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We give some versions of Hahn-Banach, sandwich, duality, Moreau--Rockafellar-type theorems, optimality conditions and a formula for the subdifferential of composite functions for order continuous vector lattice-valued operators, invariant or equivariant with respect to a fixed group G of homomorphisms. As applications to optimization problems with both convex and linear constraints, we present some Farkas and Kuhn-Tucker-type results.

Lipid disorders have been implicated in overweight and menopause. However, evidence on lipidomic analysis of fatty acids in erythrocytes of menopausal women is scarce. The aim of this study was to investigate the relationship between the body mass index within or beyond 5 years of menopause and erythrocyte fatty acid profile. This case-control study was conducted on out of 37 menopausal women total patients, 22 with body mass index ≥ 25 and 12 matched controls (body mass index 25 and non-physiological content of fatty acids, were richer in percentage of saturated fatty acids and poorer of monounsaturated fatty acids and polyunsaturated fatty acids than women with BMI25 than in women with BMI

The main purpose of this paper is to apply the theory of vector lattices and the related abstract modular convergence to the context of Mellin-type kernels and (non)linear vector lattice-valued operators, following the construction of an integral given in earlier papers., 15 pages

A "Bochner-type" integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion., 29 pages

We give some limit theorems for sequences of lattice group-valuedk-triangular set functions,in the setting of filter convergence, and some results about their equivalence. We use the toolof filter exhaustiveness to get uniform (s)-boundedness, uniform continuity and uniform regular-ity of a suitable subsequence of the given sequence, whose indexes belong to the involved filter. Furthermore we pose some open problems.

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.

In this paper we give a survey about recent versions of Korovkin-type theorems for modular function spaces, a class which includes $L^p$, Orlicz, Musielak-Orlicz spaces and many others. We consider various kinds of modular convergence, using certain summability processes, like triangular matrix statistical convergence, and filter convergence (which are generalizations of the statistical convergence). Finally, wwe consider an abstract axiomatic convergence which includes the previous ones and even almost convergence, which is not generated by any filter, as we show by an example.

We prove a Korovkin-type approximation theorem using the relative uniform convergence of a sequence of functions at a point, which is a method stronger than the classical ones. We give some examples on this new convergence method and we study also rates of convergence.

The fatty acid composition of human breast milk is relevant for the energy, immunity and eicosanoid production in infants. Additionally, the antioxidant properties of foods are essential for human health. Therefore, in the present study we aimed to investigate the relationship between maternal diet and fatty acids composition as well as the antioxidant potential of breast milk from donors to human milk bank of Perugia&rsquo, s hospital, Italy. Results were compared with infant formulas. We observed increased levels of total fatty acids and, in particular, saturated and monounsaturated fatty acids in milk from mothers fed on a vegetable and fruit-rich diet compared with a Mediterranean diet. In the same milk, a reduced antioxidant potential was found. All infant formulas resulted in richer total fatty acid content than human breast milk. Only some formulas were qualitatively similar to breast milk. Of note, the antioxidant potential of the formulas was higher or lower than the human milk with the exception of one sample. The antioxidant potential of four formulas was very high. Dietary supplementation with antioxidants has been shown to have a teratogenic effect and to increase the formation of metastases in adult. There are no data on the effects of excess antioxidants in the infants, but the possibility that they can be harmful cannot be excluded.

Affine representations for archimedean $${\ell}$$ -groups and semisimple MV-algebras via embedding theorems are presented; they are simple to work with but powerful enough to express significant properties of our studied objects. Indeed, we focus on the space of particular homomorphisms between an archimedean $${\ell}$$ -group (a semisimple MV-algebra, respectively) and a vector lattice (a Riesz MV-algebra, respectively), i.e., the set of the generalized states, providing a general framework.

We prove Hahn-Banach, sandwich and extension theorems for vector lattice-valued operators, equivariant with respect to a given group G of homomorphisms. As applications and consequences, we present some Fenchel duality and separation theorems, a version of the Moreau-Rockafellar formula and some Farkas and Kuhn-Tucker-type optimization results. Finally, we prove that the obtained results are equivalent to the amenability of G.

