on a $$\hbox {d}_{0}$$ -algebraA, the quotient of A modulo the ideal of $$\mu $$ -negligible elements is always modular.

We give examples showing that the Kolmogorov-Sinai entropy generator theorem is false for both upper and lower Riesz entropy of MV-algebraic dynamical systems, both two sided (i.e., analogous to two sided Bernoulli shifts) and one sided (i.e., analogous to one sided Bernoulli shifts).

We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on whichσ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.

We present the algebraic structures behind the approaches used to work with pairwise comparison matrices and, in general, the representation of preferences. We obtain a general definition of consistency and a universal decomposition in the space of PCMs, which allow us to define a consistency index. Also Arrow’s theorem, which is presented in a general form, is relevant. All the presented results can be seen in the main formulations of PCMs, i.e., multiplicative, additive and fuzzy approach, by the fact that each of them is a particular interpretation of the more general algebraic structure needed to deal with these theories.

Background: The mTORC1 inhibitor everolimus has been approved in combination with the aromatase inhibitor exemestane for the treatment of hormone receptor-positive (HR+) human epidermal growth factor receptor 2-negative (HER2&minus, ) metastatic breast cancer (HR+ mBC) progressing on prior therapy with a non-steroidal aromatase inhibitor. To date, no predictive biomarkers of tumor sensitivity/resistance for everolimus-based treatments have been identified. We hypothesized that precocious changes in the Standardized Uptake Volume (∆SUV%), as assessed by 18F-Fluorodeoxyglucosepositron-emission tomography (18F-FDG PET/CT), may be a marker of everolimus efficacy. Methods: This was a retrospective study including 31 HR+ HER2- patients treated with everolimus and exemestane in two Italian centers between 2013 and 2018. The objective of the study was to investigate ∆SUV% as a predictive marker of everolimus antitumor efficacy. 18F-FDG PET/CT scans were performed at baseline and after three months of treatment. Patients were defined as long responders (LRs) if disease progression occurred at least 10 months after treatment initiation and long survivors (LSs) if death occurred later than 36 months after starting therapy. ROC analysis was used to determine the optimal cut-off values of ∆SUV% to distinguish LRs from non-LRs and LSs from non-LSs. Progression-free survival (PFS) and overall survival (OS) were estimated by Kaplan&ndash, Meier method. Results: The SUVmax values decreased significantly from baseline to 3 months after therapy (p = 0.003). Dynamic changes of SUVmax (Delta SUV) had a higher accuracy in discriminating long-responders from non-long-responders (AUC = 0.67, Delta SUV cut-off = 28.8%) respects to its ability to identify long survivors from no-long survivors (AUC = 0.60, Delta SUV cut-off = 53.8%). Patients were divided into groups according to the Delta SUV cut-offs and survival outcomes were evaluated: patients with a decrease of ∆SUV% &ge, 28.8% had significantly better PFS (10 months-PFS: 63.2%, 95% CI: 37.9&ndash, 80.4% and 16.7%, 95% CI: 2.7&ndash, 41.3% respectively, p = 0.005). As regard as OS, patients with ∆SUV% &ge, 53.8% had longer OS when compared to patients with ∆SUV% &lt, 53.8% (36 month-OS: 82.5% vs. 45.9% vs. p = 0.048). Conclusion: We found two precocious ∆SUV% thresholds capable of identifying HR+ HER2-mBC patients, which would achieve long-term benefit or long-term survival during everolimus-exemestane therapy. These results warrant further validation in prospective studies and should be integrated with molecular biomarkers related to tumor metabolism and mTORC1 signaling.

Introduction Reliable biomarkers of response to mTOR inhibition are yet to be identified. As mTOR is heavily implicated in cell-metabolism, we investigated the relation between BMI variation and outcomes in metastatic breast cancer (mBC) patients treated with everolimus. Results we found a linear correlation between everolimus exposure duration and BMI/weight decrease. Patients exhibiting >2 kg weight loss or >3% BMI decrease from baseline at the end of treatment (EOT) had a statistically significant improvement in PFS. Interestingly, a similar BMI/weight decrease within the first 8 weeks of therapy identified patients at higher risk of progression. Patients and methods we performed a retrospective analysis of patients enrolled in the BALLET trial who progressed during the study. Primary end-point was progression-free survival (PFS). Secondary end-point was the identification of other predictors of response. Conclusion A >3% weight loss at EOT is associated with better outcome in mBC patients treated with everolimus. On the contrary, a significant early weight loss represents a predictor of poor survival and could therefore be used as an early negative prognostic marker. As PI3K-inhibition also converges onto mTOR, these findings might extend to patients treated with selective PI3K inhibitors and warrant further investigation.

