Geometric Basis Research Articles

Particle and light motion in Lyra–Schwarzschild spacetime

Abstract In this paper we recall the static and spherically symmetric solution derived within LyST gravity. LyST stands for “Lyra Scalar-Tensor” and is a scalar-tensor proposal for the gravitational interaction modifying general relativity by adopting Lyra manifold as the spacetime substrate. Lyra manifold is characterized both by the metric tensor gµν(x) and the scale φ(x); the latter being an integral part of the definition of reference frame. In a previous work, we have launched the formal geometrical basis of LyST, we have proposed an associated action integral for the gravitational interaction, we have derived the field equations generalizing Einstein field equation, and we have solved these equation to obtain the generalized version of Schwarzschild line element in Lyra manifold. This lead us to Lyra-Schwarzschild spacetime, which is further explored in the present paper. Herein, the trajectories of test particles is thoroughly studied. This is done for both massive particles and massless particles. The geodesic equations are built and solved. The effective potential associated to the motion of these particles around the source is determined and characterized. The possible trajectories include gravitational capture, scattering, bounded orbits, stable and unstable circular orbits, near-horizon motion. The innermost circular orbit allowed for massive particles is determined and compared to the one predicted by GR. The last photon orbit is also calculated for Lyra-Schwarzschild metric. The equation for the periastron shift in the context of Lyra-Schwarzschild solution is constructed; its application to observational data could be useful to constrain the parameters typical of LyST, telling it apart from GR.

Structure constants in equivariant oriented cohomology of flag varieties

We introduce generalized Demazure operators for the equivariant oriented cohomology of the flag variety, which have specializations to various Demazure operators and Demazure–Lusztig operators in both equivariant cohomology and equivariant K-theory. In the context of the geometric basis of the equivariant oriented cohomology given by certain Bott–Samelson classes, we use these operators to obtain formulas for the structure constants arising in different bases. Specializing to divided difference operators and Demazure operators in singular cohomology and K-theory, we recover the formulas for structure constants of Schubert classes obtained in Goldin and Knutson (Pure Appl Math Q 17(4):1345–1385, 2021). Two specific specializations result in formulas for the the structure constants for cohomological and K-theoretic stable bases as well; as a corollary we reproduce a formula for the structure constants of the Segre–Schwartz–MacPherson basis previously obtained by Su (Math Zeitschrift 298:193–213, 2021). Our methods involve the study of the formal affine Demazure algebra, providing a purely algebraic proof of these results.

Real-Time Cone-Growth Model for Determination of Pharmaceutical Powder Flow Properties.

The flow properties of pellets or granules are crucial for further processing drug dosage forms. Optimal compression or filling of multiparticulate dosage forms into capsules is influenced by forces between discrete particles, which could be partially characterized by flow properties. Several techniques have been developed to examine flowability, including static and dynamic methods applying empirical studies and up-to-date chaos theory; however, the newest methods seem only to be powerful with the supplementation of empirical principles. Our experiments try to refine both the technique of analysis and the methods, by finding new, alternative ways. Our approach to the flowability measurements was to set up a dynamic time-dependent model that combined empirical observations and chaos theory on a geometrical basis, thus finding new characteristics regarding the flow properties of pellets and granules that could be relevant for drug developers. Our findings indicate that sphericity and particle size are the most significant factors influencing the flowability of pharmaceutical multiparticular preparations. Furthermore, this study confirms that integrating chaos theory and empirical observations in a time-dependent dynamic model provides a comprehensive understanding of particle flow behavior, pivotal for optimizing manufacturing processes.

Development of an individual 3D particle reconstruction method for discrete mechanical modeling: Interpolation by Fourier composition

Reconstruction methods for granular materials are an essential step in achieving an accurate geometrical basis in the use of particle methods such as the well-known discrete element method. In this study, a novel continuous analytical method will be proposed to achieve individual 3D reconstructions of granular materials. For this purpose, the 2D Fourier Descriptor theory is used and a generalization of it to the 3D case is obtained, which is quite different from the spherical harmonics theory. Some limitations existing in the original model will be addressed by developing and calibrating a generalized approach that results in a total of four main models. Particles from a study sample are reconstructed, and the models are evaluated based on four generalized error metrics, as well as the presence of some specific limitations detected in the models. Finally, the models are compared using a decision-making method in different decisional scenarios.

