Triangle Inequality Research Articles

Failure Rates Of The Matrix Triangle Inequality

The triangle inequality is a fundamental principle in mathematics. However, its validity is not always guaranteed in matrix theory, specifically, when dealing with the matrix absolute value. We investigate how often the matrix absolute value satisfies the triangle inequality for different spaces of matrices. By using numerical methods, we quantify the frequency with which the triangle inequality holds and examine possible key factors contributing to its success or failure.

Decoupling by Local Random Unitaries without Simultaneous Smoothing, and Applications to Multi-user Quantum Information Tasks

We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old (Dupuis et al. in Commun Math Phys 328:251–284, 2014) and new methods (Dupuis in IEEE Trans Inf Theory 69:7784–7792, 2023), we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of Rényi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, which are additionally optimal for entanglement of assistance and state merging.

Applications of distance measure between dual hesitant fuzzy sets in medical diagnosis and weighted dual hesitant fuzzy sets in making decision

This paper introduces a novel distance measure for dual hesitant fuzzy sets (DHFS) and weighted dual hesitant fuzzy sets (WDHFS), with a rigorous proof of the triangular inequality to ensure its mathematical validity. The proposed measure extends the normalized Hamming, generalized, and Euclidean distance measures to dual hesitant fuzzy elements (DHFE), offering a broader framework for handling uncertainty in fuzzy environments. Additionally, the utilization of a score function is shown to simplify the computation of these distance measures. The practical relevance of the proposed measure is demonstrated through its application in medical diagnosis and decision-making processes. A comparative analysis between the newly introduced distance measure denoted as χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi$$\\end{document}, and an existing measure, χ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _1$$\\end{document} is performed to underscore the superiority and potential advantages of the new approach in real-world scenarios.

Exploring Double Composed Partial Metric Spaces: A Novel Approach to Fixed Point Theorems

This paper innovatively extends partial metric spaces to introduce double composed partial metric space (DCPMS). Unlike traditional metrics, DCPMS replaces the triangle inequality with a nuanced form, integrating control functions into the metric. Building upon Ayoob et al.’s work, this novel generalization focuses on establishing fixed point theorems for DCPMS, contributing to the evolving landscape of mathematical analysis in this unique domain.

Interactions of massless fermionic fields in three dimensions

All independent interaction vertices involving massless (Fang-Fronsdal) fermions in three dimensions are classified, completing the classification of interactions of massless fields of any spin. Similar to the bosonic case, we get no independent vertices at quartic or higher order in the fields involving fields with spin s≥3/2 and cubic vertices only for spins satisfying triangle inequalities, apart from the cases involving (matter) fields with spin s≤1. Different from the bosonic cases, we get only one vertex for each triple of spins with two Majorana fermions and one boson, which is parity even (odd) when the sum of the spins of all fields involved is odd (even). When the two Majorana fermions are identical, their coupling to an odd-spin boson is trivialized. We comment on the nontrivial holographic dictionary relating these vertices to conformal field theory correlators in two dimensions. Published by the American Physical Society 2024

Triangle inequalities in international trade: The neglected dimension

Estimating trade costs is key to understanding the welfare effects of trade liberalizations. Cost minimization implies that the triangle inequality (TI) of international trade costs must hold for any three countries to avoid cross-border arbitrage. We show that re-routing opportunities might arise when trade costs change because a shipment through an intermediary becomes cheaper. The TI captures such re-routing opportunities. However, standard approaches to calculating the gains from trade liberalizations ignore this no-arbitrage condition. We outline an estimation routine that is model-consistent and respects the TI. Counterfactual exercises suggest that the welfare gains from re-routing after trade liberalizations can be substantial.

