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This study examines how fractional derivatives affect the theory of curves and related surfaces, an application area that is expanding daily. There has been limited research on the geometric interpretation of fractional calculus. The present study applied the Caputo fractional calculation method, which has the most suitable structure for geometric computations, to examine the effect of fractional calculus on differential geometry. The Caputo fractional derivative of a constant is zero, enabling the geometric solution and understanding of many fractional physical problems. We examined flux tubes, which are magnetic surfaces that incorporate these lines of magnetic fields as parameter curves. Examples are visualized using mathematical programs for various values of Caputo fractional analysis, employing theory‐related examples. Fractional derivatives and integrals are widely utilized in various disciplines, including mathematics, physics, engineering, biology, and fluid dynamics, as they yield more numerical results than classical solutions. Also, many problems outside the scope of classical analysis methods can be solved using the Caputo fractional calculation approach. In this context, applying the Caputo fractional analytic calculation method in differential geometry yields physically and mathematically relevant findings, particularly in the theory of curves and surfaces. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear fractional Kairat-X equation is investigated on the basis of computational simulation. The nonlinear fractional Kairat-X equation is an integrable equation and is used to explain the differential geometry of curves and equivalence aspects. Several kinds of solitary wave structures of the nonlinear fractional Kairat-X equation are established successfully via the implantation of the extended simple equation method. Here, we explore the interesting, novel and general solutions in trigonometric, exponential, and rational types, which represent periodic wave solitons, mixed solitons in the shape of bright–dark solutions, kink wave solitons, peakon bright and dark solitons, anti-kink wave solutions, bright solitons, dark solitons, and solitary wave structure. The physical structures of secured results, aided by numerical simulation, have numerous applications in applied sciences such as optical fiber, geophysics, laser optics, mathematical physics, nonlinear optics, nonlinear dynamics, communication system, and engineering. This study explores the physical behavior of models through the visualization of solutions in contour, 2D and 3D plots by revealing that these solutions yield profitable results in the field of mathematical physics. The study demonstrates that the proposed technique is more reliable, efficient, and powerful in analyzing nonlinear evolution equations in various domains of science. [ABSTRACT FROM AUTHOR]

Imaging dense clusters of seismicity is crucial to many problems in seismology: to delineate complex systems of faults, provide constraints on the causes of volcanic and cryogenic swarms, and to shed light on possible means to prevent damaging induced seismicity in mining, geothermal, and oil and gas extraction activities. Current imaging methods rely upon high-resolution relative location techniques, commonly requiring arrival-time picks for seismic phases. This paper examines an alternative approach, based upon concepts drawn from differential geometry, that images directly from waveform data. It relies upon the common assumption of spatial continuity of seismic wavefield observations, which implies that a differentiable map exists between the source region to be imaged and waveform observations considered as elements of a vector space. The map creates an image of event clusters on a Riemannian manifold embedded in that vector space. The image can be visualized by projecting the observations into a tangent space of the manifold and is a distorted rendering of cluster geometry. However, the distortion can be predicted and removed if a model for wavefield propagation is available. This visualization approach is applicable to clusters of uniform events with highly similar waveforms, such as are commonly acquired with correlation detectors or other pattern matching techniques. To assess its performance, it is applied to the closely related reciprocal problem of imaging the (known) geometry of an array from observations by the array of several regional events. Differences between the original problem and its reciprocal analogue are noted and controlled for in the analysis. Chief among the differences is the necessity for aligning the waveforms in the original problem, which, to maintain consistency with the original problem, is solved in the reciprocal problem by a generalization of the VanDecar–Crosson algorithm. The VanDecar–Crosson algorithm exhibits a bias, shown through an analysis of the situation when the observed wavefields are adequately modelled as plane waves. In that circumstance, the bias can be predicted and removed. In a test using a portion of a large-N array, this imaging approach is shown to successfully reconstruct the array geometry. The method is applicable directly to infinitesimal array apertures, but is extended to a larger aperture by partitioning the image into local, effectively infinitesimal overlapping subsets. These are inverted, then assembled into a global picture of the array geometry using constraints provided by the overlapped regions. Although demonstrated in a reciprocal array context, the method appears viable for imaging clusters of events with highly similar source mechanisms and time histories. [ABSTRACT FROM AUTHOR]