Carbon nanotubes/aluminum interface structure and its effects on the strength and electrical conductivity of aluminum

The contradiction between strength and electrical conductivity of pure aluminum (Al) hinders its wide applications in electrical engineering. In this study, an attempt was made to overcome this limitation by adding carbon nanotubes (CNTs) into Al matrix using friction stir processing (FSP) method. The results showed that CNTs were singly distributed in Al matrix and strong bonded CNTs/Al interfaces without chemical reaction as well as impurity contamination were formed. Interestingly, it was found that the axis of CNTs tended to be parallel to Al (111) planes, which provided a geometric basis for the formation of semi-coherent CNTs/Al interfaces. These semi-coherent CNTs/Al interfaces possessed an extremely low electrical resistivity, which weakened the total electron scattering generated by all CNT/Al interfaces. Meanwhile, strong bonded CNTs/Al interfaces effectively hindered dislocations movement, inducing an improvement of 84 % and 50 % in yield strength and tensile strength of Al respectively. As a result, the strength of Al was significantly improved without losing its electrical conductivity. This study provides a new strategy for breaking through the contradiction between strength and electrical conductivity of Al and then producing CNTs/Al nanocomposites with high strength and high electrical conductivity.

Geometric Anatomy Basis for Safe and Effective Focal Ablation of Prostate Cancer by Irreversible Electroporation (IRE)

Irreversible electroporation (IRE) is a recent and minimally invasive method of partial prostate ablation. However, knowledge of the essential landmarks of prostate anatomy is crucial to achieving safe and effective partial ablation by IRE. High-quality imaging of the prostate is essential before the procedure. The individual morphological pattern of the prostate must be taken into account and detailed mapping with measurement of the lesion is necessary to determine optimal needle placement. The entire tumour volume must be covered while ensuring the safety of critical anatomical structures such as the rectum, urethra, nerve bundles and sphincter muscle.

ANALYSIS OF POSSIBLE GEODETIC APPROACHES FOR 3D MODEL CREATION IN BIM

Abstract. The topic of this paper is the description and analysis of technologies for generating a geometric basis for BIM. The main element is an analysis of the capabilities of mobile laser scanning and the outputs of this method compared to other methods. The selected interior space was measured by different technologies, laser scanning, classical geodetic using total station, and manual measurement. The main device tested was the ZEB-REVO. It is a handheld mobile laser scanner, carried by an operator. It has a range of 40 m, with the accuracy in measured lengths is 1–3cm per 10 m depending on the type of surface. The scanning speed is 43,000 points per second. The device is equipped with an inertial measurement unit (IMU) and uses SLAM technology for trajectory definition. The device does not have a camera in the simplest configuration and the operator cannot monitor the scanning status. Nevertheless, measurement with this scanner is very simple. The instrument is mainly used for surveying construction objects, but it can be advantageously used for documentation of underground spaces or in e.g., forestry for defining DBH or volume of stockpiles. This device is not equipped with a GNSS unit, and the typical scanning time is 20–30 min without model deformation. Accuracy and output analysis was performed for all technologies used. The results are presented in tables and show the suitability of PLS (personal laser scanner) in general.

Steady heat transfer analysis for anisotropic structures using the coupled IGA-EFG method

A numerical model of steady heat transfer analysis for anisotropic structures using coupled isogeometric analysis and element-free Galerkin (IGA-EFG) method was developed. The model uses reproduction conditions to realize the equivalence between moving least squares (MLS) shape function and equal geometric basis function, and realizes adaptive refinement based on temperature gradient. Its advantages include accurate geometric representation, convenient load application and flexible adaptive refinement. The effectiveness of the present model is verified by comparison with those of IGA and EFG models using numerical examples including anisotropic fan blades and airplane wings. The effects of knot vectors of non-uniform rational B-spline (NURBS), orders of NURBS and refinement number of mesh on the computational accuracy of the present model are investigated. The results indicate that the accuracy of coupled IGA-EFG method increases with the uniformity of knot vector distribution and decreases with the order of NURBS. Meanwhile, the higher accuracy is achieved when the orders of NURBS are the same in different directions, while the reasonable range of refinement number of mesh is 1∼2. The influence of thermal conductivity factor and off-angle are explored, and their reasonable ranges for anisotropic airplane wings are 2 ∼ 5 and 15° ∼ 30°, respectively, which can effectively improve the temperature field.

