Neutral Element Research Articles

Finite-Timeless Local Observability for Linear Control Systems on Lie Groups

In this work, we introduce finite-timeless observability for linear control systems on Lie groups. Observability problem is to distinguish the elements of the state space by looking the images of the solutions passing through those elements in the output space. For this aim, we define almost indistinguishability and show that it is an equivalence relation. We consider linear control systems on Lie groups, where the output function is a Lie group homomorphism and study properties of the image of almost indistinguishable points from the neutral element. Thus, we relate local observability of this kind of systems to its finite-timeless local observability.

Idempotent uninorms on bounded lattices with at most a single point incomparable with the neutral element: Part II

ABSTRACT This is the second part of a two-part paper which discusses the structure of idempotent uninorms on bounded lattices such that the set of all points incomparable with the neutral element contains at most one point. In Part I we have completely characterized idempotent uninorms on bounded lattices such that all points are comparable with the neutral element. Moreover, we have shown some basic properties and observations on idempotent uninorms on bounded lattices with one point incomparable with the neutral element and we have completely characterized the class of internal uninorms on such lattices. In the present Part II we characterize two special classes of idempotent uninorms on bounded lattices with one point incomparable with the neutral element, including idempotent uninorms with annihilator incomparable with the neutral element, and by composition of these special cases we show the complete characterization of idempotent uninorms on such lattices.

Idempotent uninorms on bounded lattices with at most single point incomparable with the neutral element: Part I

We discuss the structure of idempotent uninorms defined on bounded lattices such that the set of all points incomparable with the neutral element contains at most one point. We show that the structure of idempotent uninorms on lattices with one point incomparable with the neutral element is much more complex than the structure of those defined on lattices where all points are comparable with the neutral element. In Part I of this two-part paper, we completely characterize idempotent uninorms defined on bounded lattices such that all points are comparable with the neutral element. Moreover, we show some basic properties and observations on idempotent uninorms on bounded lattices with one point incomparable with the neutral element and we completely characterize the class of internal uninorms on such lattices. In Part II, we will characterize two special classes of idempotent uninorms defined on bounded lattices with one point incomparable with the neutral element, including idempotent uninorms with annihilator incomparable with the neutral element, and by the composition of these special cases, we will show the complete characterization of idempotent uninorms on such lattices.

Extremal values-based aggregation functions

We introduce and study aggregation functions based on extremal values, namely extended (l,u)-aggregation functions whose outputs only depend on a fixed number l of extremal lower input values and a fixed number u of extremal upper input values, independently of the arity of the input n-tuples (n≥l+u). We discuss several general properties of (l,u)-aggregation functions and we study special (l,u)-aggregation functions with neutral element, including t-conorms, t-norms and uninorms. We also study (l,u)-aggregation functions defined by means of integrals with respect to discrete fuzzy measures, as well as (l,u)-ordered weighted quasi-arithmetic means based on appropriate weighting vectors. We also stress some generalizations based on recently introduced new types of monotonicity. Some possible applications are sketched, too.

Mechanical properties of various thermomechanical processed Ti-Mo-Al-Zr and pseudo-binary (Ti-Zr)-Mo shape memory alloys for biomedical applications

Shape memory alloys (SMAs) have attracted much attention due to the great demand for biomedical materials. Among the SMAs, the Ti-based SMAs are promising materials owing to their high biocompatibility, non-toxicity, excellent functionalities, and proper mechanical properties. Concerning the Ti-based SMAs, adding other elements is an often-used strategy, and the combinations of the addition materials are unlimited. Among the various Ti-based alloy systems, the Ti-Mo-Al-Zr alloy is one fundamental alloy system, since Mo and Al are served as the specific metals for the calculations of the Mo equivalent and Al equivalent for the evaluation of the alloy stability. While Zr is a neutral element for fine-tuning the properties of the alloys. This fundamental quaternary alloy was thus chosen in this study. In this research series, it has been found that the Ti-5.5Mo-8Al-6Zr alloy performed proper mechanical properties. Therefore, this specific alloy was further subjected to various thermomechanical treatments to further improve its properties. In addition to the Ti-5.5Mo-8Al-6Zr alloy, pseudo-binary alloys of (Ti-Zr)-Mo were also investigated due to their intrinsic low phase transformation temperature compared to the pure Ti and Zr elements. Strategies for the development of these SMAs by conducting thermomechanical treatments have been proposed in this study.