Quantifying spatiotemporal dynamics during embryogenesis is crucial for understanding congenital diseases. We developed Spateo (https://github.com/aristoteleo/spateo-release), a 3D spatiotemporal modeling framework, and applied it to a 3D mouse embryogenesis atlas at E9.5 and E11.5, capturing eight million cells. Spateo enables scalable, partial, non-rigid alignment, multi-slice refinement, and mesh correction to create molecular holograms of whole embryos. It introduces digitization methods to uncover multi-level biology from subcellular to whole organ, identifying expression gradients along orthogonal axes of emergent 3D structures, e.g., secondary organizers such as midbrain-hindbrain boundary (MHB). Spateo further jointly models intercellular and intracellular interaction to dissect signaling landscapes in 3D structures, including the zona limitans intrathalamica (ZLI). Lastly, Spateo introduces "morphometric vector fields" of cell migration and integrates spatial differential geometry to unveil molecular programs underlying asymmetrical murine heart organogenesis and others, bridging macroscopic changes with molecular dynamics. Thus, Spateo enables the study of organ ecology at a molecular level in 3D space over time. [Display omitted] • Spateo reconstructs 3D maps and models spatiotemporal dynamics at the whole-embryo scale • Reconstruction of the molecular holograms of mouse embryos at stages E9.5 and E11.5 • 3D digitization and cell communication reveal spatial gradients and signaling pathways • Morphic vector field predicts molecular drivers of heart asymmetrical morphogenesis Spateo is a comprehensive framework for 3D reconstruction and characterization of spatial gradients and cellular interactions at whole-organ and embryo levels, using spatial transcriptomics. Importantly, Spateo also introduces morphometric vector field analyses that connect macroscopic cell morphogenesis with microscopic molecular dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we give an introduction to Drinfel’d’s associator coming from the Knizhnik–Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential geometry: we rather use limits of concrete parallel transports. [ABSTRACT FROM AUTHOR]

This study examines the spinor representations of TN (tangent and normal), NB (normal and binormal), TB (tangent and binormal) and TNB (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space E 3 . Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the C 3 complex vector space form a two-dimensional surface in the C 2 complex space. Additionally, each isotropic vector in C 3 space corresponds to two vectors in C 2 space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its (TN , NB , TB and TNB)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the dressing field method (DFM) in the context of quantum (gauge) field theories (QFT). In order to adapt this method to the functional path integral formalism of QFT, we depart from the usual differential geometry approach used so far to study the DFM which also allows to tackle the infinite dimension of the field spaces. Our main result is that gauge fixing is an instance of the application of the DFM. The Faddeev–Popov gauge fixing procedure and the so-called unitary gauge are revisited in light of this result. [ABSTRACT FROM AUTHOR]

AbstractThe planar flexible Koch fractal eddy current probe exhibits high consistency in detecting short cracks in all orientations. However, it is limited by a weak signal caused by the small number of coil turns. This paper utilizes multilayer flexible circuit board technology to address the issue. The probe is designed with a mutual inductance coil structure and is arranged on a 10-layer flexible circuit board. This board includes one layer for the excitation coil and nine layers for the pick-up coil. The probe’s performance was studied through C-scan experiments, which examined cracks of varying depths, widths, orientations, and lengths. The experimental results show that the probe can detect minimum crack sizes of 0.5 mm × 0.15 mm × 1 mm for 5083-aluminum alloy, 3 mm × 0.15 mm × 1 mm for GH4169 superalloy, and 1 mm × 0.15 mm × 1 mm for Q235 steel. The results demonstrate that the flexible fractal eddy current probe has significant advantages in detecting small cracks. [ABSTRACT FROM AUTHOR]

