In this paper, we investigate the reducibility property of semidirect products of the form V * D relatively to (pointlike) systems of equations of the form x 1 = · · · = x n , where D denotes the pseudovariety of definite semigroups. We establish a connection between pointlike reducibility of V * D and the pointlike reducibility of the pseudovariety V. In particular, for the canonical signature κ consisting of the multiplication and the (ω − 1)power, we show that V * D is pointlike κ-reducible when V is pointlike κ-reducible.

In this paper we prove that the pseudovariety LSl of local semilattices is completely κ-reducible, where κ is the implicit signature consisting of the multiplication and the ω-power. Informally speaking, given a finite equation system with rational constraints, the existence of a solution by pseudowords of the system over LSl implies the existence of a solution by κ-words of the system over LSl satisfying the same constraints.

In this paper we exhibit a type of semigroup presentations which determines a class of local groups. We show that the finite elements of this class generate the pseudovariety LG of all finite local groups and use them as test-semigroups to prove that LG and S, the pseudovariety of all finite semigroups, verify the same κ-identities involving κ-terms of rank at most 1, where κ denotes the implicit signature consisting of the multiplication and the (ω − 1)-power. of all finite semigroups S such that eSe ∈ H for each idempotent e of S, and we recall thatwhere D is the pseudovariety of all finite semigroups whose idempotents are right zeros. It is well known (see [13] for a proof) that a finite semigroup S is a local group if and only if all the idempotents of S lie in the minimal ideal of S. A proof of the generalization of this result to arbitrary semigroups can be found in Proposition 2.1 below. There, a semigroup S is characterized as being a local group if and only if S has no idempotents or S has a minimal ideal J which is a completely simple semigroup that contains all the idempotents of S. In this case, by the Rees-Suschkewitsch Theorem, J is isomorphic to a Rees matrix semigroup over a group (the maximal subgroup of S). In [9], Howie and Ruškuc showed how to find a semigroup presentation for a Rees matrix semigroup M[G; I, Λ; P ] given a semigroup presentation for the group G.In [2] (see also [1, Section 10.6]), Almeida and Azevedo showed that a semidirect product of the form V * D, with the pseudovariety V not locally trivial, is generated by a class formed by certain semigroups M k (S, ) with k ≥ 1, S ∈ V, A an alphabet and : A + → S a ksuperposition homomorphism. Therefore, possible properties of V * D may be tested on the semigroups M k (S, ) and Almeida and Azevedo applied those test-semigroups (an expression used in [1]) to obtain a representation of the free pro-(V * D) semigroup over A.

This paper deals with the reducibility property of semidirect products of the form V * D relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that, if the pseudovariety V is reducible with respect to the canonical signature κ consisting of the multiplication and the (ω − 1)-power, then V * D is also reducible with respect to κ.It is known that the semidirect product operator does not preserve decidability of pseudovarieties [20,11]. The notion of tameness was introduced by Almeida and Steinberg [7,8] as a tool for proving decidability of semidirect products. The fundamental property for tameness is reducibility. This property was originally formulated in terms of graph equation systems and latter extended to any system of equations [2,21]. It is parameterized by an implicit signature σ (a set of implicit operations on semigroups containing the multiplication), and we speak of σ-reducibility. For short, given an equation system Σ with rational constraints, a pseudovariety V is σ-reducible relatively to Σ when the existence of a solution of Σ by implicit operations over V implies the existence of a solution of Σ by σ-words over V and satisfying the same constraints. The pseudovariety V is said to be σ-reducible if it is σ-reducible with respect to every finite graph equation system. The implicit signature which is most commonly encountered in the literature is the canonical signature κ = {ab, a ω−1 } consisting of the multiplication and the (ω − 1)-power. For instance, the pseudovarieties D [9], G [10, 8], J [1, 2] of all finite J -trivial semigroups, LSl [16] and R [6] of all finite R-trivial semigroups are κ-reducible.In this paper, we study the κ-reducibility property of semidirect products of the form V * D. This research is essentially inspired by the papers [15,16] (see also [13] where a stronger form of κ-reducibility was established for LSl). We prove that, if V is κ-reducible then V * D is κreducible. In particular, this gives a new and simpler proof (though with the same basic idea) of the κ-reducibility of LSl and establishes the κ-reducibility of the pseudovarieties LG, J * D and R * D. Combined with the recent proof that the κ-word problem for LG is decidable [14], this shows that LG is κ-tame, a problem proposed by Almeida a few years ago. This also extends part of our work in the paper [15], where we proved that under mild hypotheses on an implicit signature σ, if V is σ-reducible relatively to pointlike systems of equations (i.e., systems of equations of the form x 1 = · · · = x n ) then V * D is pointlike σ-reducible as well. As in [15], we use results from [5], where various kinds of σ-reducibility of semidirect products with an order-computable pseudovariety were considered. More specifically, we know from [5] that a pseudovariety of the form V * D k is κ-reducible when V is κ-reducible, where D k is the order-computable pseudovariety defined by the identity yx 1 · · · x k = x 1 · · · x k . As V * D = k V * D k , we utilize this result as a way t...

