Background: The effects of gait training based on the positioning of affected foot muscle activity, gait parameters, and balance ability were investigated in patients with subacute stroke. Material and Methods: Forty-five patients with subacute stroke were randomly assigned to three groups: straight gait training (SGT) group (n = 15), outward curved gait training group with the paretic foot positioned laterally (n = 15), or inward curved gait training (ICGT) group with the paretic foot positioned medially (n = 15). All groups received 30-minute interventions, comprising 15 min of gait training, five times per week for four weeks. Outcomes were measured in terms of muscle activation of the gluteus maximus (GM), vastus medialis, and vastus lateralis; five gait parameters (step length, stance phase, swing phase, velocity, and maximum force); and balance ability assessed using the timed up and go (TUG) test. Results: All groups exhibited significant improvements in all variables after the 4-week intervention period (p < 0.05). However, except for comparisons between the SGT and ICGT groups in terms of GM muscle activity and TUG, no significant differences were observed between the groups for the other variables (p > 0.05). Conclusions: Although patients with subacute stroke can improve their overall physical function regardless of the gait training method, ICGT may be more effective in enhancing muscle activity and balance ability.

Running on a non-motorized, curved-deck treadmill is thought to improve gait mechanics. It is not known, though, if the change in gait carries over to running on a motorized treadmill or level ground. To determine the effect of running on a curved non-motorized treadmill (CNT) on gait characteristics measured during a subsequent bout of running on a traditional motorized treadmill (TMT). Sixteen healthy college-aged participants, age (mean±SD) 20.4±1.6 years volunteered to have their gait analyzed while running on a TMT and CNT. After familiarization and warm-up on both treadmills, each subject completed five, 4-minute bouts of running alternating between traditional motorized and curved non-motorized treadmills: TMT-1, CNT-1, TMT-2, CNT-2, and TMT-3. Variables of interest included step length (m), stride length (m), imbalance score (%), and step angle (°) and were measured using Optogait gait analysis equipment. Differences in gait char acteristics among TMT-1, TMT-2, and TMT-3 can be attributed to running on the CNT. The results show that running on a CNT resulted in significant changes in gait characteristics. These findings suggest that running on a CNT can significantly alter gait characteristics may result in improvements in running gait that persist to subsequent running on a TMT.

The recent Planck Legacy release revealed the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB). Notably, this amplitude is higher than that estimated by the lambda cold dark matter model (ΛCDM), which endorses the positive curvature of the early Universe with a confidence level greater than 99%. Although General Relativity (GR) performs accurately in the local/present Universe where spacetime is almost flat, its lost boundary term, incompatibility with quantum mechanics and the necessity of dark matter and dark energy might indicate its incompleteness. By utilising the Einstein–Hilbert action, this study presents extended field equations considering the pre-existing/background curvature and the boundary contribution. The extended field equations consist of Einstein field equations with a conformal transformation feature in addition to the boundary term, which could remove singularities from the theory and facilitate its quantisation. The extended equations have been utilised to derive the evolution of the Universe with reference to the scale factor of the early Universe and its radius of curvature.

Abstract Although we know something about single cell neuromuscular junction, It is still mysterious how multiple skeletal muscle cells coordinate to complete the intricate spatial curve movement. Here I propose a hypothesis that skeletal muscle cell populations with action potentials are alligned according to a curved manifolds on space(a curved shape on space) and the skeletal muscle also moves according to this corresponding shape(manifolds) when an specific motor nerve impulses are transmitted. the action potential of motor nerve fibers has the characteristics of time curve manifold and this time manifold curve of motor nerve fibers come from visual cortex in which a spatial geometric manifolds are formed within the synaptic connection of neurons. This spatial geometric manifolds of the synaptic connection of neurons orginate from spatial geometric manifolds in outside nature that are transmitted to brain through the cone cells and ganglion cells of the retina.Further,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, due to 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 protein three-dimensional spatial structure should also have infinite diversity.