This work explores the geometry of extremal Kerr-Newman black holes by analyzing their mass/energy relationships and the conditions ensuring black hole existence. Using differential geometry in E 3 , we examine the topology of the event horizon surface and identify two distinct families of extremal black holes, each defined by unique proportionalities between their core parameters: mass (m), charge (Q), angular momentum (L), and the irreducible mass ( m i r ). In the first family, these parameters are proportionally related to the irreducible mass by irrational numbers, with a characteristic flat Gaussian curvature at the poles. In the second family, we uncover a more intriguing structure where m, Q, and L are connected to m i r through coefficients involving the golden ratio − ϕ − . Within this family lies a unique black hole whose physical parameters converge on the golden ratio, including the irreducible mass and polar Gauss curvature. This black hole represents the highest symmetry achievable within the constraints of the Kerr-Newman metric. This remarkable symmetry invites further speculation about its implications, such as the potential determination of the dark energy density parameter Ω Λ for Kerr-Newman-de Sitter black holes. Additionally, we compute the maximum energy that can be extracted through reversible transformations. We have determined that the second, golden-ratio-linked family allows for a greater energy yield than the first. [ABSTRACT FROM AUTHOR]

The article explores geodesics on polyhedra, particularly focusing on simple closed geodesics and their properties. It discusses the classification of geodesics on regular polyhedra, the relationship between geodesics and billiards, and the uniqueness of disphenoids in having arbitrarily long geodesics. The text also addresses the existence of simple closed geodesics on tetrahedra and the properties of geodesics on nonconvex polyhedra, including the construction of long closed geodesics using seven cubes. Open questions are posed regarding geodesics on nonconvex polyhedra in different spaces and the minimal number of vertices needed for long geodesics. The author acknowledges the contributions of an anonymous referee for their feedback. [Extracted from the article]

A smooth ruled surface in 4‐space has only parabolic points or inflection points of the real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along which the parallel projection exhibits A$\mathcal {A}$‐singularities of type butterfly or worse. In particular, such parabolic points can be classified as butterfly hyperbolic, parabolic, or elliptic points depending on the value of the discriminant of a binary differential equation (BDE). Also, whenever such discriminant is positive, we ensure that the integral curves of these directions form a pair of foliations on the ruled surface. Moreover, the set of points that nullify the discriminant is a regular curve transverse to the regular curve formed by inflection points of the real type. Finally, using a particular projective transformation, we obtain a simple parametrization of the ruled surface such that the moduli of its 5‐jet identify a butterfly hyperbolic/parabolic/elliptic point, as well as we get the stable configurations of the solutions of BDE in the discriminant curve. [ABSTRACT FROM AUTHOR]

This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite-dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE (3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task. [ABSTRACT FROM AUTHOR]

In this paper we present several new contributions to the oscillation theory of linear differential equations, in particular of linear Hamiltonian systems, when the traditional Legendre condition is absent. Following our recent work (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 2023), we introduce the multiplicity of a generalized right focal point and derive the corresponding local Sturmian separation theorem. We also examine the relation between the existence of finitely many generalized right focal points, or in the special case the nonexistence of generalized right focal points, with the Legendre condition. As the main tools we use new notions of the minimal multiplicities at a given point and the dual comparative index — an object from matrix analysis or differential geometry (Maslov index theory). Furthermore, we study local limit properties of the dual comparative index and the comparative index and apply them for deriving new oscillation results phrased in terms of the generalized right and left focal points. The investigation of the interplay between generalized right and left focal points leads to conditions characterizing the situation, when in the local Sturmian separation theorem the corresponding multiplicities attain the minimal possible value. This also provides a generalization of the concepts of the right and left proper focal point defined by Kratz (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 2007) to the setting, which does not impose the Legendre condition. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm–Liouville differential equations of arbitrary even order. [ABSTRACT FROM AUTHOR]

In this expository article we revisit the Bernstein problem for several geometric PDEs including the minimal surface, Monge-Ampère, and special Lagrangian equations. We also discuss the minimal surface system where appropriate. The article is based on a lecture series given by the author for the inaugural European Doctorate School of Differential Geometry, held in Granada in June 2024. [ABSTRACT FROM AUTHOR]

