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Comparability of Metrics and Norms in terms of Basis of Exponential Vector Space arxiv.org/abs/2411.15141

Comparability of Metrics and Norms in terms of Basis of Exponential Vector Space

In this paper, we shall compare two metrics in terms of orderly dependence, a notion developed in exponential vector space in the article 'Basis and Dimension of Exponential Vector Space' by Jayeeta Saha and Sandip Jana in Transactions of A. Razmadze Mathematical Institute Vol. 175 (2021), issue 1, 101-115. Exponential vector space, in short 'evs', is a partially ordered space associated with a commutative semigroup structure and a compatible scalar multiplication. In the present paper we shall show that the collection $\mathcal{D(\mathbf X)}$ of all metrics on a non-empty set $\mathbf X$, together with the constant function zero $O$, forms a topological exponential vector space. We shall discuss the orderly dependence of two metrics through our findings of a basis of $\mathcal{D(\mathbf X)}\smallsetminus\{O\}$ in different scenario. We shall characterise orderly independence of two elements of a topological evs in terms of the comparing function, another mechanism developed in topological exponential vector space, which can measure the degree of comparability of two elements of an evs. Finally, we shall discuss existence of orderly independent norms on a linear space. For an infinite dimensional linear space we shall construct a large number of orderly independent norms depending on the dimension of the linear space. Orderly independent norms are precisely those which are totally non-equivalent, in the sense that they produce incomparable topologies.

arXiv.org

Approximability of Poisson structures for the 4-vertex model, and the higher-spin XXX chain, and Yang-Baxter algebras arxiv.org/abs/2411.15188

Approximability of Poisson structures for the 4-vertex model, and the higher-spin XXX chain, and Yang-Baxter algebras

We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.

arXiv.org

On the nonlinear programming problems subject to a system of generalized bipolar fuzzy relational equalities defined with continuous t-norms arxiv.org/abs/2411.15225

On the nonlinear programming problems subject to a system of generalized bipolar fuzzy relational equalities defined with continuous t-norms

This paper, in the first step, develops the system of bipolar fuzzy relational equations (FRE) to the most general case where the bipolar FREs are defined by an arbitrary continuous t-norm. Also, since fuzzy relational equations are special cases of the bipolar FREs, the proposed system can be also interpreted as a generalization of traditional FREs where the fuzzy compositions are generally defined by any continuous t-norm. The consistency of the continuous bipolar FREs is initially investigated and some necessary and sufficient conditions are derived for determining the feasibility of the proposed system. Subsequently, the feasible solutions set of the problem is completely characterized. It is shown that unlike FREs and those bipolar FREs defined by continuous Archimedean t-norms, the feasible solutions set of the generalized bipolar FREs is formed as the union of a finite number of compact sets that are not necessarily connected. Moreover, five techniques have also been introduced with the aim of simplifying the current problem, and then an algorithm is accordingly presented to find the feasible region of the problem. In the second step, a new class of optimization models is studied where the constraints are defined by the continuous bipolar FREs and the objective function covers a wide range of (non)linear functions such as maximum function, geometric mean function, log-sum-exp function, maximum eigenvalue of a symmetric matrix, support function of a set, etc. It is proved that the problem has a finite number of local optimal solutions and a global optimal solution can always be obtained by choosing a point having the minimum objective value compared to all the local optimal solutions. Finally, to illustrate the discussed study, a step-by-step example is described in several sections, whose constraints are a system of the bipolar FRES defined by Dubois-Prade t-norm.

arXiv.org
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