Modular Quasi-Pseudo Metrics and the Aggregation Problem

The applicability of the distance aggregation problem has attracted the interest of many authors. Motivated by this fact, in this paper, we face the modular quasi-(pseudo-)metric aggregation problem, which consists of analyzing the properties that a function must have to fuse a collection of modular quasi-(pseudo-)metrics into a single one. In this paper, we characterize such functions as monotone, subadditive and vanishing at zero. Moreover, a description of such functions in terms of triangle triplets is given, and, in addition, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions is discussed. Specifically, we show that the class of modular (quasi-)(pseudo-)metric aggregation functions coincides with that of modular (pseudo-)metric aggregation functions. The characterizations are illustrated with appropriate examples. A few methods to construct modular quasi-(pseudo-)metrics are provided using the exposed theory. By exploring the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions, we find that every modular quasi-pseudo-metric aggregation function with 0 as the neutral element is an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics is also provided. Furthermore, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions is studied. Particularly, we have proven that they are the same only when the former functions are finite. Finally, the usefulness of modular quasi-(pseudo-)metric aggregation functions in multi-agent systems is analyzed.

A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

The enumeration of logical connectives and aggregation functions defined on a finite chain has been a hot topic in the literature for the last decades. Multiple advantages can be derived from knowing a general formula about their cardinality, for instance, the ability to anticipate the computational cost required for generating operators with different properties. This is of paramount importance in image processing and decision making scenarios, where the identification of the most optimal operator is essential. Furthermore, it facilitates the examination of how constraining a certain property is in relation to its parent class. As a consequence, this paper aims to compile the main existing formulas and the methodologies with which they have been derived. Additionally, we introduce some novel formulas for the number of smooth discrete aggregation functions with neutral element or absorbing element, idempotent conjunctions, and commutative and idempotent conjunctions.

On the representation of internal uninorms on a bounded lattice

Internal uninorms on a bounded lattice always output one of the two input values and are nothing else but idempotent uninorms when the lattice is a chain. In this paper, we study the existence and representation of internal uninorms with a given neutral element e on a bounded lattice. We obtain a necessary condition for the existence of such an internal uninorm. In the case that those elements that are incomparable with the neutral element (collected in the set Ie) constitute a chain, we prove that this necessary condition also becomes a sufficient one and allows to represent any internal uninorm on a bounded lattice in terms of an internal uninorm on the bounded chain obtained by deleting Ie from the lattice and an internal quasi-uninorm on Ie connected via a family of lower sets and a family of upper sets indexed by Ie.

Correspondence of the aging behaviors and the phase constituents of the Ti-Mo-based alloys tailored by the Al and Zr additions

Ti-based alloys are promising materials for wide applications, such as aerospace community and biomedical materials. Concerning the applications, the mechanical properties are fundamental and crucial issues. Therefore, effect of the phase constituents on the aging behaviors of the Ti-Mo-Al-Zr alloys has been investigated in this study. The phase constituents of the Ti-Mo-based alloys were manipulated by the introductions of the Al and Zr elements. These metals were chosen since Mo and Al are typical β-stabilizer and α-stabilizer, respectively; while the Zr element is classified as the neutral element. Hence, these three typical addition elements were chosen for tailoring the mechanical properties of the Ti-based alloys making this system a fundamental guideline for further alloy fabrications. The phase identification, microstructure observations, and examinations of micro Vickers hardness were conducted for explaining the mechanical properties of the alloys. The typical ω-phase and α-phase, which often accompany the formation of the β-phase, were focused to clarify the influence of the phase constituents and microstructures on the micro Vickers hardness. A color mapping of micro Vickers hardness along with the recognition of phase constituents explains the alternations of the mechanical properties brought about by the phase transformations.

A Compositional Theory of Linearizability

Compositionality is at the core of programming languages research and has become an important goal toward scalable verification of large systems. Despite that, there is no compositional account of linearizability , the gold standard of correctness for concurrent objects. In this article, we develop a compositional semantics for linearizable concurrent objects. We start by showcasing a common issue, which is independent of linearizability, in the construction of compositional models of concurrent computation: interaction with the neutral element for composition can lead to emergent behaviors, a hindrance to compositionality. Category theory provides a solution for the issue in the form of the Karoubi envelope. Surprisingly, and this is the main discovery of our work, this abstract construction is deeply related to linearizability and leads to a novel formulation of it. Notably, this new formulation neither relies on atomicity nor directly upon happens-before ordering and is only possible because of compositionality, revealing that linearizability and compositionality are intrinsically related to each other. We use this new, and compositional, understanding of linearizability to revisit much of the theory of linearizability, providing novel, simple, algebraic proofs of the locality property and of an analogue of the equivalence with observational refinement . We show our techniques can be used in practice by connecting our semantics with a simple program logic that is nonetheless sound concerning this generalized linearizability.