Meshes with spherical faces and circular edges are an attractive alternative to polyhedral meshes for applications in architecture and design. Approximation of a given surface by such a mesh needs to consider the visual appearance, approximation quality, the position and orientation of circular intersections of neighboring faces and the existence of a torsion free support structure that is formed by the planes of circular edges. The latter requirement implies that the mesh simultaneously defines a second mesh whose faces lie on the same spheres as the faces of the first mesh. It is a discretization of the two envelopes of a sphere congruence, i.e., a two-parameter family of spheres. We relate such sphere congruences to torsal parameterizations of associated line congruences. Turning practical requirements into properties of such a line congruence, we optimize line and sphere congruence as a basis for computing a mesh with spherical triangular or quadrilateral faces that approximates a given reference surface. [ABSTRACT FROM AUTHOR]

Motivated by the emergence of rough surfaces in various areas of design, we address the computational design of triangle meshes with controlled roughness. Our focus lies on small levels of roughness. There, roughness or smoothness mainly arises through the local positioning of the mesh edges and faces with respect to the curvature behavior of the reference surface. The analysis of this interaction between curvature and roughness is simplified by a 2D dual diagram and its generation within so-called isotropic geometry, which may be seen as a structure-preserving simplification of Euclidean geometry. Isotropic dihedral angles of the mesh are close to the Euclidean angles and appear as Euclidean edge lengths in the dual diagram, which also serves as a tool for visualization and interactive local design. We present a computational framework that includes appearance-aware remeshing, optimization-based automatic roughening, and control of dihedral angles. [ABSTRACT FROM AUTHOR]

This paper provides computational tools for the modeling and design of quad mesh mechanisms, which are meshes allowing continuous flexions under the assumption of rigid faces and hinges in the edges. We combine methods and results from different areas, namely differential geometry of surfaces, rigidity and flexibility of bar and joint frameworks, algebraic geometry, and optimization. The basic idea to achieve a time-continuous flexion is time-discretization justified by an algebraic degree argument. We are able to prove computationally feasible bounds on the number of required time instances we need to incorporate in our optimization. For optimization to succeed, an informed initialization is crucial. We present two computational pipelines to achieve that: one based on remeshing isometric surface pairs, another one based on iterative refinement. A third manner of initialization proved very effective: We interactively design meshes which are close to a narrow known class of flexible meshes, but not contained in it. Having enjoyed sufficiently many degrees of freedom during design, we afterwards optimize towards flexibility. [ABSTRACT FROM AUTHOR]

Alignable lamella gridshells are 3D grid structures capable of collapsing into a planar strip. This feature significantly simplifies on-site assembly and also ensures compactness for efficient transport and storage. However, designing these structures still remains a challenge. This paper tackles the inverse design problem of alignable lamella gridshells leveraging concepts from differential geometry and Cartan's theory of moving frames. The study unveils that geodesic alignable gridshells, where lamellae are disposed tangentially to the surface, are limited to forming shapes isometric to surfaces of revolution. Furthermore, it demonstrates that alignable gridshells with lamellae arranged orthogonally to a surface can be realized only on a specific class of surfaces that meet a particular curvature condition along their principal curvature lines. Finally, drawing on these theoretical findings, this work introduces novel computational tools tailored for the design of these structures. [ABSTRACT FROM AUTHOR]

This paper is devoted to a description of the second‐order differential geometry of torsion‐free almost quaternionic skew‐Hermitian manifolds, that is, of quaternionic skew‐Hermitian manifolds (M,Q,ω)$(M, Q, \omega)$. We provide a curvature characterization of such integrable geometric structures, based on the holonomy theory of symplectic connections and we study qualitative properties of the induced Ricci tensor. Then, we proceed with bundle constructions over such a manifold (M,Q,ω)$(M, Q, \omega)$. In particular, we prove the existence of almost hypercomplex skew‐Hermitian structures on the Swann bundle over M and investigate their integrability. [ABSTRACT FROM AUTHOR]

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems. [ABSTRACT FROM AUTHOR]

