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A method for characterizing the topological fluctuations in liquids is proposed. This approach exploits the concept of the weighted gyration tensor of a collection of particles and permits the definition of a local configurational unit (LCU). The first principal axis of the gyration tensor serves as the director of the LCU, which can be tracked and analyzed by molecular dynamics simulations. Analysis of moderately supercooled Kob–Andersen mixtures suggests that orientational relaxation of the LCU closely follows viscoelastic relaxation and exhibits a two-stage behavior. The slow relaxing component of the LCU corresponds to the structural, Maxwellian mechanical relaxation. Additionally, it is found that the mean curvature of the LCUs is approximately zero at the Maxwell relaxation time with the Gaussian curvature being negative. This observation implies that structural relaxation occurs when the configurationally stable and destabilized regions interpenetrate each other in a bicontinuous manner. Finally, the mean and Gaussian curvatures of the LCUs can serve as reduced variables for the shear stress correlation, providing a compelling proof of the close connection between viscoelastic relaxation and topological fluctuations in glass-forming liquids. [ABSTRACT FROM AUTHOR]

[Display omitted] 3D reconstruction is an important part of digital city, high-accuracy 3D modeling method has been widely studied as an important pathway to visualizing 3D city scenes. However, the problems of image resolution, noise, and occlusion result in low quality and smooth features in the mesh model. Therefore, the model needs to be refined to improve the mesh quality and enhance the visual effect. This paper proposes a mesh refinement algorithm to fine-tune the vertices of the mesh and constrain their evolution direction on the normal vector, reducing their freedom degrees to one. The evolution of vertices only involves one motion distance parameter on the normal vector, simplifying the complexity of the energy function derivation. Meanwhile, Gaussian curvature is used as a regularization term, which is anisotropic and preserves the edge features during the reconstruction process. The mesh refinement algorithm with unary operations fully utilizes the original image information and effectively enriches the local detail features of the mesh model. This paper utilizes five public datasets to conduct comparative experiments, and the experimental results show that the proposed algorithm can better restore the detailed features of the model and has a better refinement effect in the same number of iterations compared with OpenMVS library refinement algorithm. At the same time, in the comparison of refinement results with fewer iterations, the algorithm in this paper can achieve more desirable results. [ABSTRACT FROM AUTHOR]

A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes between Gaussian curvature and both circumference and area deficits. [ABSTRACT FROM AUTHOR]

In this study, we explore the sweeping surfaces in Euclidean 3-space, utilizing the modified orthogonal frames with non-zero curvature and torsion, which allows us to consider the spine curves even if their second differentiations vanish. If the curvature of the spine curve of a sweeping surface has discrete zero points, the Frenet frame might undergo a discontinuous change in orientation. Therefore, the conventional parametrization with the Frenet frame of such a surface cannot be given. Thus, we introduce two types of modified sweeping surfaces by considering two types of spine curves; the first one's curvature is not identically zero and the second one's torsion is not identically zero. Then, we determine the criteria for classifying the coordinate curves of these two types of modified sweeping surfaces as geodesic, asymptotic, or curvature lines. Additionally, we delve into determining criteria for the modified sweeping surfaces to be minimal, developable, or Weingarten. Through our analysis, we aim to clarify the characteristics defining these surfaces. We present graphical representations of sample modified sweeping surfaces to enhance understanding and provide concrete examples that showcase their properties. [ABSTRACT FROM AUTHOR]

This paper investigates the Cauchy problems for a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system via Riemannian invariants. Based on the first a priori estimate of the solutions, the second a priori estimate of the derivation of the solutions, and the third a priori estimate of the continuous mould of the first‐order partial derivatives of the solutions to quasilinear hyperbolic system, a lower bound estimate of classical solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation is derived. [ABSTRACT FROM AUTHOR]

We develop a general approach to study three-dimensional Schrödinger operators with confining potentials depending on the distance to a surface. The main idea is to apply parallel coordinates based on the surface but outside its cut-locus in the Euclidean space. If the surface is asymptotically planar in a suitable sense, we give an estimate on the location of the essential spectrum of the Schrödinger operator. Moreover, if the surface coincides up to a compact subset with a surface of revolution with strictly positive total Gauss curvature, it is shown that the Schrödinger operator possesses an infinite number of discrete eigenvalues. [ABSTRACT FROM AUTHOR]