Memerna: Sparse RNA folding including coaxial stacking

Determining RNA secondary structure is a core problem in computational biology. Fast algorithms for predicting secondary structure are fundamental to this task.Department of Biochemistry & Biophysics, University of Rochester Medical Center, Rochester, NY, USA We describe a modified formulation of the Zuker-Stiegler algorithm with coaxial stacking, a stabilising interaction in which the ends of helices in multi-loops are stacked. In particular, optimal coaxial stacking is computed as part of the dynamic programming state, rather than in an inner loop. We introduce a new notion of sparsity, which we call replaceability. Replaceability is a more general condition and applicable in more places than the triangle inequality that is used by previous sparse folding methods. We also introduce non-monotonic candidate lists as an additional sparsification tool. Existing usages of the triangle inequality for sparsification can be thought of as an application of both replaceability and monotonicity together. The modified recurrences along with replaceability allows sparsification to be applied to coaxial stacking as well, which increases the speed of the algorithm. We implemented this algorithm in software we call memerna, which we show to have the fastest exact (non–heuristic) implementation of RNA folding under the complete Turner 2004 model with coaxial stacking, out of several popular RNA folding tools supporting coaxial stacking. We also introduce a new notation for secondary structure which includes coaxial stacking, terminal mismatches, and dangles (CTDs) information. The memerna package 0.1 release is available at https://github.com/Edgeworth/memerna/tree/release/0.1.

A new class of distances on complex projective spaces

The complex projective space P(Cn) can be interpreted as the space of all quantum pure states of size n. A distance on this space, interesting from the perspective of quantum physics, can be induced from a classical distance defined on the n-point probability simplex by the ‘earth mover problem’. We show that this construction leads to a quantity satisfying the triangle inequality, which yields a true distance on complex projective space belonging to the family of quantum 2-Wasserstein distances.

An extension to Kendall’s Tau metric to evaluate dissimilarities between data series

Data analysis is performed to examine, interpret, and extract information from data series, and it includes applying various methods and techniques to understand patterns and compare data. An approach to compare data is to use rank metrics that help identify how distinct two data series are when compared to each other according to patterns, formats, criteria, and dimensions in both data series. Among these metrics, Kendall’s Tau metric stands out, as it is robust and inexpensive, widely used in analyzing sequences and genomes, to detect errors in flash memories, and to compare distributions and top-k ranked values. However, a challenge arises when comparing lists with different lengths or when lists do not share the same elements. This happens, for example, when lists are defined by top-k elements, commonly called k-list. In this case, there is no guarantee that two k-lists share the same set of elements. Traditional metrics like Kendall’s Tau are designed to quantify differences only between shared elements in lists. Recognizing this limitation, a possible solution is to apply the metric to the shared elements of the lists. Another solution, named the generalization of Kendall’s Tau, proposed by Fagin et al., considers all elements in two lists. However, this generalization of Kendall Tau is a semi-metric, as it does not satisfy the triangular inequality. To solve this problem, we propose the Extended Kendall Tau (EKT) metric that meets all the conditions of a metric and simultaneously considers the distinct elements of the compared lists. The proposed metric was evaluated by applying conventional Kendall’s Tau and the extended Kendall’s Tau over 40 text files divided into five different languages (eight files per language). We compared KT and EKT measures within the ”same language” and across ”other language” files for the two scenarios. The results revealed that both methods could accurately identify the differences between the groups of texts of the ”same language” and ”other language”. However, the numerical results show that EKT is able to more significantly highlight the difference between groups of texts of different languages.

Analysis of Functional Properties of Fuzzy \(\tilde{L}\) \(^p\Omega\) Spaces

This paper proposes a study of fuzzy \(\tilde{L}\)p (Ω) spaces, incorporating functions with triangular fuzzy coefficients. These fuzzy functional spaces provide a better adaptation to fuzzy or imprecise functions. We establish the theoretical foundations of these spaces by examining key functional properties, such as the fuzzy scalar product and the fuzzy norm. To do this, we checked the bilinearity, symmetry, positivity, homogeneity and triangular inequality in a fuzzy environment and in the presence of functions whose coefficients are triangular fuzzy numbers, by the \(\alpha\) -cut Dubois and Prade approach. The aim of this paper is to address observations identified in the existing literature, where some functional properties of \(\tilde{L}\)p (Ω) fuzzy spaces are often addressed in a too general manner, without specifying the types of fuzzy functions. This study aims to provide a more detailed and rigorous analysis, thus enriching mathematical understanding and paving the way for practical applications in diverse fields such as: fuzzy differential equations, artificial intelligence, information processing, and decision making in uncertain environments.