Krueger Design and Motion Requirements

The industrial valid design of a folding bull nose Krueger leading edge device will be presented. It serves as a geometrical basis for the UHURA project, which investigates the retraction and deployment process with numerical and experimental means.A set of realistic aerodynamic and kinematic requirements was applied, such as deflection angle range and deployed target position to enable insect shielding. In order to achieve an industrial relevant design solution of a Krueger device, realistic space allocation constraints and structural keep-out-zones were defined within the fixed leading edge.Numerical shape optimization of the Krueger device was performed to balance size, shape and deployed position with optimal aerodynamic performance, respecting the given requirements and constraints.Based on that, layout and sizing of the kinematic structural and connecting elements was carried out. Industrial viability was ensured by applying aerospace design rules, as well as relevant material selection, deformation and tolerance analysis.

A review of the geometrical basis and the principles underlying the use and interpretation of the video head impulse test (vHIT) in clinical vestibular testing.

This paper is concerned mainly with the assumptions underpinning the actual testing procedure, measurement, and interpretation of the video head impulse test-vHIT. Other papers have reported in detail the artifacts which can interfere with obtaining accurate eye movement results, but here we focus not on artifacts, but on the basic questions about the assumptions and geometrical considerations by which vHIT works. These matters are crucial in understanding and appropriately interpreting the results obtained, especially as vHIT is now being applied to central disorders. The interpretation of the eye velocity responses relies on thorough knowledge of the factors which can affect the response-for example the orientation of the goggles on the head, the head pitch, and the contribution of vertical canals to the horizontal canal response. We highlight some of these issues and point to future developments and improvements. The paper assumes knowledge of how vHIT testing is conducted.

Geometric Outlines of the Gravitational Lensing and Its Astronomic Applications

Gravitational lensing is a topic of great application value in the field of astronomy. The properties and research methods of gravitational lensing are closely related to the geometric and relativistic characteristics of the background universe. This review focuses on the theoretical research and application of strong lenses and weak lenses. We first introduce the basic principles of gravitational lensing, focusing on the geometric basis of geometric lensing, the representation of deflection angles, and the curvature relationship in different geometric spaces. In addition, we summarize the wide range of applications of gravitational lensing, including the application of strong gravitational lensing in Schwarzschild black holes, time delay, the cosmic shearing based on weak lensing, the applications in signal extraction, dark matter, and dark energy. In astronomy, through the use of advanced astronomical instruments and computers, analyzing gravitational lensing effects to understand the structure of galaxies in the universe is an important topic at present. It is foreseeable that gravitational lensing will continue to play an important role in the study of cosmology and will enrich our understanding of the universe.

Shelling the \(m=1\) amplituhedron

The amplituhedron $\mathcal{A}_{n,k,m}$ was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is a projection inside the Grassmannian $\text{Gr}_{k,k+m}$ of the totally nonnegative part of $\text{Gr}_{k,n}$. Karp and Williams (2019) studied the $m=1$ amplituhedron $\mathcal{A}_{n,k,1}$, giving a regular CW decomposition of it. Its face poset $R_{n,l}$ (with $l := n-k-1$) consists of all projective sign vectors of length $n$ with exactly $l$ sign changes. We show that $R_{n,l}$ is EL-shellable, resolving a problem posed by Karp and Williams. This gives a new proof that $\mathcal{A}_{n,k,1}$ is homeomorphic to a closed ball, which was originally proved by Karp and Williams. We also give explicit formulas for the $f$-vector and $h$-vector of $R_{n,l}$, and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset $P_{n,l}$ introduced by Machacek (2019), consisting of all projective sign vectors of length $n$ with at most $l$ sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.