The arranged paths of dominoes have many shapes. The scaling law for the propagation speed of domino toppling has been extensively investigated. However, in all previous investigations, the scaling law for the velocity of domino toppling motion in curved lines was not taken into account. In the present work, the finite-element analysis (FEA) program ABAQUS was used to study the velocity of domino toppling motion in curved lines. It is shown that the domino propagation speed has a rising trend with increasing domino spacing in a straight line. It is also found that domino propagation speed is linearly proportional to the square root of domino separation. This research proved that the scaling law for the speed of domino toppling motion given by Sun (2020) is true [B-H. Sun, 2020. Scaling law for the propagation speed of domino toppling. AIP Advances, 10(9),095124.]. Moreover, the shape of domino arrangement paths has no influence on the scaling law for the propagation speed of dominoes but can affect the coefficient of the scaling law for the velocity. Therefore, the amendatory function for the propagation speed of dominoes in curved lines was formulated by the FEA data. The fitted amendatory function, $\varphi_{revise}$, provides the simple method for a domino player to quickly estimate the propagation speed of dominoes in curved lines.

In this work we demonstrate non-flat metasurface holograms with applications in imaging, sensing, and anti-counterfeiting. In these holograms the image and its symmetry properties, with respect to the polarization of the light, depend on the specific shape of the substrate. Additionally, the sensitivity of the holographic image to the substrate shape can be engineered by distributing the phase information into determined areas of the metasurface.

The design of curved surface sliders (CSS) based on the elastic response spectrum is done by iteration to find the combination of friction coefficient and displacement capacity which satisfies the condition that the maximum horizontal CSS force is equal to the horizontal force of the structure. Although this CSS design is valid it does not necessarily minimize structural acceleration. This paper therefore describes the optimum CSS design for minimum structural acceleration. All valid CSS designs and the optimum CSS design are represented by their associated trajectory in the elastic response spectrum plane which visualizes the optimization problem. The results demonstrate that the optimum CSS design is not obtained at maximum tolerated effective damping ratio. The subsequent sensitivity analysis describes how much the structural acceleration increases if the actual friction coefficient of the real CSS deviates from its optimum design value. The analysis points out that the increase in structural acceleration is approximately one order of magnitude smaller than the deviation in friction. The sensitivity data may be used by structural engineer to determine tolerable deviations in friction coefficient which still results in acceptable structural accelerations.

The mechanical effects of membrane compositional inhomogeneities are analyzed in a process analogous of neck formation in cellular membranes. We cast on the Canham-Helfrich model of fluid membranes with both the spontaneous curvature and the surface tension being non-homogeneous functions along the cell membrane. The inhomogeneous distribution is determined by the equilibrium mechanical equations and the boundary conditions as considered in the axisymmetric setting compatible with the necking process. To establish the role played by mechanical inhomogeneity, we focus on the catenoid, a surface of zero mean curvature. Analytic solutions are shown to exist for the spontaneous curvature and the constrictive forces in terms of the border-radii. Our theoretical analysis shows that the inhomogeneous distribution of spontaneous curvature in a mosaic-like neck constrictional forces potentially contributing to the membrane scission under minimized work in living cells.

This paper proposes a curved link-slab (CLS) structure, simplified into a hingeless-arch model, to address the current cracking phenomenon of CLS concrete. The stress formula of the hingeless arch under various loads is derived based on the classical mechanic's method. Based on an actual bridge example, the mechanical properties of CLS are analyzed under different loads and load combinations. The results show that: (1) the CLS stress is significantly lower than that of the flat link-slab structure (FLS), (2) its stress values are less than the concrete tensile limit, and (3) the CLS can effectively solve the concrete cracking phenomenon on the link-slab. The rationality of the stress formula derived from the simplified model of the hingeless arch is verified using the finite element method (FEM). The parametric sensitivity analysis shows that variation of the reinforcement ratio of the CLS has a limited impact on it. Considering both the concrete tensile and compressive limit, the thickness of the CLS should be 15 cm to 20 cm, and its design span should be about 5% to7.5% of the main beam length.

It is well-known that surface plasmon wave propagates along a straight line, but this common sense was broken by the artificial curved light – plasmon Airy beam. In this paper, we introduce a new class of curved surface plasmon wave - the photonic hook plasmon. It propagates along wavelength scaled curved trajectory with radius less than surface plasmon polariton wavelength, and can exist despite the strong energy dissipation at the metal surface.

