Search Results (150)

In this work, we undertake a comprehensive study of the functional measure of gravitational path integrals within a general framework involving non-trivial configuration spaces. As in Riemannian geometry, the integration over non-trival configuration spaces requires a metric. We examine the interplay between the functional measure and the dynamics of spacetime for general configuration-space metrics. The functional measure gives an exact contribution to the effective action at the one-loop level. We discuss the implications and phenomenological consequences of this correction, shedding light on the role of the functional measure in quantum gravity theories. In particular, we describe a mechanism in which the graviton acquires a mass from the functional measure without violating the diffeomorphism symmetry nor including Stückelberg fields. Since gauge invariance is not violated, the number of degrees of freedom goes as in general relativity. For the same reason, Boulware–Deser ghosts and the vDVZ discontinuity do not show up. The graviton thus becomes massive at the quantum level while avoiding the usual issues of massive gravity. Full article

A brain–computer interface (BCI) provides a direct communication pathway between the human brain and external devices, enabling users to control them through thought. It records brain signals and classifies them into specific commands for external devices. Among various classifiers used in BCI, those directly classifying covariance matrices using Riemannian geometry find broad applications not only in BCI, but also in diverse fields such as computer vision, natural language processing, domain adaption, and remote sensing. However, the existing Riemannian-based methods exhibit limitations, including time-intensive computations, susceptibility to disturbances, and convergence challenges in scenarios involving high-dimensional matrices. In this paper, we tackle these issues by introducing the Bures–Wasserstein (BW) distance for covariance matrices analysis and demonstrating its advantages in BCI applications. Both theoretical and computational aspects of BW distance are investigated, along with algorithms for Fréchet Mean (or barycenter) estimation using BW distance. Extensive simulations are conducted to evaluate the effectiveness, efficiency, and robustness of the BW distance and barycenter. Additionally, by integrating BW barycenter into the Minimum Distance to Riemannian Mean classifier, we showcase its superior classification performance through evaluations on five real datasets. Full article

Automated handwritten signature verification continues to pose significant challenges. A common approach for developing writer-independent signature verifiers involves the use of a dichotomizer, a function that generates a dissimilarity vector with the differences between similar and dissimilar pairs of signature descriptors as components. The Dichotomy Transform was applied within a Euclidean or vector space context, where vectored representations of handwritten signatures were embedded in and conformed to Euclidean geometry. Recent advances in computer vision indicate that image representations to the Riemannian Symmetric Positive Definite (SPD) manifolds outperform vector space representations. In offline signature verification, both writer-dependent and writer-independent systems have recently begun leveraging Riemannian frameworks in the space of SPD matrices, demonstrating notable success. This work introduces, for the first time in the signature verification literature, a Riemannian dichotomizer employing Riemannian dissimilarity vectors (RDVs). The proposed framework explores a number of local and global (or common pole) topologies, as well as simple serial and parallel fusion strategies for RDVs for constructing robust models. Experiments were conducted on five popular signature datasets of Western and Asian origin, using blind intra- and cross-lingual experimental protocols. The results indicate the discriminative capabilities of the proposed Riemannian dichotomizer framework, which can be compared to other state-of-the-art and computationally demanding architectures. Full article

In a previous study we investigated the spherically symmetric Schwarzschild and Schwarzschild–de Sitter solutions within a Finsler–Randers-type geometry. In this work, we extend our analysis to charged and rotating solutions, focusing on the Reissner–Nordström and Kerr-like metrics in the Finsler–Randers gravitational framework. In particular, we extract the modified gravitational field equations and we examine the geodesic equations, analyzing particle trajectories and quantifying the deviations from their standard counterparts. Moreover, we compare the results with the predictions of general relativity, and we discuss how potential deviations from Riemannian geometry could be reached observationally. Full article