The preface to "Geometry and Topology with Applications" highlights the historical significance and evolution of geometry as a research field in mathematics. It discusses the contributions of prominent mathematicians like Euclid, Euler, Gauss, Lobachevsky, Riemann, Hilbert, Poincaré, Thurston, and Gromov to the development of various geometries and topologies. The document also focuses on the work of Thurston and Gromov in geometric topology and geometric group theory, respectively, and their impact on modern mathematics. The Special Issue attracted submissions from researchers worldwide, with 12 papers accepted for publication, including survey papers on discrete groups and topology in the fourth dimension. [Extracted from the article]

Galilean and Carrollian algebras acting on two-dimensional Newton-Cartan and Carrollian manifolds are isomorphic. A consequence of this property is a duality correspondence between one-dimensional Galilean and Carrollian fluids. We describe the dynamics of these systems as they emerge from the relevant limits of Lorentzian hydrodynamics, and explore the advertised duality relationship. This interchanges longitudinal and transverse directions with respect to the flow velocity, and permutes equilibrium and out-of-equilibrium observables, unveiling specific features of Carrollian physics. We investigate the action of local hydrodynamic-frame transformations in the Galilean and Carrollian configurations, i.e. dual Galilean and Carrollian local boosts, and comment on their potential breaking. Emphasis is laid on the additional geometric elements that are necessary to attain complete systems of hydrodynamic equations in Newton-Cartan and Carroll spacetimes. Our analysis is conducted in general Cartan frames as well as in more explicit coordinates, specifically suited to Galilean or Carrollian use. [ABSTRACT FROM AUTHOR]

To explore the meshing mechanism of classic conic worm drive and provide a theoretical basis for creating other types of worm drive, a complete meshing theory on the proposed worm pair and its meshing performance are investigated. A new mathematical model containing a full set of geometric parameters on this drive is established using differential geometry. The formulae for calculating the reference point coordinates in different coordinate systems are attained, and then the blank size of the conical worm wheel is given. To build tooth surface boundaries, the meshing theory based on reference point is applied, which ensures the location of this point on the tooth surface. By solving multivariate nonlinear equation system in MATLAB environment, these conjugate regions and instantaneous contact lines are plotted within the axial section of the conic worm pair, at the same time, these values of sliding angle and induced principal curvature at meshing points are calculated. Finally, a numerical study is performed and the results verify that the conic worm pair has a relatively high tooth surface utilization ratio and favorable conditions for contact stress and lubrication, which is consistent with the practical application. [ABSTRACT FROM AUTHOR]

The purpose of this work is to present the atomic model proposed by Sommerfeld. We outline the classical calculation of the elliptical orbit and then apply the Wilson–Sommerfeld quantization rules to obtain expressions for the quantization of energy and orbit semi-axes. We then apply the relativistic theory to solve the problem of degenerate orbits. Thus, we see that the Sommerfeld atom is a very rich topic, involving the integration of different subjects ranging from Lagrangian mechanics to relativity, as well as plane geometry and differential equations. Editor's Note: The authors present Sommerfeld s atomic model as a valuable teaching tool, both because it incorporates many important topics in physics instruction and because it helps students appreciate the development of scientific ideas. Readers will appreciate the careful presentation of the physics. [ABSTRACT FROM AUTHOR]

In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ( r + G ′ G )-expansion method and the simple ( G ′ G )-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena. [ABSTRACT FROM AUTHOR]

The geometry processing of a point cloud 2-manifold (or point cloud surface) heavily depends on the discretization of differential geometry properties such as Gaussian curvature, mean curvature, principal curvature, and principal directions. Most of the existing algorithms indirectly compute these differential geometry properties by seeking a local approximation surface or fitting point clouds with certain polynomial functions and then applying the curvature formulas in classical differential geometry. This paper initially proposed a new discretized Laplace-Beltrami operator by applying an inherent distance parameter, which acts as the foundation for precisely estimating the mean curvature. Subsequently, the estimated mean curvature was taken as a strong constraint condition for estimating the Gaussian curvatures, principal curvatures, and principal directions by determining an optimal ellipse. The proposed methods are mainly based on the heat kernel function and do not require local surface reconstruction, thus belonging to truly mesh-free methods. We demonstrated the correctness of the estimated curvatures in both analytic and non-analytic models. Various experiments indicated that the proposed methods have high accuracy. As an exemplary application, we utilized the mean curvature for detecting features of point clouds. [ABSTRACT FROM AUTHOR]