The geodesic motion in a Lorentzian spacetime can be described by trajectories in a 3-dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions of its Killing vectors. The resulting Riemannian metric inherits the geodesics for asymptotically flat spacetimes including the stationary and axisymmetric ones. The method allows us to find Riemannian metrics in three and two dimensions plus the radial geodesic equation when we project over three different directions. The 3-dimensional Riemannian metric reduces to the Jacobi metric when static, spherically symmetric and asymptotically flat spacetimes are considered. However, it can be calculated for a larger variety of metrics in any number of dimensions. We show that the geodesics of the original spacetime metrics are inherited by the projected Riemannian metric. We calculate the Gaussian and geodesic curvatures of the resulting 2-dimensional metric, we study its near horizon and asymptotic limit. We also show that this technique can be used for studying the violation of the strong cosmic censorship conjecture in the context of general relativity. [ABSTRACT FROM AUTHOR]

In this paper, the authors generalize the fractional curvature of plane curves introduced by Rubio et al. in 2023, to regular curves in the Euclidean space R 3 , and study the geometric properties of the curve using Caputo's fractional derivative. Furthermore, we introduce a new definition of fractional curvature and fractional mean curvature of a regular surface, using fractional principal curvatures; and prove that such concepts are invariant under isometries; i.e., they belong to the intrinsic geometry of the regular surface. Also, a geometric interpretation is given to Caputo's fractional derivative of algebraic polynomials. [ABSTRACT FROM AUTHOR]

Gaussian dual curvature measure is introduced and Gaussian dual Minkowski problem is studied. This problem amounts to solving a class of Monge-Ampère type equations on the unit sphere. Existence and uniqueness of solutions to the relevant Monge-Ampère type equations are obtained in the smooth category when q ≤ 0 , respectively. For q < 0 , a complete solution to existence part of the Gaussian dual Minkowski problem is presented. For the case of q = 0 , a weak solution to the Monge-Ampère type equation related to this problem is provided when given measure has the density f which is sandwiched between two positive constants belonging to the interval 0 to 1. [ABSTRACT FROM AUTHOR]

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman's result [12] of the Riemannian case cannot be extended to the Finsler geometry as such. [ABSTRACT FROM AUTHOR]

In this study, we investigate the deflection angle of a torus-like regular charged black hole in the limit approximation of a weak field to check the effects of non-magnetic plasma and non-plasma medium. Using spacetime optical geometry, we first compute the Gaussian optical curvature. We study the light deflection angle from a charged torus-like black hole using the Gibbons and Werner approach. By taking into account the well-known Keeton–Petters approach, we validate our results. Moreover, we examine the shadow of a charged torus-like black hole by using the ray-tracing algorithm under the influence of pressureless and non-magnetized plasma and study the graphical effects of cosmological constant and charge for both anti-de Sitter and de Sitter cases on the shadow of a black hole. [ABSTRACT FROM AUTHOR]

Multi-focus image fusion has wide applications in the field of computer vision. This research work presents a two-fold multi-focus image fusion framework in spatial domain. Firstly, the salient features of the focused regions in multi-focus images are detected using Gaussian curvature filter (GCF) and range filtering. The initial focus detection map is then refined using morphological filters. Experiments have shown that the proposed method is fast and robust when compared with latest multi-focus fusion schemes. [ABSTRACT FROM AUTHOR]

This paper is motivated to study the weak gravitational lensing of the black hole in the context of Starobinsky–Bel–Robinsion gravity. We deduce the deflection angle of light in the weak field limits. For this purpose, we find the Gaussian curvature and apply the Gauss–Bonnet theorem. We also investigate the deflection angle at which light is deflected by a plasma medium. Furthermore, we analyze the graphical behavior of the deflection angle in the framework of SBR gravity. We also calculate the energy extracted from the black hole in SBR gravity, Einstein ring, and lens equation in the non-plasma and plasma framework. We conclude this study with a discussion of how light beams behave when they approach such a four-dimensional black hole by calculating the deflection angle. [ABSTRACT FROM AUTHOR]

