Search Results (60)

This study evaluates hierarchical clustering methodologies to identify patterns associated with timeout requests for EuroLeague basketball games. Using play-by-play data from 3743 games spanning the 2008–2023 seasons (over 1.9 million instances), we applied Principal Component Analysis to reduce dimensionality and tested multiple agglomerative and divisive clustering techniques (e.g., Ward and DIANA) with different distance metrics (Euclidean, Manhattan, and Minkowski). Clustering quality was assessed using internal validation indices such as Silhouette, Dunn, Calinski–Harabasz, Davies–Bouldin, and Gap statistics. The results show that Ward.D and Ward.D2 methods using Euclidean distance generate well-balanced and clearly defined clusters. Two clusters offer the best overall quality, while four clusters allow for meaningful segmentation of game situations. The analysis revealed that teams that did not request timeouts often exhibited better scoring efficiency, particularly in the advanced game phases. These findings offer data-driven insights into timeout dynamics and contribute to strategic decision-making in professional basketball. Full article

Background: Traditional neonatal assessments rely on anthropometric measures such as weight, body size, and head circumference. However, recent studies suggest that objective movement quantification may serve as a complementary clinical indicator of development in preterm infants. Methods: This study evaluates non-invasive computer vision-based quantification of neonatal movement using contactless pose tracking based on computer vision. We analyzed approximately 800,000 postural data points from ten preterm infants to identify reliable algorithms, optimal recording duration, and whether whole-body or regional tracking is sufficient. Results: Our findings show that 30 s video segments are adequate for consistent motion quantification. Optical flow methods produced inconsistent results, while distance-based algorithms—particularly Chebyshev and Minkowski—offered greater stability, with coefficients of variation of 5.46% and 6.40% in whole-body analysis. Additionally, Minkowski and Mahalanobis metrics applied to the lower body yielded results similar to full-body tracking, with minimal differences of 0.89% and 1%. Conclusions: The results demonstrate that neonatal movement can be quantified objectively and without physical contact using computer vision techniques and reliable computational methods. This approach may serve as a complementary clinical indicator of neonatal progression, alongside conventional measures such as weight and size, with applications in continuous monitoring and early clinical decision-making for preterm infants. Full article

The cosmic time dilation observed in Type Ia supernova light curves suggests that the passage of cosmic time varies throughout the evolution of the Universe. This observation implies that the rate of proper time is not constant, as assumed in the standard FLRW metric, but instead is time-dependent. Consequently, the commonly used FLRW metric should be replaced by a more general framework, known as the Conformal Cosmology (CC) metric, to properly account for cosmic time dilation. The CC metric incorporates both spatial expansion and time dilation during cosmic evolution. As a result, it is necessary to distinguish between comoving and proper (physical) time, similar to the distinction made between comoving and proper distances. In addition to successfully explaining cosmic time dilation, the CC metric offers several further advantages: (1) it preserves Lorentz invariance, (2) it maintains the form of Maxwell’s equations as in Minkowski spacetime, (3) it eliminates the need for dark matter and dark energy in the Friedmann equations, and (4) it successfully predicts the expansion and morphology of spiral galaxies in agreement with observations. Full article

We analyse linearised field equations around the Minkowski metric, with its standard flat parallel transport structure, in models of newer GR, which refers to quadratic actions in terms of a nonmetricity tensor. We show that half of the freedom in choosing the model parameters is immediately fixed by asking for reasonable properties of tensors and vectors, defined with respect to spatial rotations, and we accurately describe the much more complicated sector of scalars. In particular, we show that, from the teleparallel viewpoint, the STEGR model with an additional term of a gradient squared of the metric determinant exhibits one and a half new dynamical modes, and not just one new dynamical mode as it was previously claimed. Full article

The intersection of fractals, non-Euclidean geometry, spatial autocorrelation, and urban structure offers valuable theoretical and practical application insights, which echoes the overarching goal of this paper. Its research question asks about connections between graph theory adjacency matrix eigenfunctions and certain non-Euclidean grid systems; its explorations reflect accompanying synergistic influences on modern urban design. A Minkowski metric with an exponent between one and two bridges Manhattan and Euclidean spaces, supplying an effective tool in these pursuits. This model coalesces with urban fractal dimensions, shedding light on network density and human activity compression. Unlike Euclidean geometry, which assumes unique shortest paths, Manhattan geometry better represents human movements that typically follow multiple equal-length network routes instead of unfettered straight-line paths. Applying these concepts to urban spatial models, like the Burgess concentric ring conceptualization, reinforces the need for fractal analyses in urban studies. Incorporating a fractal perspective into eigenvector methods, particularly those affiliated with spatial autocorrelation, provides a deeper understanding of urban structure and dynamics, enlightening scholars about city evolution and functions. This approach enhances geometric understanding of city layouts and human behavior, offering insights into urban planning, network density, and human activity flows. Blending theoretical and applied concepts renders a clearer picture of the complex patterns shaping urban spaces. Full article

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in $2+1$ dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras. Full article