A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u = ( u n ) n ∈ N 0 in N such that ‖ u n x ‖ → 0 uniformly on A . In particular, A is contained in the subgroup t u ( T ) : = { x ∈ T : ‖ u n x ‖ → 0 } , which is the subgroup of T characterized by u . Using strictly increasing sequences u in N such that u n divides u n + 1 for every n ∈ N , we find in T a family of closed perfect D -sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D -sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Finally, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.

A subgroup H of R is characterized if H = τ u ( R ) : = { x ∈ R : u n x → 0 mod Z } for some sequence u in R . Given two sequences u and v in R , we find conditions under which τ u ( R ) is contained or not in τ v ( R ) . As a by-product of our main theorems, we find a known result by Elias on inclusions of characterized subgroups of T , motivated by problems in harmonic analysis.

In this article, we will describe the embedded case study of the European project “E-Magic” carried out within the Erasmus+ project and which involved 12-16 years old students and teachers from three nations: “Acharnes Special Education Vocational School” in Athens (Greece), IIS “L. Da Vinci-Nitti”, a high school in Potenza (Italy), and from Gabinete de Modernizao das Tecnologias Educativas (G.M.T.E.) in Madeira Islands (Portugal). The project is finalized to enhance Mathematical Competencies by Serious Games learning. We carried out both empirical qualitative and quantitative methods to highlight Serious Games’ effectiveness, as a teaching methodology, in motivating students to learn mathematics. We also hypothesize that prompting a strategic way of thinking within a didactical intervention by game-approach could improve students’ proving processes and support them in mathematical thinking production. Rabardel’s cognitive ergonomics and the Technological Pedagogical Content Knowledge model are the lenses for reading educational experiments’ data. Data seems to show that simulation helps students find alternative and creative solutions to problems. Therefore, Serious Games’ use in education promotes significant changes and mental resources activation.

We prove an algebraic and a topological decomposition theorem for complete pseudo-D-lattices (i.e. lattice-ordered pseudo-effect algebras). As a consequence, we obtain a Hammer–Sobczyk type decomposition theorem for group-valued modular measures defined on pseudo-D-lattices and compactness of the range of every \(\mathbb {R}^{n} \)-valued σ-additive modular measure on a σ-complete pseudo-D-lattice.

We investigate some properties of (uniformly) absolutely continuous, singular and non-concentrated k-subadditive lattice group-valued capacities, and we examine some relations between them. Furthermore, we pose some open problems.

We prove a Lyapunov type theorem for closed pseudo-non-injective modular measures on pseudo-D-lattices and we prove that any closed modular measure can be decomposed into the sum of a Lyapunov modular measure and an anti-Lyapunov modular measure.

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete l-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete l-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for l-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.

We prove an extension theorem for modular measures on lattice ordered effect algebras. This is used to obtain a representation of these measures by the classical ones. With the aid of this theorem we transfer control theorems, Vitali-Hahn-Saks, Nikodým theorems and range theorems to this setting.

We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a Hammer�Sobczyk type decomposition theorem for modular measures on D-lattices.

For a recursively defined sequence u := (un) of integers, we describe the subgroup tu(T) of the elements x of the circle group T satisfying limn unx =0 . More attention is dedicated to the sequences satisfying a second- order recurrence relation. In this case, we show that the size and the free-rank of tu(T) is determined by the asymptotic behaviour of the ratios qn = u n un! 1 and we extend previous results of G. Larcher, C. Kraaikamp, and P. Liardet obtained from continued fraction expansion.