An isogeometric design-analysis-optimization workflow of stiffened thin-walled structures via multilevel NURBS-based free-form deformations (MNFFD)

Stiffened thin-walled structures are widely used in various fields as load-carrying components, but the process of design, analysis and optimization still remains challenging. In this paper, a new design-analysis-optimization workflow is proposed based on isogeometric paradigms for stiffened structures, which is on the basis of a simpler and more efficient method, called multi-level NURBS-based free-form deformations (MNFFD). In terms of modeling, different from the single-level mapping of traditional FFD, MNFFD allows to establish multi-level and composite mapping, which provides strong flexibility for stiffened structures with complex assembly relationships. In terms of analysis, in order to achieve the integration of design and analysis, we propose a MNFFD-based IGA and derive a series of mapped elements. Furthermore, we implemented an isogeometric formulation of the 6-DOFs degenerated shell, and it is more convenient for the coupling of non-conforming NURBS surface patches in the frame of IGA, especially for the patches with kinks. In terms of optimization, all involved models, i.e., geometric models, analysis models, and optimization models, use the same geometric basis. We propose a simultaneous optimization framework of shape optimization and stiffener layout for lightweight design of stiffened thin-walled structures. Because the interaction between skin shape and stiffener layout is considered, the simultaneous optimization expands design space and may obtain better design Based on the proposed isogeometric design-analysis-optimization workflow, the integration of design-analysis-optimization for complex thin-walled structures is achieved. Several numerical examples demonstrate the accuracy, flexibility and robustness of the proposed design-through-analysis workflow centered on MNFFD.

Novel Mechanically-Alloyed Cu–La–P Ternary Alloy Electrocatalysts for the Alkaline Hydrogen Evolution Reaction

Transition metal phosphides, such as Cu3P, are of research interest as hydrogen evolution electrocatalysts due to a combination of good intrinsic activity and good stability. Rare earth-transition metal alloying is known to improve electrocatalytic performance, especially by the formation of intermetallic phases. Current transition metal phosphide electrocatalyst manufacturing methods are not capable of forming these intermetallic phases. Mechanical alloying is a promising technique to synthesize these intermetallic phases. Alloy powders of Cu3P, Cu19La5P12, and a novel multi-phase Cu74La4P22 composition were prepared using mechanical alloying and evaluated as electrocatalysts for the alkaline hydrogen evolution reaction on a geometric and intrinsic area basis. On an intrinsic basis, the novel Cu–La–P composition demonstrated excellent Tafel performance of 69.4 mV dec−1. Tafel slope, exchange current density, and overpotential data demonstrated the importance of spillover effects in multiphase Cu3P/Cu19La5P12 surface structures. These results suggest that Cu–La–P alloys are promising potential catalysts for electrochemical hydrogen production, and mechanical alloying of rare earth elements is an effective technique for improving the electrochemical performance of transition metal phosphides.

Computational study on the prototropic tautomerism between simple oxo-, thio-, carbon-, aza-hydrazones, and their respective azines.

Most hydrazone compounds prefer the azine tautomeric states. However, oxygen-/sulfur-substituted compounds prefer hydrazone tautomers. In this study, density functional theory at M062X level with the basis set of 6-311 + + g(2d, 2p), with MP2/cc-pVTZ for reference, was used to investigate the different tautomeric mechanisms between hydrazone and azine forms with oxygen, sulfur, carbon, and nitrogen as negative centers. The energetic stabilities are in the order as oxygen- < sulfur- < imine- < amidino- < carbene-substituted hydrazones with respect to their azine tautomeric structures. Resonance of the molecular structures might be the geometrical basis for their energy stabilities and were estimated based on HOMED indices. Further, the increased proton affinities in the trend as hydroxyl < sulfhydryl < imine terminal groups account for their increasing azine preferences. Proton release was examined for the reversible equilibrium from azine to hydrazone by calculating the transition energy barrier of H transfer. It is favorable to form hydrazone tautomers for oxygen and sulfur containing groups, while it is less favorable for amino-substituted ones. Azine form is the most stable tautomer for methyl substituted.