Introduction: Early detection of suspicious skin lesions is critical to prevent skin malignancies, particularly the melanoma, which is the most dangerous form of human skin cancer. In the last decade, image processing techniques have been an increasingly important tool for early detection and mathematical models play a relevant role in mapping the progression of lesions. Methods: This work presents an algorithm to describe the evolution of the border of the skin lesion based on two main measurable markers: the symmetry and the geometric growth path of the lesion. The proposed methodology involves two dermoscopic images of the same melanocytic lesion obtained at different moments in time. By applying a mathematical model based on planar linear transformations, measurable parameters related to symmetry and growth are extracted. Results: With this information one may compare the actual evolution in the lesion with the outcomes from the geometric model. First, this method was tested on predefined images whose growth was controlled and the symmetry known which were used for validation. Then the methodology was tested in real dermoscopic melanoma images in which the parameters of the mathematical model revealed symmetry and growth rates consistent with a typical melanoma behavior. Conclusions: The method developed proved to show very accurate information about the target growth markers (variation on the growth along the border, the deformation and the symmetry of the lesion trough the time). All the results, validated by the expected phantom outputs, were similar to the ones on the real images.

No abstract

O objetivo do presente estudo foi associar o consumo alimentar e dados antropométricos com o risco de disbiose intestinal em mulheres com sobrepeso e obesidade de uma cidade do Nordeste brasileiro. Pesquisa transversal de natureza descritiva e quantitativa, com delineamento de pesquisa de campo. A amostra do presente estudo incluiu 62 indivíduos, todos do sexo feminino com média de idade ± 39,7 anos. A média de peso corporal obtida foi de 82,37kg (DP 13,59), a altura média das mulheres participantes do estudo foi de 1,56 (DP 0,06), tais resultados refletiram diretamente nos valores de IMC que ficou em uma média de 33,72 (DP 4,72), caracterizando a maioria da amostra do estudo na classificação de obesidade grau I. 98,4% das mulheres apresentaram risco metabólico altíssimo. Sobre o consumo alimentar, 87,1% relataram não seguir nenhum tipo de dieta de déficit calórico ou mudanças de hábitos alimentares com acompanhamento profissional do nutricionista, a ingestão de alimentos era de acordo com a disponibilidade do alimento na casa e a vontade de ingestão, sendo que 41,9% relataram fazer apenas 4 refeições ao dia. A análise mostrou associação estatística significativa entre o IMC e o consumo de café adicionado de açúcar. Também foi observada associação entre o consumo de farofa, cuscuz e tapioca, carne de boi/carneiro/suína com o risco de disbiose intestinal. O presente estudo apresentou informações importantes sobre o estado de saúde relacionado ao consumo alimentar, disbiose intestinal e obesidade em mulheres.

No abstract