Search Results (11)

In this paper, we investigate some properties of L-fuzzy ideals in MV-algebras from a new perspective by taking the completely distributive lattice L as the lattice-valued context. Firstly, we introduce the notion of the L-fuzzy ideal degree in an MV-algebra, which can illustrate the extent to which an L-fuzzy subset qualifies as an L-fuzzy ideal. Secondly, we further use four kinds of cut sets to characterize the L-fuzzy ideal degree. Furthermore, we show that the L-fuzzy ideal degree is precisely an L-fuzzy convex structure in an MV-algebra and discuss its related properties. Finally, we demonstrate that the collection of all L-fuzzy ideals in an MV-algebra forms an L-convex structure and present its L-convex hull formula. Full article

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of $\epsilon$-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et al. provided a possible way of constructing $\epsilon$-nets for $\left({\mathcal{K}}^n,d_{BM}\right)$ based on finite subsets of ${\mathbb{Z}}^n$ theoretically. In this work, we present an algorithm to construct $\epsilon$-nets for $\left({\mathcal{K}}^2,d_{BM}\right)$ and a $(1/4)$-net for $\left({\mathcal{C}}^2,d_{BM}\right)$ is constructed. To the best of our knowledge, this is the first concrete $\epsilon$-net for $\left({\mathcal{C}}^2,d_{BM}\right)$ for such a small $\epsilon$. Full article

The discrete Legendre transform is a powerful tool for analyzing the properties of convex lattice sets. In this paper, for $t>0$, we study a class of convex lattice sets and establish a relationship between vertices of the polar of convex lattice sets and vertices of the polar of its $t-$dilation. Subsequently, we show that there exists a class of convex lattice sets such that its polar is itself. In addition, we calculate upper and lower bounds for the discrete Mahler product of a class of convex lattice sets. Full article

A lattice-filled multicellular square tube features a regular cross-sectional shape, good energy consumption, and good crashworthiness, which is suitable for the design of energy absorbers in various protection fields such as automobiles, aerospace, bridges, etc. Based on the super folding theory, two reference planes are set to refine the energy consumption zone of the super folding element in this study. The energy consumption calculation of convex panel stretching is involved, and the critical crushing force formula is introduced in this study. Meanwhile, the calculation method from a single-cell square tube to a multicellular thin-walled square tube is extended and the structural optimization is investigated, in which the NSGAII algorithm is used to obtain the Pareto front (PF) of the crashworthiness performance index of the square multicellular tubes, the Normal Boundary Intersection (NBI) method is adopted to select knee points, and the influence of different cross-sectional widths on the number, as well as the thickness, of cells are discussed. This study’s results indicate that the theoretical value is consistent with that obtained from the numerical simulation, meaning that the improved theoretical model can be applied to predict the crashworthiness of multicellular square cross-sectional tubes. Also, the optimization method and study results proposed in this study can provide a reference for the design of square lattice multicellular tubes. Full article

The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some metric, to the original data points. In this way, when the parameters are appropriately chosen, one can approximate the shape of every rough surface. To achieve this, we first find the convex hull of each subset of data points in every subdomain of the original lattice, calculate the volume of each convex polyhedron and find the pairwise intersections between two convex polyhedra, i.e., the convex hull of the subdomain and the transformed one within this subdomain. Then, based on the proposed methodology for parameter identification, we minimise the symmetric difference between bounding volumes of an appropriately selected set of points. A methodology for constructing continuous fractal interpolation surfaces by using iterated function systems is also presented. Full article

This paper aims to study fuzzy order bounded linear operators between two fuzzy Riesz spaces. Two lattice operations are defined to make the set of all bounded linear operators as a fuzzy Riesz space when the codomain is fuzzy Dedekind complete. As a special case, separation property in fuzzy order dual is studied. Furthermore, we studied fuzzy norms compatible with fuzzy ordering (fuzzy norm Riesz space) and discussed the relation between the fuzzy order dual and topological dual of a locally convex solid fuzzy Riesz space. Full article