We present a review of recent developments in cosmological models based on Finsler geometry, as well as geometric extensions of general relativity formulated within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend not only on position but also on an additional internal degree of freedom, typically represented by a vector field at each point of the spacetime manifold. We examine in detail the possibility that Finsler-type geometries can describe the physical properties of the gravitational interaction, as well as the cosmological dynamics. In particular, we present and review the implications of a particular implementation of Finsler geometry, based on the Barthel connection, and of the $(\alpha ,\beta )$ geometries, where $\alpha$ is a Riemannian metric, and $\beta$ is a one-form. For a specific construction of the deviation part $\beta$, in these classes of geometries, the Barthel connection coincides with the Levi–Civita connection of the associated Riemann metric. We review the properties of the gravitational field, and of the cosmological evolution in three types of geometries: the Barthel–Randers geometry, in which the Finsler metric function F is given by $F=\alpha +\beta$, in the Barthel–Kropina geometry, with $F={\alpha }^2/\beta$, and in the conformally transformed Barthel–Kropina geometry, respectively. After a brief presentation of the mathematical foundations of the Finslerian-type modified gravity theories, the generalized Friedmann equations in these geometries are written down by considering that the background Riemannian metric in the Randers and Kropina line elements is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equations are also presented, and they are interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. We investigate the cosmological properties of the Barthel–Randers and Barthel–Kropina cosmological models in detail. In these scenarios, the additional geometric terms arising from the Finslerian structure can be interpreted as an effective geometric dark energy component, capable of generating an effective cosmological constant. Several cosmological solutions—both analytical and numerical—are obtained and compared against observational datasets, including Cosmic Chronometers, Type Ia Supernovae, and Baryon Acoustic Oscillations, using a Markov Chain Monte Carlo (MCMC) analysis. A direct comparison with the standard $\Lambda$CDM model is also carried out. The results indicate that Finslerian cosmological models provide a satisfactory fit to the observational data, suggesting they represent a viable alternative to the standard cosmological model based on general relativity. Full article

Large Language models have shown a remarkable ability to “converse” with humans in a natural language across myriad topics. Despite the proliferation of these models, a deep understanding of how they work under the hood remains elusive. The core of these Generative AI models is composed of layers of neural networks that employ the Transformer architecture. This architecture learns from large amounts of training data and creates new content in response to user input. In this study, we analyze the internals of the Transformer using Information Theory. To quantify the amount of information passing through a layer, we view it as an information transmission channel and compute the capacity of the channel. The highlight of our study is that, using Information-Theoretical tools, we develop techniques to visualize on an Information plane how the Transformer encodes the relationship between words in sentences while these words are projected into a high-dimensional vector space. We use Information Geometry to analyze the high-dimensional vectors in the Transformer layer and infer relationships between words based on the length of the geodesic connecting these vector distributions on a Riemannian manifold. Our tools reveal more information about these relationships than attention scores. In this study, we also show how Information-Theoretic analysis can help in troubleshooting learning problems in the Transformer layers. Full article

Planning a multi-joint minimum-effort coordinated human movement to achieve a visual goal is computationally difficult: (i) The number of anatomical elemental movements of the human body greatly exceeds the number of degrees of freedom specified by visual goals; and (ii) the mass–inertia mechanical load about each elemental movement varies not only with the posture of the body but also with the mechanical interactions between the body and the environment. Given these complications, the amount of nonlinear dynamical computation needed to plan visually-guided movement is far too large for it to be carried out within the reaction time needed to initiate an appropriate response. Consequently, we propose that, as part of motor and visual development, starting with bootstrapping by fetal and neonatal pattern-generator movements and continuing adaptively from infancy to adulthood, most of the computation is carried out in advance and stored in a motor association memory network. From there it can be quickly retrieved by a selection process that provides the appropriate movement synergy compatible with the particular visual goal. We use theorems of Riemannian geometry to describe the large amount of nonlinear dynamical data that have to be pre-computed and stored for retrieval. Based on that geometry, we argue that the logical mathematical sequence for the acquisition of these data parallels the natural development of visually- guided human movement. Full article