On p-numerical radius inequalities for commutators of operators

In this work, we give some p-numerical radius inequalities for operators as well as for commutators of operators. As applications, when p = ∞, we improve some earlier numerical radius inequalities for operators. Also, we refine the triangle inequalities for the numerical radius and the operator norm.

Imperfect discrimination, similarity, and stochastic transitivity

We study the relationship between imperfect discrimination, similarity, and stochastic transitivity in generalized versions of Perturbed Utility (Fudenberg et al. in Econometrica 83(6):2371–2409, 2015) and Fechnerian models (Debreu in Econometrica 26(3):440–444, 1958). We show that these models are equivalent and that, within them, the properties of a similarity function can characterize all notions of stochastic transitivity (as Weak, Moderate, and Strong). Specifically, He and Natenzon (Am Econ Rev Insights 6(2):176–195, 2024) have recently shown that choice probabilities are moderately transitive if and only if the similarity function is a metric. We provide a counterpoint and show that unless choice probabilities are strongly transitive, the similarity function can violate the triangle inequality.

Basis-independent quantum coherence and its distribution under relativistic motion

Recent studies have increasingly focused on the effect of relativistic motion on quantum coherence. Prior research predominantly examined the influence of relative motion on basis-dependent quantum coherence, underscoring its susceptibility to decoherence under accelerated conditions. Yet, the effect of relativistic motion on basis-independent quantum coherence, which is critical for understanding the intrinsic quantum features of a system, remains an interesting open question. This paper addresses this question by examining how total, collective, and localized coherence are affected by acceleration and coupling strength. Our analysis reveals that both total and collective coherence significantly decrease with increasing acceleration and coupling strength, ultimately vanishing at high levels of acceleration. This underscores the profound impact of Unruh thermal noise. Conversely, localized coherence exhibits relative stability, decreasing to zero only under the extreme condition of infinite acceleration. Moreover, we demonstrate that collective, localized, and basis-independent coherence collectively satisfy the triangle inequality. These findings are crucial for enhancing our understanding of quantum information dynamics in environments subjected to high acceleration and offer valuable insights on the behavior of quantum coherence under relativistic conditions.

GVP-RRT: a grid based variable probability Rapidly-exploring Random Tree algorithm for AGV path planning

In response to the issues of low solution efficiency, poor path planning quality, and limited search completeness in narrow passage environments associated with Rapidly-exploring Random Tree (RRT), this paper proposes a Grid-based Variable Probability Rapidly-exploring Random Tree algorithm (GVP-RRT) for narrow passages. The algorithm introduced in this paper preprocesses the map through gridization to extract features of different path regions. Subsequently, it employs random growth with variable probability density based on the features of path regions using various strategies based on grid, probability, and guidance to enhance the probability of growth in narrow passages, thereby improving the completeness of the algorithm. Finally, the planned route is subjected to path re-optimization based on the triangle inequality principle. The simulation results demonstrate that the planning success rate of GVP-RRT in complex narrow channels is increased by 11.5–69.5% compared with other comparative algorithms, the average planning time is reduced by more than 50%, and the GVP-RRT has a shorter average planning path length.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

The ultrametric backbone is the union of all minimum spanning forests

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node reachability, with applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e. those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all shortest paths and De Morgan's law consistency.