Isogeometric and NURBS-enhanced boundary element formulations for steady-state heat conduction with volumetric heat source and nonlinear boundary conditions

Boundary Element Method (BEM) is a numerical tool that is applied to many different types of engineering problems. BEM possesses several advantages over other numerical methods such as boundary-only discretization and its semi-analytical nature. Application of Isogeometric Analysis (IGA) to BEM is a recently proposed computational method where the main purpose is to use geometrical basis functions as the shape functions of field variables. Key features of combining these two techniques are the exact representation of the geometry, elimination or suppression of the meshing procedure, reduction of the problem dimension and obtaining highly accurate results especially for the flux values on the boundaries compared to conventional BEM. In this study, steady-state heat conduction problems with nonlinear boundary condition and surface heat source are analyzed where geometries are formed by NURBS, and field variables are described with Lagrangian (constant and quadratic) basis functions and NURBS basis functions. Convergence rates and CPU time of different types of element representations are discussed which eventually reveals the performance and capabilities of IGABEM.

Geometric basis of action potential of skeletal muscle cells and neurons.

Although we know something about single-cell neuromuscular junctions, it is still unclear how multiple skeletal muscle cells coordinate to complete intricate spatial curve movement. Here, we hypothesize that skeletal muscle cell populations with action potentials are aligned according to curved manifolds in space (a curved shape in space). When a specific motor nerve impulse is transmitted, the skeletal muscle also moves according to the corresponding shape (manifolds). The action potential of motor nerve fibers has the characteristics of a time curve manifold, and this time-manifold curve of motor nerve fibers comes from the visual cortex in which spatial geometric manifolds are formed within the synaptic connection of neurons. This spatial geometric manifold of the synaptic connection of neurons originates from spatial geometric manifolds outside nature that are transmitted to the brain through the cone cells and ganglion cells of the retina. The essence of life is that life is an object that can move autonomously, and the essence of life’s autonomous movement is the movement of proteins. Theoretically, because of the infinite diversity of geometric manifold shapes in nature, the arrangement and combination of 20 amino acids should have infinite diversity, and the geometric manifold formed by the protein three-dimensional spatial structure should also have infinite diversity.

An effective knowledge transfer method based on semi-supervised learning for evolutionary optimization

Effective knowledge transfer has been proven to achieve superior performance in evolutionary optimization. Evolutionary multitasking optimization (EMT), which can solve several optimization tasks simultaneously using evolutionary algorithms, continues to be a young research field but is growing rapidly. Due to the parallelism of population-based search, the performance of component tasks can be improved through effective knowledge transfer between different tasks. The main challenge in the EMT field is addressing the negative transfer. Aiming to overcome this challenge, this paper proposes an EMT algorithm that transfers effective knowledge through semi-supervised learning. In addition, a semi-supervised classification method is designed based on the cluster assumption, which is part of the geometric basis of semi-supervised learning. By using both labeled and unlabeled samples generated in the optimization process, the proposed method can identify individuals that contain valuable knowledge and select them to transfer the knowledge between tasks. In this way, the performance of the EMT algorithm can be significantly improved. The effectiveness of the proposed method is verified by empirical tests and comparison with two benchmarks. Further, a case study is conducted. The results indicate that the proposed algorithm can achieve highly competitive performance compared with the state-of-the-art EMT algorithms.

A Hexagonal Pattern in the Paraninfo at the Universidad de Alcalá

In the sixteenth century, the assembly hall, or Paraninfo, at the recently inaugurated Universidad de Alcalá was known as the pieça del theatro. This study focuses on the layouts and proportions designed for this hall, which could be the key to understanding the whole project as a Renaissance theater that was singularly inspired by Roman models. Specifically, the hexagonal pattern of the coffered ceiling represents a genuine formal exploration in comparison with other wooden ceilings having similar geometric bases. All this led us to study this uncommon masterpiece as a proposal for a new model toward an intellectual and architectural recovery of the Antiquity, in the core of the Universidad, conceived for the representation of academic ceremonies and humanistic theater.

Accurate computations with Gram and Wronskian matrices of geometric and Poisson bases

In this paper we deduce a bidiagonal decomposition of Gram and Wronskian matrices of geometric and Poisson bases. It is also proved that the Gram matrices of both bases are strictly totally positive, that is, all their minors are positive. The mentioned bidiagonal decompositions are used to achieve algebraic computations with high relative accuracy for Gram and Wronskian matrices of these bases. The provided numerical experiments illustrate the accuracy when computing the inverse matrix, the eigenvalues or singular values or the solutions of some linear systems, using the theoretical results.