We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss–Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our approach allows us to solve problems that are impossible to solve in the pointwise conformal case. Moreover, we obtain a deep and more delicate information on pointwise conformal deformations. We prove new existence and nonexistence results for metrics with prescribed curvature in the conformal setting, which depend on the Euler characteristic. [ABSTRACT FROM AUTHOR]

The current work deals with the mechanical buckling behavior of saturated porous toroidal shell segments sandwiched by piezoelectric actuator layers. Material properties of the porous shells with nonuniform distributed porosities are assumed to vary as a specific function of the thickness and porosity parameters. The nonlinear equations of the sandwich toroidal shell having either positive or negative Gaussian curvature are obtained on the basis of Biot's poroelasticity theory. The energy method as the minimum potential energy principle is applied to derive the governing equations of the sandwich shells. The buckling equations of the piezo-porous sandwich shells under various types of mechanical loading are established by employing the adjacent equilibrium criterion. The governing equations as a set of the coupled partial differential equations are solved using an analytical method including the Airy stress function. Closed-form solutions are derived for shallow segments of the sandwich toroidal shell with positive/negative Gaussian curvature under lateral pressure, axial compression, and hydrostatic pressure loadings. Novel numerical results are presented in this work to show the effects of important factors such as the piezoelectric layer thickness, applied voltage, shell geometrical ratio, and variation of the porosity coefficient on the buckling behavior of these structures. [ABSTRACT FROM AUTHOR]

Spherical product surfaces are obtained with the help of a special product by considering two curves in n−dimensional space. One of their special cases is rotational surface. The reason why the present study is significant that the spherical product is used to construct hypersurfaces. (n−1)−curves are needed during this construction. Firstly, the spherical product hypersurfaces are defined in E4, Gaussian and mean curvature are yielded and then conditions being flat or minimal are examined. Moreover, superquadrics, which are associated with spherical product, are handled for the first time in hypersurface form and give some examples. Finally, spherical product hypersurfaces are generalized to n−dimensional Euclidean space and contribute to literature. [ABSTRACT FROM AUTHOR]

Surfaces with concentric K-contours and parallel K-contours in Euclidean 3-space are defined. Crucial examples are presented and characterization of them are given. [ABSTRACT FROM AUTHOR]

Understanding bone surface curvatures is crucial for the advancement of bone material design, as these curvatures play a significant role in the mechanical behavior and functionality of bone structures. Previous studies have demonstrated that bone surface curvature distributions could be used to characterize bone geometry and have been proposed as key parameters for biomimetic microstructure design and optimization. However, understanding of how bone surface curvature distributions correlate with bone microstructure and mechanical properties remains limited. This study hypothesized that bone surface curvature distributions could be used to predict the microstructure as well as mechanical properties of trabecular bone. To test the hypothesis, a convolutional neural network (CNN) model was trained and validated to predict the histomorphometric parameters (e.g., BV/TV, BS, Tb.Th, DA, Conn.D, and SMI), geometric parameters (e.g., plate area PA, plate thickness PT, rod length RL, rod diameter RD, plate-to-plate nearest neighbor distance NNDPP, rod-to-rod nearest neighbor distance NNDRR, plate number PN, and rod number RN), as well as the apparent stiffness tensor of trabecular bone using various bone surface curvature distributions, including maximum principal curvature distribution, minimum principal curvature distribution, Gaussian curvature distribution, and mean curvature distribution. The results showed that the surface curvature distribution-based deep learning model achieved high fidelity in predicting the major histomorphometric parameters and geometric parameters as well as the stiffness tenor of trabecular bone, thus supporting the hypothesis of this study. The findings of this study underscore the importance of incorporating bone surface curvature analysis in the design of synthetic bone materials and implants. [ABSTRACT FROM AUTHOR]