We consider two different SU (2) 2 -invariant cohomogeneity one manifolds, one non-compact M = R 4 × S 3 and one compact M = S 4 × S 3 , and study the existence of coclosed SU (2) 2 -invariant G 2 -structures constructed from half-flat SU (3) -structures. For R 4 × S 3 , we prove the existence of a family of coclosed (but not necessarily torsion-free) G 2 -structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed G 2 -structure constructed from a half-flat SU (3) -structure is in this family. For S 4 × S 3 , we prove that there are no SU (2) 2 -invariant coclosed G 2 -structures constructed from half-flat SU (3) -structures. [ABSTRACT FROM AUTHOR]

Abstract Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton’s Ricci flow—a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of ‘global Ricci network flow’ that can be used to assess a DNN’s ability to disentangle complex data geometries to solve classification problems. By training more than 1500 DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

Differential geometry of a kind of helical hypersurface family that depends on six parameters within the seven‐dimensional Euclidean space is explored. The curvatures of these hypersurfaces are determined, and their minimality is examined, along with illustrative examples being provided. Lastly, the Laplace–Beltrami operator for that kind of hypersurfaces is calculated. [ABSTRACT FROM AUTHOR]

The covariant form of the multivariable diffusion-drift process is described by the covariant Fokker–Planck equation using the standard toolbox of Riemann geometry. The covariant form of the adapted Langevin stochastic differential equation is long sought after in both physics and mathematics. We show that the simplest covariant Stratonovich stochastic differential equation depending on the local orthogonal frame (cf. vielbein) becomes the desired covariant Langevin equation provided we impose an additional covariant constraint: the vectors of the frame must be divergence-free. [ABSTRACT FROM AUTHOR]

In this review work, we outline a conceptual path that, starting from the numerical investigation of the transition between weak chaos and strong chaos in Hamiltonian systems with many degrees of freedom, comes to highlight how, at the basis of equilibrium phase transitions, there must be major changes in the topology of submanifolds of the phase space of Hamiltonian systems that describe systems that exhibit phase transitions. In fact, the numerical investigation of Hamiltonian flows of a large number of degrees of freedom that undergo a thermodynamic phase transition has revealed peculiar dynamical signatures detected through the energy dependence of the largest Lyapunov exponent, that is, of the degree of chaoticity of the dynamics at the phase transition point. The geometrization of Hamiltonian flows in terms of geodesic flows on suitably defined Riemannian manifolds, used to explain the origin of deterministic chaos, combined with the investigation of the dynamical counterpart of phase transitions unveils peculiar geometrical changes of the mechanical manifolds in correspondence to the peculiar dynamical changes at the phase transition point. Then, it turns out that these peculiar geometrical changes are the effect of deeper topological changes of the configuration space hypersurfaces  ∑ v = V N − 1 (v)  as well as of the manifolds  { M v = V N − 1 ((− ∞ , v ]) } v ∈ R  bounded by the ∑v. In other words, denoting by vc the critical value of the average potential energy density at which the phase transition takes place, the members of the family  { ∑ v } v < v c  are not diffeomorphic to those of the family  { ∑ v } v > v c ; additionally, the members of the family  { M v } v > v c  are not diffeomorphic to those of  { M v } v > v c . The topological theory of the deep origin of phase transitions allows a unifying framework to tackle phase transitions that may or may not be due to a symmetry-breaking phenomenon (that is, with or without an order parameter) and to finite/small N systems. [ABSTRACT FROM AUTHOR]

In differential geometry, the concept of golden structure represents a compelling area with wide-ranging applications. The exploration of golden Riemannian manifolds was initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the principles of the golden structure. Subsequently, numerous researchers have contributed significant insights with respect to golden Riemannian manifolds. The purpose of this paper is to provide a comprehensive survey of research on golden Riemannian manifolds conducted over the past decade. [ABSTRACT FROM AUTHOR]