Using the Maeda–Ogasawara–Vulikh representation theorem and sliding hump-type techniques, we prove some Dieudonne-type theorems for k-triangular set functions, taking values in lattice groups.

We introduce Kuelbs&ndash, Steadman-type spaces ( K S p spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of L q -type spaces into K S p spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals.

We give a direct proof of sandwich-type theorems for linear invariant partially ordered vector space operators in the setting of convexity. As consequences, we deduce equivalence results between sandwich, Hahn-Banach, separation and Krein-type extension theorems, Fenchel duality, Farkas and Kuhn-Tucker-type minimization results and subdifferential formulas in the context of invariance. As applications, we give Tarski-type extension theorems and related examples for vector lattice-valued invariant probabilities, defined on suitable kinds of events.

Introduction: In this paper, we deal with the demosaicing problem when the Bayer pattern is used. We propose a fast heuristic algorithm, consisting of three parts. Methods: In the first one, we initialize the green channel by means of an edge-directed and weighted average technique. In the second part, the red and blue channels are updated, thanks to an equality constraint on the second derivatives. The third part consists of a constant-hue-based interpolation. Results: We show experimentally how the proposed algorithm gives in mean better reconstructions than more computationally expensive algorithms.

Some versions of Dieudonne-type convergence and uniform boundedness theorems are proved, for k-triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case.

We prove sandwich, Hahn-Banach, Fenchel duality theorems and a version of the Moreau-Rockafellar formula for invariant partially ordered vector space-valued operators. As consequences and applications, we give some versions of Farkas and Kuhn-Tucker-type optimization results and separation theorems, we prove the equivalence of these results and give a further application to Tarski-type theorems and probability measures defined on suitable product spaces.

We investigate some of the main properties of lattice group-valued k-triangular set functions and we prove some Brooks–Jewett, Nikodým, Vitali–Hahn–Saks and Schur-type theorems and their equivalence. A Drewnowski-type theorem on existence of continuous restrictions of (s)-bounded set functions is given.

We investigate some properties of lattice group-valued positive, monotone and k-subadditive set functions, and in particular, we give some comparisons between regularity and continuity from above. Moreover, we prove different kinds of limit theorems with respect to filter convergence. Furthermore, some open problems are posed.

We give some Korovkin-type theorems on convergence and estimates of rates of approximations of nets of functions, satisfying suitable axioms, whose particular cases are filter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some applications to Mellin-type convolution and bivariate Kantorovich-type discrete operators.

The theory of Henstock—Kurzweil integral is generalized to the case of functions ranging in complex Riesz space R and defined on any zero-dimensional compact Abelian group. The constructed integral is used to solve the problem of recovering the R-valued coefficients of series in systems of characters of these groups by using generalized Fourier formulas.

We introduce a concept of convergence of order ?, with 0 < ? ? 1, with respect to a summability matrix method A for sequences (which generalizes the notion of statistical convergence of order ?), taking values in (?)-groups. Some main properties and differences with the classical A-convergence are investigated. A Cauchy-type criterion and a closedness result for the space of convergent sequences according our notion is proved.

A Vitali-type theorem for vector lattice-valued modulars with respect to filter convergence is proved. Some applications are given to modular convergence theorems for moment operatorsin the vector lattice setting, and also for the Brownian motion and related stochastic processes., 19 pages, 2 figures

In this paper we present some different types of ideal convergence/divergence and of ideal continuity for Riesz space-valued functions, and prove some basic properties and comparison results. We investigate the relations among different modes of ideal continuity and present a characterization of the (AP)-property for ideals of an abstract set Λ. Finally we pose some open problems.

We present some convergence and boundedness theorems with respect to filter convergence for lattice group-valued measures. We give a direct proof, based on the sliding hump argument. Furthermore we pose some open problems.

We study the problem of approximating a real-valued function f by considering sequences of general operators of sampling type, which include both discrete and integral ones. This approach gives a unitary treatment of various kinds of classical operators, such as Urysohn integral operators, in particular convolution integrals, and generalized sampling series.