Soft and biological matter come in a variety of shapes and geometries. When soft surfaces that do not fit into each other due to a mismatch in Gaussian curvatures form an interface, beautiful geometry-induced patterns are known to emerge. In this paper, we study the effect of geometry on the dynamical response of soft surfaces moving relative to each other. Using a simple experimental scheme, we measure friction between a highly bendable thin polymer sheet and a hydrogel substrate. At this soft and low-friction interface, we find a strong dependence of friction on the relative geometry of the two surfaces--a flat sheet experiences significantly larger friction on a spherical substrate than on flat or cylindrical substrate. We show that the stress developed in the sheet due to its geometrically incompatible confinement is responsible for the enhanced friction. This mechanism also leads to a transition in the nature of friction as the sheet radius is increased beyond a critical value. Our finding reveals a hitherto unnoticed mechanism based on an interplay between geometry and elasticity that may influence friction significantly in soft, biological, and nanoscale systems. In particular, it provokes us to reexamine our understanding of phenomena such as the curvature dependence of biological cell mobility. [ABSTRACT FROM AUTHOR]

To investigate the suspension and diffusion mechanisms of sediment particles under the action of fluids from both particle and flow field scales, a Gaussian curvature model is introduced in the discrete element method to characterize the nonlinear contact properties of non-spherical particles. Additionally, improvements are made to the drag force model and porosity models for non- spherical particles. By doing so, the polydispersed non-spherical particle CFD-DEM coupling numerical method is established. The accuracy and superiority of this model are validated through comparisons with physical experimental results. Subsequently, the suspension and diffusion processes of sediment under fluids are simulated. Results show that the sediment settling rate is positively correlated with the initial discharge quantity of particles and negatively correlated with flow velocity, while the escape rate shows an opposite trend. At high initial discharge, particle clustering effects promote fluid kinetic energy dissipation, causing gravity to surpass inertial forces and leading to rapid particle settling. At low initial discharge, higher fluid kinetic energy enhances horizontal particle transport. At lower flow velocities, particle shape significantly affects the spatial distribution of sediment particles. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

We consider solutions of some two-dimensional fourth-order equation with a biharmonic operator and exponential nonlinearity of a counterpart of the classical Gauss–Bieberbach–Rademacher second-order equation, which was previously inspected by many authors in connection with problems of the geometry of surfaces with negative Gaussian curvature, rarefied gas dynamics, and the theory of automorphic functions. We obtain some conditions for the absence of a solution in a disk of sufficiently large radius and show that global solutions on the plane can exist only if the coefficient of nonlinearity decays at infinity at the rate at least . Otherwise the mean value of the solution on a circle of radius would tend to with exponential rate as . The Pokhozhaev–Mitidieri nonlinear capacity method, based on the choice of appropriate cutoff test functions, proves the impossibility of the existence of such global solution. Also, for the solutions in , periodic in all but variables, the absence of global solutions is obtained by similar methods when the nonlinearity coefficient decays at rate slower than . [ABSTRACT FROM AUTHOR]

Research suggests structural and connectivity abnormalities in patients with treatment-resistant schizophrenia (TRS) compared to first-line responders and healthy-controls. However, measures of these abnormalities are often influenced by external factors like nicotine and antipsychotics, limiting their clinical utility. Intrinsic-cortical-curvature (ICC) presents a millimetre-scale measure of brain gyrification, highly sensitive to schizophrenia differences, and associated with TRS-like traits in early stages of the disorder. Despite this evidence, ICC in TRS remains unexplored. This study investigates ICC as a marker for treatment resistance in TRS, alongside structural indices for comparison. We assessed ICC in anterior cingulate, dorsolateral prefrontal, temporal, and parietal cortices of 38 first-line responders, 30 clozapine-resistant TRS, 37 clozapine-responsive TRS, and 52 healthy-controls. For comparative purposes, Fold and Curvature indices were also analyzed. Adjusting for age, sex, nicotine-use, and chlorpromazine equivalence, principal findings indicate ICC elevations in the left hemisphere dorsolateral prefrontal (p < 0.001, η2partial = 0.142) and temporal cortices (LH p = 0.007, η2partial = 0.060; RH p = 0.011, η2partial = 0.076) of both TRS groups, and left anterior cingulate cortex of clozapine-resistant TRS (p = 0.026, η2partial = 0.065), compared to healthy-controls. Elevations that correlated with reduced cognition (p = 0.001) and negative symptomology (p < 0.034) in clozapine-resistant TRS. Fold and Curvature indices only detected group differences in the right parietal cortex, showing interactions with age, sex, and nicotine use. ICC showed interactions with age. ICC elevations were found among patients with TRS, and correlated with symptom severity. ICCs relative independence from sex, nicotine-use, and antipsychotics, may support ICC's potential as a viable marker for TRS, though age interactions should be considered. [ABSTRACT FROM AUTHOR]