IBPF-RRT*: An improved path planning algorithm with Ultra-low number of iterations and stabilized optimal path quality

Due to its asymptotic optimality, the Rapidly-exploring Random Tree star (RRT*) algorithm is widely used for robotic operations in complex environments. However, the RRT* algorithm suffers from poor path quality, slow convergence, and unstable generation of high-quality paths in the path planning process. This paper proposes an Improved Bi-Tree Obstacle Edge Search Artificial Potential Field RRT* algorithm (IBPF-RRT*) to address these issues. First, based on the RRT* algorithm, this paper proposes a new obstacle edge search artificial potential field strategy (ESAPF), which speeds up the path search and improves the path quality simultaneously. Second, a bi-directional pruning strategy is designed to optimize the bi-directional search tree branch nodes and combine the bi-directional search strategy to reduce the number of iterations for convergence speed significantly. Third, a novel path optimization strategy is proposed, which enables high-quality paths to be generated stably by creating an entirely new node between two path nodes and then optimizing the paths using a pruning strategy based on triangular inequalities. Experimental results in three different scenarios show that the proposed IBPF-RRT* algorithm outperforms other methods in terms of optimal path quality, algorithm stability, and the number of iterations when compared to the RRT*, Q-RRT*, PQ-RRT*, F-RRT* and CCPF-RRT* algorithms, and proves the effectiveness of the proposed three strategies.

On the Restricted Steiner Multi Cycle Problem

We introduce a new problem, called Steiner Multi Cycle Problem, in which we are given a complete weighted graph G=(V,E), which respects the triangle inequality, a collection of terminal sets {T1,…,Tk}, where for each a in [k] we have a subset Ta of V and these terminal sets are pairwise disjoint. The goal is to find a set of disjoint cycles of minimum cost such that for each a in [k], all vertices of Ta belong to a same cycle. Our main interest is in a restricted case where |Ta|=2, for each a in [k], which models a collaborative problem with pickup and delivery. We show that even the restricted problem is NP-Hard, and present three heuristics to solve it: a dynamic programming algorithm called Refinement Search Heuristic, which explores geometric and laminar set properties; a heuristic based on the Gonzales Clustering algorithm; and a GRASP Heuristic with Path Relinking. We performed computational experiments with 525 instances to compare these methods, which achieved results close to the optimum, highlighting the performance of the Refinement Search Heuristic both in quality and time.

A note on the generalized Babuška–Brezzi theory: Revisiting the proof of the associated Strang error estimates

In this note we simplify the derivation of the error estimates for the generalized Babuška–Brezzi theory with Galerkin schemes defined in terms of approximate bilinear forms and functionals. More precisely, we provide a straight proof that makes no use of any translated continuous or discrete kernel nor of the distance between them, but of suitable upper bounds of the distances of each component of the Galerkin solution to any other member of the respective finite element subspace. In this way, the Strang error estimates are obtained simply by applying the aforementioned bounds along with the triangle inequality, so that they become cleaner and with fully explicit constants. The case in which the discrete bilinear forms can be evaluated at the continuous solution is also considered, which yields the consistency terms to appear separately from the distances to the subspaces, thus allowing the former to be handled independently from the latter.

A new distance measure between two basic probability assignments based on penalty coefficient

Uncertainty information processing is a challenging topic in information science. Dempster-Shafer evidence theory shows excellent performance when dealing with uncertainty in information representation, fusion, and processing. The measurement of the distance between the basic probability assignments (BPA) in Dempster-Shafer evidence theory is a powerful tool that deals with uncertainty in evidential information processing. Most of the existing methods are based on the difference in the assessment of evidential degree for different hypotheses, with a notable example Belief-Jensen-Shannon (BJS) divergence. However, these methods ignore the information that is carried by different units in the power sets that represent the hypotheses, i.e., the difference on the hypotheses themselves that include in the evidential degree assessment of different BPA should be considered in the distance measurement. Therefore, in this paper, a novel method grounded in BJS divergence and incorporating a penalty coefficient to rectify the distance between basic probability assignments is proposed. This methodology showcases commendable properties, including symmetry, boundedness, and adherence to the triangle inequality. To demonstrate the efficiency of the proposed distance measurement method between BPAs, we apply it to two common-used machine learning data sets: Iris data set and Dry Beans data set. Compared to the represented existing evidence distance measurement approaches, the proposed method yields superior performance in both data sets with respect to classification accuracy.