As a natural generalization of a measure space, Butnariu and Klement introduced T-tribes of fuzzy sets with T-measures. They made the first steps towards a characterization of monotonic real-valued T-measures for a Frank triangular norm T. Later on, Mesiar and the authors of this paper found independently two generalizations, one for vector-valued T-measures with respect to Frank t-norms (in particular for nonmonotonic ones) [3], the other for monotonic real-valued T-measures with respect to general strict t-norms [15]. Here we present a common generalization --- a characterization of nonmonotonic T-measures with respect to an arbitrary strict t-norm. Moreover, we prove this for vector-valued T-measures. Using this characterization, we generalize Ljapunov Theorem to this context.

A lattice ordered group valued measure is extended from a D-lattice into a σ-complete D-lattice. A D-lattice is a lattice with a greatest element 1 and a smallest element 0 endowed with an order-compatible operation, called a difference, which satisfies a list of axioms. The result generalizes the classical result known for measures on Boolean algebras.

We deal with measures on Δ-l-semigroups, in particular on MV-algebras. Examples of such measures are T∞-valuations on clans of fuzzy sets. We first provide the Hahn decomposition theorem for measures on Δ-l-semigroups. This is then used to obtain a representation theorem for such measures, which itself is a basic tool in the proof of Liapounoff type theorems.

We extend Liapunov theorem to R n -valued finitely additive measures of fuzzy sets. Moreover, we study the convexity of the range of B-convex-valued measures.

We prove that every R N -valued σ -order continuous modular measure μ , defined on a σ -complete D-lattice L , has a closed range, and is open if L is μ -chained.

It is proved that if X is innite-dimension al, then there exists an innite- dimensional space of X-valued measures which have innite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that ca(B;;X ) n M , the measures with non- -nite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Mu~ noz Fern andez et al. (Linear Algebra Appl. 428 (2008)).

One of the most loved and celebrated theorems of measure theory is “Lyapunov’s theorem” which states that the range of a nonatomic σ -additive measure on a σ -algebra with values in a finite dimensional vector space is convex. In this paper I extend Lyapunov’s theorem to measures defined on D-posets. My proof follows Halmos’ idea and reduces the proof to the semi-convex case. Indeed, I extend Lyapunov’s theorem to measures defined on a weaker algebraic structure which is endowed with an ordering and a partial operation compatible with order as explained in Section 4. Every D-poset satisfies these axioms and so does every complemented modular lattice. So Theorem 4.10 generalizes Avallone’s version, (Avallone, 1995) Theorem 2.3, of Lyapunov’s theorem valid for modular function on complemented lattices. In (Avallone and Basile, 2003), Avallone and Basile have applied the version of Lyapunov’s theorem valid for D-posets, Theorem 3.6 in an economic context. D-posets have been introduced in Chovanec and Kopka (1994) as a generalization of many structures, as orthomodular posets, orthoalgebras and MV-algebras. Therefore the study of measures on D-posets allows us to unify the study of measures on orthomodular posets in noncommutative measure theory and measures on MV-algebras in fuzzy measure theory. I wish to express my gratitude to Professor Hans Weber for his helpful suggestions.

Let L be a complemented or sectionally complemented lattice and let X be a locally convex linear space. We prove the existence of an X-valued control for uniformly exhaustive sequences of X -valued modular functions on L. Moreover, if L is cr-complete, we prove that every countable family of cr-o.c. nonatomic group-valued modular functions on L admits separating points.

A topological approach to the study of fuzzy measures is developed. To do so we need (instead of a clan of fuzzy sets) a more general structure of the domain of the fuzzy measures. This structure is defined by means of some equations. Our general setting allows us to treat simultaneously fuzzy measures, group-valued measures on Boolean rings, and linear operators on Riesz spaces. We deal with extension and decomposition theorems. Also we study connected, totally disconnected, and compact MV-algebras.

We answer a question of Raczkowski on totally bounded Hausdorff group topologies on the integers with a convergent sequence (un). More specifically, we show that for fast growing and for slowly growing sequences (un) the asymptotic behaviour of the ratio un+1/un leads to rather specific properties of the topologies in question. (a) If un+1/un→∞, then there exists a totally bounded Hausdorff group topology of weight c on Z that makes (un) converging to 0. (b) If un+1/un is bounded, then every group topology as in (a) must be metrizable (i.e., has weight ℵ 0 ). We also show (under the assumption of Martin's Axiom) that there exists a precompact group topology τ on Z without non-trivial convergent sequences generated by a measure-zero subgroup H of T .