We consider the problem of producing a ruled Möbius band by subjecting an unstretchable, homogeneous, isotropic, elastic material surface material surface in a circular helicoidal reference configuration to a deformation that is isometric and chirality preserving. We find that such a Möbius band is completely determined by the unit binormal of the Frenet frame of its midline, which must be a geodesic and must have uniform torsion inversely proportional to the pitch of the helicoidal reference configuration. Granted that the energy density of the material surface depends quadratically on the mean curvature of its deformed configuration, we show that the total energy stored in producing a ruled Möbius band as described reduces, in closed form and without approximation, to an integral over the midline of the Möbius band. We formulate and numerically solve a constrained variational problem for finding relative minima of the dimensionally reduced bending energy and construct corresponding stable Möbius bands. The only input parameter entering our variational problem is the number ν of turns in a helicoidal reference configuration. We only find solutions if ν exceeds a certain threshold, which we compute to machine precision. Above that threshold, an interplay between the operative constraints leads to a multiplicity of coexisting stable solutions with n ≥ 3 half twists. For each n ≥ 3 , we construct an energetically optimal Möbius band which exhibits n -fold rotational symmetry. All other energy minima yield Möbius bands which lack symmetry. To our knowledge, this study contains the first examples of stable Möbius bands produced by isometrically deforming reference configurations that are not flat. [ABSTRACT FROM AUTHOR]

The present paper aims to study a quarter symmetric metric connection in the tangent bundle and investigate an induced metric and connection on a submanifold of co-dimension 2 and hypersurface by means of mathematical operators concerning the QSM connection in the tangent bundle TM. Totally geodesic (TG) and totally umbilical (TU) concerning the QSM connection on the submanifold of co-dimension 2 and hypersurface in TM are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

We establish that Grunsky norm of any normalized univalent function on a quasidisk is completely determined by the squares of holomorphic abelian differentials (in contrast to Teichmüller norm, which relates to all integrable holomorphic quadratic differentials). The result has important interesting applications in several fields of geometric function theory: it provides an explicit representation of Fredholm eigevalues of all quasiconformal curves, sheds a new light on the distortion theory for univalent functions with quasiconformal extension, etc. [ABSTRACT FROM AUTHOR]

Ruffled surfaces that appear in biological forms, such as coral and lettuce, are a great source of inspiration for architectural and furniture design. This paper proposes a mechanism based on a bending-active scissors grid that effectively reproduces the process of differential growth. The structure can deploy from a linearly folded state with rotational symmetry to a complex ruffled surface without rotational symmetry. The deployed shape further exhibits a wave-like motion similar to the swimming gait of flatworms or cuttlefish. First, we propose a design method for the mechanism computed from the surface of constant negative Gaussian curvature. We then numerically analyze the symmetry-breaking process of deployment and zero-stiffness wave-like motion after deployment. We built two demonstrators to verify the deployment and the transformation. The first demonstrator with 1.5m diameter was fabricated to verify the symmetry-breaking deployment motion. The second demonstrator with 0.9m diameter was fabricated to demonstrate the wave-like motion by controlled pulling of the group of threads. [ABSTRACT FROM AUTHOR]

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away from characteristic points in the Heisenberg group with the semi-symmetric metric connection. Finally, we derive the expressions of those curvatures and prove the Gauss–Bonnet theorem related to the semi-symmetric metric connection in the Heisenberg group. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the sub-Riemannian properties of the roto-translation group R T . At the same time, we compute the sub-Riemannian limits of Gaussian curvature associated with two kinds of canonical connections for a C 2 -smooth surface in the roto-translation group away from characteristic points and signed geodesic curvature associated with two kinds of canonical connections for C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss-Bonnet theorem in the R T . [ABSTRACT FROM AUTHOR]

