Search Results (70)

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (represented by the unit sphere of ${\mathbb{R}}^3$), which generate the isometries of the sphere under the rotation group $SO\left(3\right)$. These fields, realized as functions $K_v:{\mathbb{S}}^2\to {\mathbb{R}}^3$, defined by $K_v\left(q\right)=v\times q$ for a fixed $v\in {\mathbb{R}}^3$ and any $q\in {\mathbb{S}}^2$, generate a three-dimensional Lie algebra isomorphic to $\mathfrak{so}\left(3\right)$. We establish an isomorphism $K:{\mathbb{R}}^3\to \mathcal{K}\left({\mathbb{S}}^2\right)$, mapping vectors $v=au$ (with $u\in {\mathbb{S}}^2$) to scaled Killing vector fields $a{K}_u$, and analyze its relationship with $SO\left(3\right)$ through the exponential map. Subsequently, at a fixed point $e\in {\mathbb{S}}^2$, we construct a smooth orthonormal right-handed tangent frame $f_e:{\mathbb{S}}^2\backslash \{e,-e\}\to T{\left({\mathbb{S}}^2\right)}^2$, defined as $f_e\left(u\right)=({\widehat{K}}_e\left(u\right),u\times {\widehat{K}}_e\left(u\right))$, where ${\widehat{K}}_e$ is the unit vector field of the Killing field $K_e$. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on ${\mathbb{S}}^2\backslash \{e,-e\}$ is either $f_e$ or a rotation thereof by a smooth map $\rho :{\mathbb{S}}^2\backslash \{e,-e\}\to SO\left(3\right)$, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via $SO\left(3\right)$ symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

Background/Objectives: Transcatheter aortic valve replacement (TAVR) has been introduced as an optimal treatment for patients with severe aortic stenosis, offering a minimally invasive alternative to surgical aortic valve replacement. Predicting these outcomes following TAVR is crucial. Artificial intelligence (AI) has emerged as a promising tool for improving post-TAVR outcome prediction. In this systematic review and meta-analysis, we aim to summarize the current evidence on utilizing AI in predicting post-TAVR outcomes. Methods: A comprehensive search was conducted to evaluate the studies focused on TAVR that applied AI methods for risk stratification. We assessed various ML algorithms, including random forests, neural networks, extreme gradient boosting, and support vector machines. Model performance metrics—recall, area under the curve (AUC), and accuracy—were collected with 95% confidence intervals (CIs). A random-effects meta-analysis was conducted to pool effect estimates. Results: We included 43 studies evaluating 366,269 patients (mean age 80 ± 8.25; 52.9% men) following TAVR. Meta-analyses for AI model performances demonstrated the following results: all-cause mortality (AUC = 0.78 (0.74–0.82), accuracy = 0.81 (0.69–0.89), and recall = 0.90 (0.70–0.97); permanent pacemaker implantation or new left bundle branch block (AUC = 0.75 (0.68–0.82), accuracy = 0.73 (0.59–0.84), and recall = 0.87 (0.50–0.98)); valve-related dysfunction (AUC = 0.73 (0.62–0.84), accuracy = 0.79 (0.57–0.91), and recall = 0.54 (0.26–0.80)); and major adverse cardiovascular events (AUC = 0.79 (0.67–0.92)). Subgroup analyses based on the model development approaches indicated that models incorporating baseline clinical data, imaging, and biomarker information enhanced predictive performance. Conclusions: AI-based risk prediction for TAVR complications has demonstrated promising performance. However, it is necessary to evaluate the efficiency of the aforementioned models in external validation datasets. Full article

We introduce para-associative algebroids as vector bundles whose sections form a ternary algebra with a generalised form of associativity. We show that a necessary and sufficient condition for local triviality is the existence of a differential connection, i.e., a connection that satisfies the Leibniz rule over the ternary product. Full article

The rapid growth of smart home technologies, driven by the expansion of the Internet of Things (IoT), has introduced both opportunities and challenges in automating daily routines and orchestrating device interactions. Traditional rule-based automation systems often fall short in adapting to dynamic conditions, integrating heterogeneous devices, and responding to evolving user needs. To address these limitations, this study introduces a novel smart home orchestration framework that combines generative Artificial Intelligence (AI), Retrieval-Augmented Generation (RAG), and the modular OSGi framework. The proposed system allows users to express requirements in natural language, which are then interpreted and transformed into executable service bundles by large language models (LLMs) enhanced with contextual knowledge retrieved from vector databases. These AI-generated service bundles are dynamically deployed via OSGi, enabling real-time service adaptation without system downtime. Manufacturer-provided device capabilities are seamlessly integrated into the orchestration pipeline, ensuring compatibility and extensibility. The framework was validated through multiple use-case scenarios involving dynamic device discovery, on-demand code generation, and adaptive orchestration based on user preferences. Results highlight the system’s ability to enhance automation efficiency, personalization, and resilience. This work demonstrates the feasibility and advantages of AI-driven orchestration in realising intelligent, flexible, and scalable smart home environments. 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