Semilinear De Morgan monoids and epimorphisms

A representation theorem is proved for De Morgan monoids that are (i) semilinear, i.e., subdirect products of totally ordered algebras, and (ii) negatively generated, i.e., generated by lower bounds of the neutral element. Using this theorem, we prove that the De Morgan monoids satisfying (i) and (ii) form a variety—in fact, a locally finite variety. We then prove that epimorphisms are surjective in every variety of negatively generated semilinear De Morgan monoids. In the process, epimorphism-surjectivity is established for several other classes as well, including the variety of all semilinear idempotent commutative residuated lattices and all varieties of negatively generated semilinear Dunn monoids. The results settle natural questions about Beth-style definability for a range of substructural logics.

Dispersive problem and control sets of linear control systems on Lie groups

This paper is dedicated to the study of dispersiveness and controllability of linear control systems on Lie groups. Dispersiveness means absence of recursiveness, that is contrary to the existence of control set. A linear control system on a Lie group associates with a derivation operator. For a linear system with stable derivation, it is shown that the system is dispersive if and only if the trajectories through the neutral element have no limit at infinity. As a consequence, a nonempty limit set at the neutral element is a necessary condition for the linear system to admit a control set or to be controllable. In the nondispersive case, the control sets are described by the Lie subgroup of all recurrent points of the automorphism flow of the system. If the derivation operator is asymptotically stable, the central limit set at the neutral element is the unique control set of the system.

Water Element as an Aesthetic Aspect of Landscape Design

Abstract: Researchers have considered and landscape architects have employed the aquatic element as a differentiating aesthetic aspect. Its meaning and perception in Both physical (visual and nonvisual) landscapes and virtual landscapes are affected by the idiosyncrasy and the observer's perceptual filter. Methodologies both theoretical and empirical have been presented on the role of water, whether lentic or lotic, to a great or small extent, sweet or sour. Seawater, whether turbulent or calm. In general, the aquatic element serves as a link between scenery, literature, or other arts, as well as a way to alleviate the monotony imposed by architects or artists uses solid materials. Water serves as a foundation for literary and artistic perceptions of landscape, as well as a neutral architectural element that cannot be properly classified as "natural" or "built." The diversity of its usage and perception, as well as its diachronic and extensive application and resonance, show the universality of the aquatic element as an aesthetic aspect and architectural means.

Characterizations of monotone right continuous functions which generate associative functions

Associativity of a two-place function T:[0,1]2→[0,1] defined by T(x,y)=f(−1)(T⁎(f(x),f(y))) where T⁎:[0,1]2→[0,1] is an associative function with neutral element in [0,1], f:[0,1]→[0,1] is a monotone right continuous function and f(−1):[0,1]→[0,1] is the pseudo-inverse of f depends only on properties of the range of f. The necessary and sufficient conditions for the T to be associative are presented by applying the properties of the monotone right continuous function f.

A hybrid incomplete decision system using fuzzy sets and rough set theory with varying object sets and values

Researchers in science and engineering face various obstacles due to a lack of specific and full data. Many different approaches have been devised to deal with these restrictive requirements, but two notable schools of thought are the fuzzy set (FS) theory and the rough set (RS) theory, both of which have spawned many extensions and hybridizations. Although RS theory originated from an indiscernibility relation (also known as an equivalence relation in mathematics), emphasis rapidly shifted to similarity or coverings (and their fuzzy analogues). Many other hybrid schemes were suggested with this goal in mind. The gap between those concepts shrank because to this thorough analysis. Fuzzy set theory is a legitimate way to convey the ambiguity of assessment data, yet it is still inadequate for dealing with certain intricate problems in the actual world. In reality, decision makers will undoubtedly provide different kinds of ambiguous and nuanced assessments. Atanassov’s intuitionistic fuzzy set theory broadened the application of fuzzy set theory by imbuing it with an element of uncertainty. Sometimes in real life, you have to deal with a neutral element on top of the indeterminate one. Picture fuzzy sets were developed specifically for this purpose. Membership roles may be positive, neutral, or negative/refusal. In contrast, hesitant fuzzy sets and its hybrid models are useful when decision makers are on the fence about which option to choose. As a binary relation on a set, a graph is symmetric. It is a staple in mathematical modelling and is used in almost every scientific and technological discipline. Graph theory has been essential in the mathematical modelling and subsequent resolution of several real-world situations. Information about connections between things is often best represented using graph theory, which uses vertices to stand in for the items and edges for the relationships between them. The suggested dynamic algorithm is better to the static approach in dealing with the multidimensional dynamic changes of the hybrid incomplete decision system, according to a series of experiments carried out on nine UCI datasets.