The Van Lake region is situated within the boundary of the collision zone between the Arabian and Eurasian plates, which has resulted in the formation of a variety of geological structures, including faults, volcanoes, and uplifted mountain ranges. A comprehensive understanding of these structures can yield significant information pertaining to the region's tectonic history, as well as potential tectonic hazards. In the present study, several structural mapping techniques, including lineament detection techniques [Horizontal Gradient Magnitude (HGM), Analytic Signal (AS), and Tilt Angle] and curvature analysis attributes (Differential, Gaussian, Mean, and Shape Index), were analyzed and compared the results obtained using both Earth Gravity Model 2008 (EGM2008) Bouguer gravity and seismological data, with the goal of determining the most effective (suitable) edge detection method in order to gain a better understanding of high-stress regions and areas of potential seismic hazard in the vicinity of Van Lake. It is of great importance to investigate the behavior of faults and the curvatures of structural features within the crust, as they play a significant role in determining the distribution of stress, and subsequently, its impact on surface structures. The results show that HGM and AS are effective in defining fault strikes and trends, while Gaussian curvature attributes can detect lineaments and structural edges. Differential curvature attributes were found to be best suited for complex tectonic regions. The study also reveals a coherent relationship between fold orientation and fault strikes with the regional principal stress direction. The current study has focused on a region characterized by a significant negative anomaly, which appears to be a valley-like fold structure with a highly inclined interlimb angle. This area, referred to as the "silent" zone, is seismically quiet with regard to earthquakes and faults. However, focal mechanism results support the orientation of the interlimb angle fold in the vicinity of this region. The existence of a strong negative anomaly in the eastern part of the study area highlights the requirement for additional geological field observations and seismic surveys to evaluate the potential earthquake hazards. In the course of comparing various structural mapping techniques, it was concluded that the application of second-order vertical derivative of gravity potential is a suggested approach for identifying the edges of causative structures. [ABSTRACT FROM AUTHOR]

We have proposed the "Standard 4‐edge Curved Model" for the first time, which contains a standard geometric model based on superelliptic curves and a mapping relationship from a plane to a three‐dimensional surface. In this model, we can quantitatively compare the wrinkling risks of almost all known 4‐edge curved displays and provide the optimizations for the geometric parameters. [ABSTRACT FROM AUTHOR]

In this paper, we examine q-Bernstein-Bézier surfaces in Minkowski space- R13 with q as the shape parameter. These surfaces, a generalization of Bézier surfaces, have applications in mathematics, computer-aided geometric design, and computer graphics for the surface formation and modeling. We analyze the timelike and spacelike cases of q-Bernstein-Bézier surfaces using known boundary control points. The mean curvature and Gaussian curvature of these q-Bernstein-Bézier surfaces are computed by finding the respective fundamental coefficients. We also investigate the shape operator dependency for timelike and spacelike q-Bernstein-Bézier surfaces in Minkowski space- R13 , and provide biquadratic and bicubic q-Bernstein-Bézier surfaces as illustrative examples for different values of the shape controlling parameter q. [ABSTRACT FROM AUTHOR]

Cardiac magnetic resonance imaging (MRI) usually requires a long acquisition time. The movement of the patients during MRI acquisition will produce image artifacts. Previous studies have shown that clear MR image texture edges are of great significance for pathological diagnosis. In this paper, a motion artifact reduction method for cardiac MRI based on edge enhancement network is proposed. Firstly, the four-plane normal vector adaptive fractional differential mask is applied to extract the edge features of blurred images. The four-plane normal vector method can reduce the noise information in the edge feature maps. The adaptive fractional order is selected according to the normal mean gradient and the local Gaussian curvature entropy of the images. Secondly, the extracted edge feature maps and blurred images are input into the de-artifact network. In this network, the edge fusion feature extraction network and the edge fusion transformer network are specially designed. The former combines the edge feature maps with the fuzzy feature maps to extract the edge feature information. The latter combines the edge attention network and the fuzzy attention network, which can focus on the blurred image edges. Finally, extensive experiments show that the proposed method can obtain higher peak signal-to-noise ratio and structural similarity index measure compared to state-of-art methods. The de-artifact images have clear texture edges. [ABSTRACT FROM AUTHOR]