The aim here is to describe all isomorphism classes of $\mathrm{SU}(n+1)$-equivariant Hermitian holomorphic vector bundles on the complex projective space $\mathbb{C}{\mathbb{P}}^n$. If $G\subset \mathrm{SU}(n+1)$ is the isotropy subgroup of a chosen point $x_0\in \mathbb{C}{\mathbb{P}}^n$, and $\rho :G⟶\mathrm{GL}\left(V\right)$ is a unitary representation, we obtain $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$. Next, given any $v\in \mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ satisfying certain conditions, a new structure of an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on this underlying $C^{\infty }$ holomorphic Hermitian bundle is obtained. It is shown that all $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$ arise this way. Full article

Motion analysis is of great interest to a variety of applications, such as virtual and augmented reality and medical diagnostics. Hand movement tracking systems, in particular, are used as a human–machine interface. In most cases, these systems are based on optical or acceleration/angular speed sensors. These technologies are already well researched and used in commercial systems. In special applications, it can be advantageous to use magnetic sensors to supplement an existing system or even replace the existing sensors. The core of a motion tracking system is a localization unit. The relatively complex localization algorithms present a problem in magnetic systems, leading to a relatively large computational complexity. In this paper, a new approach for pose estimation of a kinematic chain is presented. The new algorithm is based on spatially rotating magnetic dipole sources. A spatial feature is extracted from the sensor signal, the dipole direction in which the maximum magnitude value is detected at the sensor. This is introduced as the “maximum vector”. A relationship between this feature, the location vector (pointing from the magnetic source to the sensor position) and the sensor orientation is derived and subsequently exploited. By modelling the hand as a kinematic chain, the posture of the chain can be described in two ways: the knowledge about the magnetic correlations and the structure of the kinematic chain. Both are bundled in an iterative algorithm with very low complexity. The algorithm was implemented in a real-time framework and evaluated in a simulation and first laboratory tests. In tests without movement, it could be shown that there was no significant deviation between the simulated and estimated poses. In tests with periodic movements, an error in the range of 1° was found. Of particular interest here is the required computing power. This was evaluated in terms of the required computing operations and the required computing time. Initial analyses have shown that a computing time of 3 $\mathsf{\mu }$s per joint is required on a personal computer. Lastly, the first laboratory tests basically prove the functionality of the proposed methodology. Full article

Mirror symmetry was first observed in worldsheet string constructions, and was shown to have profound implications in the Effective Field Theory (EFT) limit of string compactifications, and for the properties of Calabi–Yau manifolds. It opened up a new field in pure mathematics, and was utilised in the area of enumerative geometry. Spinor–Vector Duality (SVD) is an extension of mirror symmetry. This can be readily understood in terms of the moduli of toroidal compactification of the Heterotic String, which includes the metric the antisymmetric tensor field and the Wilson line moduli. In terms of the toroidal moduli, mirror symmetry corresponds to mappings of the internal space moduli, whereas Spinor–Vector Duality corresponds to maps of the Wilson line moduli. In the past few of years, we demonstrated the existence of Spinor–Vector Duality in the effective field theory compactifications of string theories. This was achieved by starting with a worldsheet orbifold construction that exhibited Spinor–Vector Duality and resolving the orbifold singularities, hence generating a smooth, effective field theory limit with an imprint of the Spinor–Vector Duality. Just like mirror symmetry, the Spinor–Vector Duality can be used to study the properties of complex manifolds with vector bundles. Spinor–Vector Duality offers a top-down approach to the “Swampland” program, by exploring the imprint of the symmetries of the ultra-violet complete worldsheet string constructions in the effective field theory limit. The SVD suggests a demarcation line between (2,0) EFTs that possess an ultra-violet complete embedding versus those that do not. Full article

We construct the smooth vector bundle of tensor densities of arbitrary weight in a coordinate-independent way. We prove the general existence of a globally smooth tensor density field, as well as the existence of a globally smooth metric density for a pseudo-Riemannian manifold, specifically. We study the coordinate description of a covariant derivative over densities, and define a natural extension of affine connections to densities. We provide an equivalent characterization, in the case of a pseudo-Riemannian manifold. Full article