We reformulate holographic space–time models in terms of Hilbert bundles over the space of the time-like geodesics in a Lorentzian manifold. This reformulation resolves the issue of the action of non-compact isometry groups on finite-dimensional Hilbert spaces. Following Jacobson, I view the background geometry as a hydrodynamic flow, whose connection to an underlying quantum system follows from the Bekenstein–Hawking relation between area and entropy, generalized to arbitrary causal diamonds. The time-like geodesics are equivalent to the nested sequences of causal diamonds, and the area of the holoscreen (The holoscreen is the maximal $d-2$ volume (“area”) leaf of a null foliation of the diamond boundary. I use the term area to refer to its volume.) encodes the entropy of a certain density matrix on a finite-dimensional Hilbert space. I review arguments that the modular Hamiltonian of a diamond is a cutoff version of the Virasoro generator $L_0$ of a $1+1$-dimensional CFT of a large central charge, living on an interval in the longitudinal coordinate on the diamond boundary. The cutoff is chosen so that the von Neumann entropy is $\ln {\mathrm{D}}_{\diamond },$ up to subleading corrections, in the limit of a large-dimension diamond Hilbert space. I also connect those arguments to the derivation of the ’t Hooft commutation relations for horizon fluctuations. I present a tentative connection between the ’t Hooft relations and $U\left(1\right)$ currents in the CFTs on the past and future diamond boundaries. The ’t Hooft relations are related to the Schwinger term in the commutator of the vector and axial currents. The paper in can be read as evidence that the near-horizon dynamics for causal diamonds much larger than the Planck scale is equivalent to a topological field theory of the ’t Hooft CR plus small fluctuations in the transverse geometry. Connes’ demonstration that the Riemannian geometry is encoded in the Dirac operator leads one to a completely finite theory of transverse geometry fluctuations, in which the variables are fermionic generators of a superalgebra, which are the expansion coefficients of the sections of the spinor bundle in Dirac eigenfunctions. A finite cutoff on the Dirac spectrum gives rise to the area law for entropy and makes the geometry both “fuzzy” and quantum. Following the analysis of Carlip and Solodukhin, I model the expansion coefficients as two-dimensional fermionic fields. I argue that the local excitations in the interior of a diamond are constrained states where the spinor variables vanish in the regions of small area on the holoscreen. This leads to an argument that the quantum gravity in asymptotically flat space must be exactly supersymmetric. Full article

: A substantial amount of research has demonstrated the robustness and accuracy of the Riemannian minimum distance to mean (MDM) classifier for all kinds of EEG-based brain–computer interfaces (BCIs). This classifier is simple, fully deterministic, robust to noise, computationally efficient, and prone to transfer learning. Its training is very simple, requiring just the computation of a geometric mean of a symmetric positive-definite (SPD) matrix per class. We propose an improvement of the MDM involving a number of power means of SPD matrices instead of the sole geometric mean. By the analysis of 20 public databases, 10 for the motor-imagery BCI paradigm and 10 for the P300 BCI paradigm, comprising 587 individuals in total, we show that the proposed classifier clearly outperforms the MDM, approaching the state-of-the art in terms of performance while retaining the simplicity and the deterministic behavior. In order to promote reproducible research, our code will be released as open source. Full article

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in different areas of differential geometry, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as Golden space forms, Sasakian space forms; diffeological and affine connection spaces; Weingarten and Delaunay surfaces; Chen-type inequalities for submanifolds; statistical submersions; manifolds endowed with different geometric structures (Sasakian, weak nearly Sasakian, weak nearly cosymplectic, LP-Kenmotsu, paraquaternionic); solitons (almost Ricci solitons, almost Ricci–Bourguignon solitons, gradient r-almost Newton–Ricci–Yamabe solitons, statistical solitons, solitons with semi-symmetric connections); vector fields (projective, conformal, Killing, 2-Killing) [...] Full article