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on S 6 was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on S 6 . We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem. [ABSTRACT FROM AUTHOR]

Lightweight electric vehicles (LEVs) possess great advantages in the viewpoint of fuel consumption, environment protection, and traffic mobility. However, due to the drastic reduction of vehicle weights and body size, the effects of inertial parameter variation in LEV control system become much more pronounced and have to be systematically estimated. This paper presents a dual adaptive unscented Kalman filter (AUKF) where two Kalman filters run in parallel to synchronously estimate vehicle inertial parameters and additional dynamic states such as vehicle mass, vehicle yaw moment of inertia, the distance from front axle to centre of gravity and vehicle sideslip angle. The proposed estimation only integrates and utilizes real-time measurements of in-wheel-motor information and other standard in-vehicle sensors in LEV. The LEV dynamics estimation model including vehicle payload parameter analysis, Pacejka model, wheel and motor dynamics model is developed, the observability of the observer is analysed and derived via Lie derivative and differential geometry theory. To address nonlinearities and undesirable noise oscillation in estimation system, the dual noise adaptive unscented Kalman filter (DNAUKF) and dual unscented Kalman filter (DUKF)are also investigated and compared. Simulation with various manoeuvres are carried out to verify the effectiveness of the proposed method using MATLAB/Simulink-Carsim®. The simulation results show that the proposed DNAUKF method can effectively estimate vehicle inertial parameters and dynamic states despite the existence of payload variations. [ABSTRACT FROM AUTHOR]

Let $f\colon M^{2n}\to \mathbb {R}^{2n+p}$ , $2\leq p\leq n-1$ , be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng (2013, Michigan Mathematical Journal 62, 421–441) conjectured that if the codimension is $p\leq 11$ , then, along any connected component of an open dense subset of $M^{2n}$ , the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces in the kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\mathbb {R}^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension 3 and then by Yan and Zheng for codimension 4. In this paper, we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step for a possible classification of the nonholomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate. [ABSTRACT FROM AUTHOR]

A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Fréchet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags. [ABSTRACT FROM AUTHOR]

A crucial ingredient in the theory of theta liftings of Kudla and Millson is the construction of a $q$ -form $\varphi_{KM}$ on an orthogonal symmetric space, using Howe's differential operators. This form can be seen as a Thom form of a real oriented vector bundle. We show that the Kudla-Millson form can be recovered from a canonical construction of Mathai and Quillen. A similar result was obtaind by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to arbitrary signature. [ABSTRACT FROM AUTHOR]

We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes. [ABSTRACT FROM AUTHOR]

Kang Feng proposed the symplectic geometric algorithms for Hamiltonian systems at Beijing "International Symposium on Differential Geometry and Differential Equations" in August 1984, opening up a new field of research. Over the past 40 years, geometric numerical integration for dynamical systems has been irreplaceably applied in numerous fields, including celestial mechanics, plasma physics, fluid mechanics, biochemistry, meteorological forecasting, seismic exploration, data science and artificial intelligence. This paper mainly summarizes the achievements of geometric numerical integration for dynamical systems in China over the past 40 years. [ABSTRACT FROM AUTHOR]

This paper considers the possibility that abstract thinking and advanced synthesis skills might encourage extraterrestrial civilizations to accept communication with mankind on Earth. For this purpose, a notation not relying upon the use of alphabet and numbers is proposed, in order to denote just some basic geometric structures of current physical theories: vector fields, 1 -form fields, and tensor fields of arbitrary order. An advanced civilization might appreciate the way here proposed to achieve a concise description of electromagnetism and general relativity, and hence it might accept the challenge of responding to our signals. The abstract symbols introduced in this paper to describe the basic structures of physical theories are encoded into black and white bitmap images that can be easily converted into short bit sequences and modulated on a carrier wave for radio transmission. [ABSTRACT FROM AUTHOR]