Uninorms internal on one or more non-trivial cuts

Each uninorm is trivially internal on the cut in the neutral element and uninorms internal on the boundary were already completely characterized. We discuss uninorms which are internal on some non-trivial cut and show that each such uninorm can be expressed as a non-trivial ordinal sum. In individual cases we discuss the structure of such conjunctive (disjunctive) uninorm and show that the corresponding ordinal sum contains uninorms, t-conorms, t-superconorms and generalized sub-uninorms (t-norms, t-subnorms and generalized super-uninorms). Moreover, we discuss the ordinal sum decomposition of uninorms with several internal cuts and show their decomposition into semigroups that have only constant internal cuts, and possibly an internal cut in the neutral element.

Generating methods of some classes of uninorms and associated structures

Uninorms are an important class of fuzzy logic connectives that have, so far, neither generating methods nor algebraic structures. In this work, we attempt to propose some novel generating methods of some well established classes of uninorms (with a fixed but arbitrary neutral element e) and investigate the algebraic structures that these generating methods can impose on those specific classes of uninorms. Towards this, we show that the operations point-wise maximum (∨) and the point-wise minimum (∧) on the set of idempotent uninorms are closed and make it a distributive lattice. Also, we show that the operations ∧ and ∨ on some sub-classes of idempotent uninorms with other classes of uninorms generate uninorms. Since the operations ∨ and ∧ on the set of uninorms are not closed in general, we propose some novel generating methods of uninorms that come from various classes such as representable uninorms, pseudo-continuous uninorms, uninorms continuous in the open unit square, uninorms with underlying continuous Archimedean t-norms and t-conorms and for each class of these uninorms, we investigate the algebraic structure imposed on it by the proposed generating methods with the help of results obtained recently in Vemuri et al. (2022) [51].

Biunit pairs in semiheaps and associated semigroups

The notion of neutral element generalizes to a pair of elements in ternary algebras. Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of Mal’cev element. In order to generalize the known correspondences between semiheaps and certain kinds of semigroups, families of functions generalizing involutions and conjugations, called switches and warps, are investigated. The main theorem establishes that there is a one-to-one correspondence between monoids equipped with a particular switch and semiheaps with a fixed biunit pair. This generalizes the celebrated result in semiheap theory that gives a one-to-one correspondence between involuted monoids and semiheaps with a fixed biunit element. A novel, previously undocumented, algebra is motivated by this result: diheaps are introduced as semiheaps whose elements belong to biunit pairs, which generalize the well-known case of heaps. Diheaps are of great interest since they are shown to be isomorphic to heaps only when they are heaps themselves and explicit non-heap examples are constructed from abelian groups and hypermatrices.

Abstract 6526: Investigation levels of trace elements in breast cancer types using laser spectroscopic techniques

Abstract Breast cancer (BC) is the 2nd leading cause of death from cancer in women. Women have a lifetime risk of breast cancer of around 31%, which makes it the first most common malignancy in women. Diagnosing cancer and classifying its types is a complex process that relies heavily on sample analysis. Early detection of cancer diseases is one of the important and main factors in determining treatment and saving many patients before the disease worsens and reaches dangerous stages. Alterations in the levels of trace elements have been observed in breast cancer (BC) patients. However, the relationships between these alterations and the metabolic and clinical consequences of BC are unknown. The aim of the present work is to study in vitro possibilities and the effectiveness of introducing an efficient optical spectroscopic technique such as LIBS for the identification types of breast cancer, as well as determining the disease grade and severity. The ultimate goal of this work is to investigate the concentrations of several trace elements in BC patients, and their relationships with the molecular subtypes of tumors. The atomic transition of the neutral element lines of Fe, Al, Na, Ca, K, V, Co and Pb was chosen to determine the trace element levels of four breast cancer types. We found that the measured average level of trace elements can differentiate between different types of breast cancer. Citation Format: Ola Sayed Ahmed, hisham imam. Investigation levels of trace elements in breast cancer types using laser spectroscopic techniques [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 6526.

Distributivity Conditions of Idempotent Uninorms and Two Special Kinds of Aggregation Functions

Recently, some authors studied the distributive equations for continuous t-norms and some families of usual classes of uninorms over overlap functions in Refs. 25 and 32, but lacked a complete characterization of the distributivity on idempotent uninorms and overlap or grouping functions widely used in image processing. As a supplement to the previous results, in this article we fully characterize the distributivity equations of idempotent uninorms over these two functions by virtue of the associated functions of idempotent uninorms. Moreover, we also discuss the distributivity conditions of above two special functions over idempotent uninorms, yet find in this case that the associated functions of idempotent uninorms must satisfy particular conditions and those two functions usually are a class of special aggregation functions with a constant domain whose value equals to the neutral element of the idempotent uninorm.