This editorial presents 24 research articles published in the Special Issue entitled Geometry of Manifolds and Applications of the MDPI Mathematics journal, which covers a wide range of topics from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in many branches of theoretical and applied mathematical studies, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as complex space forms, metallic Riemannian space forms, Hessian manifolds of constant Hessian curvature; optimal inequalities for submanifolds, such as generalized Wintgen inequality, inequalities involving $\delta$-invariants; homogeneous spaces and Poisson–Lie groups; the geometry of biharmonic maps; solitons (Ricci solitons, Yamabe solitons, Einstein solitons) in different geometries such as contact and paracontact geometry, complex and metallic Riemannian geometry, statistical and Weyl geometry; perfect fluid spacetimes [...] Full article

Resorting to microcanonical ensemble Monte Carlo simulations, we study the geometric and topological properties of the state space of a model of a network glass-former. This model, a Lennard-Jones binary mixture, does not crystallize due to frustration. We have found two peaks in specific heat at equilibrium and at low energy, corresponding to important changes in local ordering. These singularities were accompanied by inflection points in geometrical markers of the potential energy level sets—namely, the mean curvature, the dispersion of the principal curvatures, and the variance of the scalar curvature. Pinkall’s and Overholt’s theorems closely relate these quantities to the topological properties of the accessible state-space manifold. Thus, our analysis provides strong indications that the glass transition is associated with major changes in the topology of the energy level sets. This important result suggests that this phase transition can be understood through the topological theory of phase transitions. Full article

The rapid evolution of mega-constellation networks and 6G satellite communication systems has ushered in an era of ubiquitous connectivity, yet their sustainability is threatened by the energy-computation dilemma inherent in high-throughput data transmission. Polar codes, as a coding scheme capable of achieving Shannon’s limit, have emerged as one of the key candidate coding technologies for 6G networks. Despite the high parallelism and excellent performance of their Belief Propagation (BP) decoding algorithm, its drawbacks of numerous iterations and slow convergence can lead to higher energy consumption, impacting system energy efficiency and sustainability. Therefore, research on efficient early termination algorithms has become an important direction in polar code research. In this paper, based on information geometry theory, we propose a novel geometric framework for BP decoding of polar codes and design two early termination algorithms under this framework: an early termination algorithm based on Riemannian distance and an early termination algorithm based on divergence. These algorithms improve convergence speed by geometrically analyzing the changes in soft information during the BP decoding process. Simulation results indicate that, when $E_b/N_0$ is between 1.5 dB and 2.5 dB, compared to three classical early termination algorithms, the two early termination algorithms proposed in this paper reduce the number of iterations by 4.7–11% and 8.8–15.9%, respectively. Crucially, while this work is motivated by the unique demands of satellite networks, the geometric characterization of polar code BP decoding transcends specific applications. The proposed framework is inherently adaptable to any communication system requiring energy-efficient channel coding, including 6G terrestrial networks, Internet of Things (IoT) edge devices, and unmanned aerial vehicle (UAV) swarms, thereby bridging theoretical coding advances with real-world scalability challenges. Full article

Fundamentally, duality gives two different points of view of looking at the same object. It appears in many subjects in mathematics (geometry, algebra, analysis, PDEs, Geometric Measure Theory, etc.) and in physics. For example, Connections on Fiber Bundles in mathematics, and Gauge Fields in physics are exactly the same. In n-dimensional geometry, a fundamental notion is the “duality” between chains and cochains, or domains of integration and the integrands. In this paper, we extend ideas given in our earlier articles and connect seemingly unrelated areas of F-harmonic maps, f-harmonic maps, and cohomology classes via duality. By studying cohomology classes that are related with p-harmonic morphisms, F-harmonic maps, and f-harmonic maps, we extend several of our previous results on Riemannian submersions and p-harmonic morphisms to F-harmonic maps and f-harmonic maps, which are Riemannian submersions. Full article

In this paper, we investigate warped products on super quasi-Einstein manifolds under affine connections. We explore their fundamental properties, establish conditions for their existence, and prove that these manifolds can also be nearly quasi-Einstein and pseudo quasi-Einstein. To illustrate, we provide examples in both Riemannian and Lorentzian geometries, confirming their existence. Finally, we construct and analyze an explicit example of a warped product on a super quasi-Einstein manifold with respect to affine connections. Full article