TC11 impeller blades are typical complex curved surface components. Due to the complex machining load variations, these types of components have always been difficult to effectively solve issues such as significant machining deformation and large machining errors. This article characterizes complex surface features based on Gaussian curvature and mean curvature. Different machining methods are employed for different regions of the TC11 material after partitioning, and a comparative experiment is conducted between traditional methods and surface slicing methods for machining. Different machining methods are applied to process the different regions of TC11 material after surface partitioning, and a comparative experiment is conducted between traditional methods and surface slicing methods for machining. [ABSTRACT FROM AUTHOR]

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E 3 . We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis. [ABSTRACT FROM AUTHOR]

In this paper we study minimal Legendrian surfaces \Sigma immersed in tangent sphere bundle T_1{\mathbb {S}}^3. We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to one and complete minimal Legendrian surfaces with non-negative Gauss curvature. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate osculating type ruled surfaces with a type 2-Bishop frame in E 3 . We accomplish this by employing the symmetry of osculating curves. We examine osculating type ruled surfaces by taking into account the curvatures of the base curve. We investigate the geometric properties of these surfaces, focusing on their cylindrical and developable characteristics. Moreover, we calculate the Gaussian and mean curvatures and provide the requirements for the surface to be flat and minimal. We determine the requirements for the curves lying on this surface to be geodesic, asymptotic curves, or lines of curvature. Furthermore, relations between osculating type ruled surfaces with central tangent and central normal vectors are given. Finally, some examples of these surfaces are presented. [ABSTRACT FROM AUTHOR]

We prove that if the geodesic flow on a surface has an integral which is fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for the existence of fractional-linear integrals: such an integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves. [ABSTRACT FROM AUTHOR]

The star sensor is the most accurate measurement instrument in the spacecraft attitude measurement system, and the accurate centroid of the star point is the basis for ensuring the performance of the star sensor. Currently, the centroid of the gray method is the most widely used centroid extraction method in practice. Systematic errors caused by the centroid of the gray method and random noise in the detector imaging process are the main factors contributing to the deviation of the star centroiding coordinates. Considering the relationship between the point spread function and the pixel gray value, this paper proposes a centroiding method to reduce the star point centroiding error by fusing a priori information and energy distribution. The star charts are first preprocessed using a curvature filter and Gaussian blur to reduce the random noise. Then the complexity of the point spread function is considered, and the pixel gray values are corrected based on a priori information and gray value fuzzy processing. Finally, the symmetry of the one-dimensional energy distribution is used to quickly determine the sub-pixel deviation to get the star centroid coordinates. Through simulation and physical simulation experiments, the method was verified to be effective, and the extraction accuracy met the requirements of high-precision star sensors. The night sky observation test results demonstrate that the method in this paper can improve the measurement accuracy of the star sensors. [ABSTRACT FROM AUTHOR]

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space \mathbb {H}^{n+1} (n\geq 2) with the speed given by arbitrary positive power \alpha of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time t\in [0,\infty). Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of the Bianchi transform for surfac­es of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Mining top, the Minding coil, the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuens surface and the Dinis surface. The study of surfaces of constant negative Gaussian curvature (pseudospheri­cal surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudo­spherical surfaces with the theory of networks, with the theory of solitons, with non-linear differential equations and sin-Gordon equations is estab­lished. The sin-Gordon equation plays an important role in modern phys­ics. Bianchi transformations make it possible to obtain new pseudospheri­cal surfaces from a given pseudospherical surface. The Bianchi transform for the Minding coil is constructed. Using a mathematical package, the Minding coil and its Bianchi transform are constructed.

This paper presents an algorithmic approach for the conceptual design of architectural surfaces represented by triangulated meshes. Specifically, we propose a method to optimise a surface according to user-specified geometric properties including the distribution of the Gaussian curvature and preferable boundary location. Designing a surface manually with specific Gaussian curvatures can be a time-consuming task, and the proposed method automates this task. Also, in the proposed approach, the resulting mesh could be encouraged to form a regular tessellation or kept close to those of the initial one. Our method relies on the idea in computational conformal geometry called circle packing and the discrete Ricch energy, which have been used for surface modelling. We develop a least-squares-based optimisation scheme by introducing a variant of the Ricci energy to accommodate flexibility in specifying design constraints such as boundary locations and convexity of the spanned surface, which are essential to architectural applications. We provide an open-source implementation of our method in Python.