A detailed knowledge of the influence of a particle’s shape on its settling behavior is useful for the prediction and design of separation processes. Models in the available literature usually fit a given function to experimental data. In this work, a constructive and data-driven approach is presented to obtain new drag correlations. To date, the only considered shape parameters are derivatives of the axis lengths and the sphericity. This does not cover all relevant effects, since the process of settling for arbitrarily shaped particles is highly complex. This work extends the list of considered parameters by, e.g., convexity and roundness and evaluates the relevance of each. The aim is to find models describing the drag coefficient and settling velocity, based on this extended set of shape parameters. The data for the investigations are obtained by surface resolved simulations of superellipsoids, applying the homogenized lattice Boltzmann method. To closely study the influence of shape, the particles considered are equal in volume, and therefore cover a range of Reynolds numbers, limited to [9.64, 22.86]. Logistic and polynomial regressions are performed and the quality of the models is investigated with further statistical methods. In addition to the usually studied relation between drag coefficient and Reynolds number, the dependency of the terminal settling velocity on the shape parameters is also investigated. The found models are, with an adjusted coefficient of determination of 0.96 and 0.86, in good agreement with the data, yielding a mean deviation below 5.5% on the training and test dataset. Full article

The detection of redundant or irrelevant variables (attributes) in datasets becomes essential in different frameworks, such as in Formal Concept Analysis (FCA). However, removing such variables can have some impact on the concept lattice, which is closely related to the algebraic structure of the obtained quotient set and their classes. This paper studies the algebraic structure of the induced equivalence classes and characterizes those classes that are convex sublattices of the original concept lattice. Particular attention is given to the reductions removing FCA’s unnecessary attributes. The obtained results will be useful to other complementary reduction techniques, such as the recently introduced procedure based on local congruences. Full article

Dilation and erosion are two elementary operations from mathematical morphology, a non-linear lattice computing methodology widely used for image processing and analysis. The dilation-erosion perceptron (DEP) is a morphological neural network obtained by a convex combination of a dilation and an erosion followed by the application of a hard-limiter function for binary classification tasks. A DEP classifier can be trained using a convex-concave procedure along with the minimization of the hinge loss function. As a lattice computing model, the DEP classifier assumes the feature and class spaces are partially ordered sets. In many practical situations, however, there is no natural ordering for the feature patterns. Using concepts from multi-valued mathematical morphology, this paper introduces the reduced dilation-erosion (r-DEP) classifier. An r-DEP classifier is obtained by endowing the feature space with an appropriate reduced ordering. Such reduced ordering can be determined using two approaches: one based on an ensemble of support vector classifiers (SVCs) with different kernels and the other based on a bagging of similar SVCs trained using different samples of the training set. Using several binary classification datasets from the OpenML repository, the ensemble and bagging r-DEP classifiers yielded mean higher balanced accuracy scores than the linear, polynomial, and radial basis function (RBF) SVCs as well as their ensemble and a bagging of RBF SVCs. Full article

The convexity in triangular norm (for short, ⊗−convexity) is a generalization of Zadeh’s quasiconvexity. The aggregation of two ⊗−convex sets is under the aggregation operator ⊗ is also ⊗−convex, but the aggregation operator ⊗ is not unique. To solve it in complexity, in the present paper, we give some sufficient conditions for aggregation operators preserve ⊗−convexity. In particular, when aggregation operators are triangular norms, we have that several results such as arbitrary triangular norm preserve ${\otimes }_D-$ convexity and ${\otimes }_a-$ convexity on bounded lattices, ${\otimes }_M$ preserves ${\otimes }_H-$ convexity in the real unite interval $[0,1]$ . Full article

In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex spaces to formalize the notion of quantiles. We also put in evidence that many important models of decision problems can be viewed as convex spaces-based models. Several properties of aggregation operators are translated into this general setting, and independence and invariance are used to provide axiomatic characterizations of quantiles. Full article