This paper employs the fractional derivative to investigate the effect of curves in Lorentz–Minkowski space, which is of crucial significance in geometry and physics. In the method of examining this effect, the conformable fractional derivative is chosen because it best fits the algebraic structure of differential geometry. Therefore, with the aid of conformable fractional derivatives, numerous special curves and the Frenet frame that were previously derived using classical derivatives have been reinterpreted in Lorentz–Minkowski three-space. [ABSTRACT FROM AUTHOR]

Highlights In this paper, a universal mechanical model for the torsional behavior of fiber bundle was established, grounded in the principles of spatial geometry and continuum mechanics, incorporating differential geometry formulas. The model was validated through self‐designed experimental equipment and finite element simulations using Ansys Workbench, showing strong agreement between predicted torsional fracture behavior and experimental results. The study further examined the contributions of fiber bending, torsion, and tension to the overall torque, revealing that torsion plays the dominant role. Additionally, the effect of filament count on the torsional performance of fiber bundles was analyzed, demonstrating that while the number of fibers significantly influences stress distribution, its impact on overall torque is minimal. A universal mechanical model for fiber bundle torsional is proposed. High consistency between simulation and experimental results. Examined filament count effect on torsional strength. More filaments reduce stress concentration, improving performance. [ABSTRACT FROM AUTHOR]

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'. [ABSTRACT FROM AUTHOR]

Geometrical methods in quantum information are very promising for both providing technical tools and intuition into difficult control or optimization problems. Moreover, they are of fundamental importance in connecting pure geometrical theories, like GR, to quantum mechanics, like in the AdS/CFT correspondence. In this paper, we first make a survey of the most important settings in which geometrical methods have proven useful to quantum information theory. Then we lay down a geometric theory of entanglement by a principle of action, discussing a simple example with two qubits and consequences for a quantum theory of space-time. [ABSTRACT FROM AUTHOR]

This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it N -bein. Analogous to the vielbein (orthonormal frame) used in the Cartan formalism, the N -bein behaves like a 'square root' of the quantum geometric tensor. Using it, we present a quantum geometric tensor of two states that measures the possibility of moving from one state to another after two consecutive parameter variations. This new tensor determines the commutativity of such variations through its anti-symmetric part. In addition, we define a connection different from the Berry connection, and combining it with the N -bein allows us to introduce a notion of torsion and curvature à la Cartan that satisfies the Bianchi identities. Moreover, the torsion coincides with the anti-symmetric part of the two-state quantum geometric tensor previously mentioned, and thus, it is related to the commutativity of the parameter variations. We also describe our formalism using differential forms and discuss the possible physical interpretations of the new geometrical objects. Furthermore, we define different gauge invariants constructed from the geometrical quantities introduced in this work, resulting in new physical observables. Finally, we present two examples to illustrate these concepts: a harmonic oscillator and a generalized oscillator, both immersed in an electric field. We found that the new tensors quantify correlations between quantum states that were unavailable by other methods. [ABSTRACT FROM AUTHOR]

The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the Minkowski problem; however, the numerical solution of this problem has been largely left behind, with only a few methods available to achieve that goal. In this article, focusing on the two-dimensional Minkowski problem with Dirichlet boundary conditions, we introduce two solution methods, both based on operator-splitting. One of these two methods deals directly with the Dirichlet condition, while the other one uses an approximation à la Robin of this Dirichlet condition. The relaxation of the Dirichlet condition makes the second method better suited than the first one to treat those situations where the Minkowski equation (of Monge-Ampère type) and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge--Ampère equation discussed by Glowinski et al. [J. Sci. Comput., 81 (2019), pp. 2271-2302]; as such they take advantage of a divergence formulation of the Minkowski problem, which makes it well suited to both a mixed finite-element approximation and the time-discretization via an operator-splitting scheme of an associated initial value problem. Our methodology can be easily implemented on convex domains of rather general shape (with curved boundaries, possibly). The numerical experiments validate both methods, showing that if one uses continuous piecewise affine finite-element approximations of the solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second-order accuracy for the L² and L∞ norms of the approximation error, where the Minkowski--Dirichlet problem is assumed to have a smooth solution. One can easily extend the methods discussed in this article to address the solution of three-dimensional Minkowski problems. [ABSTRACT FROM AUTHOR]