Waldemar W. Koczkodaj’s research while affiliated with Laurentian University and other places

A Monte Carlo study of the pairwise comparisons method has been designed to validate the accuracy improvement by the pairwise comparisons method for 3D objects. For this, not-so-irregular objects were randomly selected. It is important to emphasize that this study focuses on testing the accuracy of the method rather than the users’ skills. The users’ inability to assess the volume of unrestricted random objects (e.g., a porcupine) would only deviate the results. As a side product, semi-randomly generated 3D objects can also be useful in many other research areas, such as software validation and verification, microeconomics (consumer preferences for products), computer entertainment, and even agriculture (selecting of fruits and vegetables). Further generalizations incorporating additional dimensions, as a comparison of different investment opportunities, can be useful, for example in enhancing financial decision-making processes.

Symmetry and antisymmetry are fundamental concepts in many strict sciences. Pairwise comparisons (PC) matrices are fundamental tools for representing pairwise relations in decision making. In this theoretical study, we present a novel framework that embeds additive skew-symmetric PC matrices into the Grassmannian manifold $G(2, n)$. This framework leverages Pl\"ucker coordinates to provide a rigorous geometric interpretation of their structure. Our key result demonstrates that the algebraic consistency condition $a_{ij} + a_{jk} - a_{ki} = 0$ is equivalent to the geometric consistency of 2-planes in $G(2, n)$, satisfying the Pl\"ucker relations. This connection reveals that the algebraic properties of PC matrices can be naturally understood through their geometric representation. Additionally, we extend this framework by interpreting PC matrices as differential 2-forms, providing a new perspective on their consistency as a closedness condition. Our framework of linear algebra, differential geometry, and algebraic geometry, placing PC matrices in a broader mathematical context. Rather than proposing a practical alternative to existing methods, our study aims to offer a theoretical foundation for future research by exploring new insights into higher-dimensional geometry and the geometric modeling of pairwise comparisons.

Physical properties such as shape, volume, and size influence the dynamics of biological systems. In this context, we focus on the geometric properties of limb movements and their physiological and biomechanical effects. Using hand grasping as a paradigmatic example, we describe how dynamic changes in geometric configuration can affect the pathophysiology of grasping. We focus on precision pinch, the simplest and most basic form of grasping, where one finger remains stable while another moves against it (Brand and Hollister, 1999). The thumb and index finger come together to grasp small objects with great precision, such as a pen. During the precision pinch, the geometric configuration of the hand changes, modifying the distribution of forces and the functional connectivity between the fingers. These changes may offer a new approach to the biomechanics of artificial arms. While current approaches consider grasps as homogeneous structures, we argue that local variations in hand geometry can lead to new biomedical effects with intriguing operational implications.

In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced. The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.