In this article, we study the properties of 4-by-4 metric matrices and characterize their dependence and independence by ${\mathcal{M}}^{4\times 4}=({\mathcal{M}}^{4\times 4}-D{\mathcal{M}}^{4\times 4})\cup D{\mathcal{M}}^{4\times 4}$, where $D{\mathcal{M}}^{4\times 4}$ is the set of all dependent metric matrices. $D{\mathcal{M}}^{4\times 4}$ is further characterized by $D{\mathcal{M}}^{4\times 4}=D{\mathcal{M}}_1^{4\times 4}\cup D{\mathcal{M}}_2^{4\times 4}$, where $D{\mathcal{M}}_2^{4\times 4}$ is characterized by $D{\mathcal{M}}_2^{4\times 4}=D{\mathcal{M}}_{21}^{4\times 4}\cup D{\mathcal{M}}_{22}^{4\times 4}$. These characterizations provide some insightful findings that go beyond the Euclidean distance or Euclidean distance matrix and link the distance functions to vector spaces, which offers some theoretical and application-related advantages. In the application parts, we show that the metric matrices associated with all Minkowski distance functions over four different points are linearly independent, and that the metric matrices associated with any four concyclic points are also linearly independent. Full article

In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space ${\mathbb{R}}^{3,1}$ leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ${\coprod }_{\alpha }{H}_{\alpha }$ of ${\mathbb{R}}^{3,1}$ by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ${\coprod }_{\alpha }{H}_{\alpha }$ for the Minkowski metric of the ambient space. Full article

Recent discoveries of massive galaxies existing in the early universe, as well as apparent anomalies in Ω_m and H₀ at high redshift, have raised sharp new concerns for the ΛCDM model of cosmology. Here, we address these problems by using new solutions for the Einstein field equations of relativistic compact objects originally found by Ni. Applied to the universe, the new solutions imply that the universe’s mass is relatively concentrated in a thick outer shell. The interior space would not have a flat, Minkowski metric, but rather a repulsive gravitational field centered on the origin. This field would induce a gravitational redshift in light waves moving inward from the cosmic shell and a corresponding blueshift in waves approaching the shell. Assuming the Milky Way lies near the origin, within the KBC Void, this redshift would make H₀ appear to diminish at high redshifts and could thus relieve the Hubble tension. The Ni redshift could also reduce or eliminate the requirement for dark energy in the ΛCDM model. The relative dimness of distant objects would instead arise because the Ni redshift makes them appear closer to us than they really are. To account for the CMB temperature–redshift relation and for the absence of a systematic blueshift in stars closer to the origin than the Milky Way, it is proposed that the Ni redshift and blueshift involve exchanges of photon energy with a photonic spacetime. These exchanges in turn form the basis for a cosmic CMB cycle, which gives rise to gravity and an Einsteinian cosmological constant, Λ. Black holes are suggested to have analogous Ni structures and gravity/Λ cycles. Full article

While there are well-known synthetic methods in the literature for finding the image of a point under circular inversion in $l_2$-normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in creating a synthetic construction of the circular inversion in $l_1$-normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the $l_1$-norm. Full article

This paper studies the Bondi–Metzner–Sachs group in homogeneous projective coordinates because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, the Bondi–Metzner–Sachs metric, asymptotic Killing vectors, generators of supertranslations as well as boosts and rotations of Minkowski spacetime are all re-expressed in homogeneous projective coordinates. Lastly, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work paves the way for more advanced applications of the geometry of asymptotically flat spacetime in projective coordinates by virtue of the tools provided from complex analysis in several variables and projective geometry. Full article

Motivated by two definitions of distance, “pre-symmetric w-distance” and “w-cone distance”, we define the concept of a pre-symmetric w-cone distance in a TVS-CMS and introduce its properties and examples. Also, we discuss the TVS-cone version of the recent results obtained by Romaguera and Tirado. Meanwhile, using Minkowski functionals, we show the equivalency between some consequences concerning a pre-symmetric w-distance in a usual metric space and a pre-symmetric w-cone distance in a TVS-CMS. Then, some types of various w-cone-contractions and the relations among them are investigated. Finally, as an application, a characterization of the completeness of TVS-cone metric regarding pre-symmetric concept is performed, which differentiates our results from former characterizations. Full article

The objective of this work is to derive the structure of Minkowski spacetime using a Hermitian spin basis. This Hermitian spin basis is analogous to the Pauli spin basis. The derived Minkowski metric is then employed to obtain the corresponding Lorentz factors, potential Lie algebra, effects on gamma matrices and complex representations of relativistic time dilation and length contraction. The main results, a discussion of the potential applications and future research directions are provided. Full article

In this overview, we discuss the (Schwartz) distributional stress–energy quadrupole and show it is a source of gravitational waves. We provide an explicit formula for the metric of linearised gravity in the case of a background Minkowski spacetime. We compare and contrast the two different representations for quadrupoles taken by Dixon and Ellis, present the formula for the dynamics of the quadrupole moments, and determine the number of free components. We review other approaches to the dynamics of quadrupoles, comparing our results. Full article

This paper redefines picture fuzzy soft matrices (pfs-matrices) because of some of their inconsistencies resulting from Cuong’s definition of picture fuzzy sets. Then, it introduces several distance measures of pfs-matrices. Afterward, this paper proposes a new kNN-based classifier, namely the Picture Fuzzy Soft k-Nearest Neighbor (PFS-kNN) classifier. The proposed classifier utilizes the Minkowski’s metric of pfs-matrices to find the k-nearest neighbor. Thereafter, it performs an experimental study utilizing four UCI medical datasets and compares to the suggested approach using the state-of-the-art kNN-based classifiers. To evaluate the performance of the classification, it conducts ten iterations of five-fold cross-validation on all the classifiers. The findings indicate that PFS-kNN surpasses the state-of-the-art kNN-based algorithms in 72 out of 128 performance results based on accuracy, precision, recall, and F1-score. More specifically, the proposed method achieves higher accuracy and F1-score results compared to the other classifiers. Simulation results show that pfs-matrices and PFS-kNN are capable of modeling uncertainty and real-world problems. Finally, the applications of pfs-matrices to supervised learning are discussed for further research. Full article