PURPOSE: We aim to compare the performance of three different radiomics models (logistic regression (LR), random forest (RF), and support vector machine (SVM)) and clinical nomograms (Briganti, MSKCC, Yale, and Roach) for predicting lymph node involvement (LNI) in prostate cancer (PCa) patients. MATERIALS AND METHODS: The retrospective study includes 95 patients who underwent mp-MRI and radical prostatectomy for PCa with pelvic lymphadenectomy. Imaging data (intensity in T2, DWI, ADC, and PIRADS), clinical data (age and pre-MRI PSA), histological data (Gleason score, TNM staging, histological type, capsule invasion, seminal vesicle invasion, and neurovascular bundle involvement), and clinical nomograms (Yale, Roach, MSKCC, and Briganti) were collected for each patient. Manual segmentation of the index lesions was performed for each patient using an open-source program (3D SLICER). Radiomic features were extracted for each segmentation using the Pyradiomics library for each sequence (T2, DWI, and ADC). The features were then selected and used to train and test three different radiomics models (LR, RF, and SVM) independently using ChatGPT software (v 4o). The coefficient value of each feature was calculated (significant value for coefficient ≥ ±0.5). The predictive performance of the radiomics models and clinical nomograms was assessed using accuracy and area under the curve (AUC) (significant value for p ≤ 0.05). Thus, the diagnostic accuracy between the radiomics and clinical models were compared. RESULTS: This study identified 343 features per patient (330 radiomics features and 13 clinical features). The most significant features were T2_nodulofirstordervariance and T2_nodulofirstorderkurtosis. The highest predictive performance was achieved by the RF model with DWI (accuracy 86%, AUC 0.89) and ADC (accuracy 89%, AUC 0.67). Clinical nomograms demonstrated satisfactory but lower predictive performance compared to the RF model in the DWI sequences. CONCLUSIONS: Among the prediction models developed using integrated data (radiomics and semantics), RF shows slightly higher diagnostic accuracy in terms of AUC compared to clinical nomograms in PCa lymph node involvement prediction. Full article

Gerbes and higher gerbes are geometric cocycles representing higher degree cohomology classes, and are attracting considerable interest in differential geometry and mathematical physics. We prove that a 2-gerbe has a torsion Dixmier–Douady class if and only if the gerbe has locally constant cocycle data. As an application, we give an alternative description of flat twisted vector bundles in terms of locally constant transition maps. These results generalize to n-gerbes for $n=1$ and $n\ge 3$, providing insights into the structure of higher gerbes and their applications to the geometry of twisted vector bundles. Full article

We study fiber bundles where the fibers are not a group G but a free G-space with disjoint orbits. The fibers are then not torsors but disjoint unions of these; hence, we like to call them semi-torsors. Bundles of semi-torsors naturally generalize principal bundles, and we call these semi-principal bundles. These bundles admit parallel transport in the same way that principal bundles do. The main difference is that lifts may end up in another group orbit, meaning that the change cannot be described by group translations alone. The study of such effects is facilitated by defining the notion of a basis of a G-set, in analogy with a basis of a vector space. The basis elements serve as reference points for the orbits so that parallel transport amounts to reordering the basis elements and scaling them with the appropriate group elements. These two symmetries of the bases are described by a wreath product group. The notion of basis also leads to a frame bundle, which is principal and so allows for a conventional treatment. In fact, the frame bundle functor is found to be a retraction from the semi-principal bundles to the principal bundles. The theory presented provides a mathematical framework for a unified description of geometric phases and exceptional points in adiabatic quantum mechanics. Full article

In this study, we investigate the tangent bundle $TM$ of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection $\tilde{\nabla }$. Our primary goal is to establish the necessary and sufficient conditions for $TM$ to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for $TM$ to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to $TM$, alongside specific constraints on the conformal factor $\lambda$ and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve $\lambda$ and the Hessian of the potential function. Similarly, for $TM$ to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of $\lambda$, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry. Full article

We examine the heap of linear connections on anchored vector bundles and Lie algebroids. Naturally, this covers the example of affine connections on a manifold. We present some new interpretations of classical results via this ternary structure of connections. Endomorphisms of linear connections are studied, and their ternary structure, in particular the endomorphism truss, is explicitly presented. We remark that the use of ternary structures in differential geometry is novel and that the endomorphism truss of linear connections provides a concrete geometric example of a truss. Full article

One of the most relevant topics in web theory is linearization. A particular class of linearizable webs is the Grassmannizable web. Akivis gave a characterization of such a web, showing that Grassmannizable webs are equivalent to isoclinic and transversally geodesic webs. The obstructions given by Akivis that characterize isoclinic and transversally geodesic webs are computed locally, and it is difficult to give them an interpretation in relation to torsion or curvature of the unique Chern connection associated with a web. In this paper, using Nagy’s web formalism, Frölisher—Nejenhuis theory for derivation associated with vector differential forms, and Grifone’s connection theory for tensorial algebra on the tangent bundle, we find invariants associated with almost-Grassmann structures expressed in terms of torsion, curvature, and Nagy’s tensors, and we provide an interpretation in terms of these invariants for the isoclinic, transversally geodesic, Grassmannizable, and parallelizable webs. Full article