When the minimal length approach emerging from noncommutative Heisenberg algebra, generalized uncertainty principle (GUP), and thereby integrating gravitational fields to this fundamental theory of quantum mechanics (QM) is thoughtfully extended to Einstein field equations, the possible deformation of the metric tensor could be suggested. This is a complementary term combining the effects of QM and general relativity (GR) and comprising noncommutative algebra together with maximal spacelike four–acceleration. This deformation compiles with GR as curvature in relativistic eight–dimensional spacetime tangent bundle, generalization of Riemannian spacetime, is the recipe applied to derive the deformed metric tensor. This dictates how the affine connection on Riemannian manifold is straightforwardly deformed. We have discussed the symmetric property of deformed metric tensor and affine connection. Also, we have evaluated the dependence of a parallel transported tangent vector on the spacelike four–acceleration given in units of L, where L = rℏcG3 is a universal constant, c is speed of light, and ℏ is Planck constant, and G is Newton’s gravitational constant.

We study the resonant dipole-dipole interaction energy between two uniformly accelerated identical atoms, one excited and the other in the ground state, prepared in a correlated Bell-type state, and interacting with the scalar field or the electromagnetic field nearby a perfectly reflecting plate. We suppose the two atoms moving with the same uniform acceleration, parallel to the plane boundary, and that their separation is constant during the motion. We separate the contributions of vacuum fluctuations and radiation reaction field to the resonance energy shift of the two-atom system, and show that Unruh thermal fluctuations do not affect the resonance interaction, which is exclusively related to the radiation reaction field. However, nonthermal effects of acceleration in the radiation-reaction contribution, beyond the Unruh acceleration-temperature equivalence, affect the resonance interaction energy. By considering specific geometric configurations of the two-atom system relative to the plate, we show that the presence of the mirror significantly modifies the resonance interaction energy between the two accelerated atoms. In particular, we find that new and different features appear with respect to the case of atoms in the free space, related to the presence of the boundary and to the peculiar structure of the quantum electromagnetic field vacuum in the locally inertial frame. Our results suggest the possibility to exploit the resonance interaction between accelerated atoms, as a probe for detecting the elusive effects of atomic acceleration on radiative processes.

Imperfection sensitivity of large amplitude vibration of curved single-walled carbon nanotubes (SWCNTs) is considered in this study. The SWCNT is modeled as a Timoshenko nano-beam and its curved shape is included as an initial geometric imperfection term in the displacement field. Geometric nonlinearities of von Kármán type and nonlocal elasticity theory of Eringen are employed to derive governing equations of motion of nano-beam. Spatial discretization of governing equations and associated boundary conditions is performed using differential quadrature (DQ) method and the corresponding nonlinear eigenvalue problem is iteratively solved. Effects of value and location of the geometric imperfection, and the nonlocal small-scale parameter on the nonlinear frequency ratio and imperfection sensitivity of a curved SWCNT for various boundary conditions are investigated. The results show that the geometric imperfection plays a significant role in the nonlinear vibration characteristics of curved SWCNTs.

Curved continuous steel box girders are extensively utilized in bridge construction due to its efficiency and environmental benefits. However, in regions with significant temperature fluctuations, temperature effects can result in cumulative deformation and stress concentration, which may severely impact structural safety and durability. This study examines the structural response of curved continuous steel box girders under diverse temperature conditions and also develops a comprehensive parameterized thermodynamic numerical model. The model assesses the influence of cross-sectional shape parameters including; the number of cross-sectional box chambers, diaphragm thickness, and height-to-width ratio, as well as longitudinal structural parameters such as planar configurations, width-to-span ratio, and support arrangements, along with the arrangement of stiffening ribs on the temperature-induced effects in the girders. Results indicate that optimizing the width and eccentricity of support stiffeners to 30% and 25% in the support plate size, respectively, can significantly alleviate local temperature-induced stresses. Additionally, variations in longitudinal and transverse stiffeners manifest minimal impact on thermal performance. These findings provide a theoretical foundation for improved design and construction practices, offering practical design recommendations to mitigate temperature effects and enhance the longevity and safety of such structures.

Despite the successful reconciliation of special relativity and quantum mechanics (QM), a good century ago, general relativity (GR) in its current classical formulation still does not compile with QM. While GR gives precise sensical predictions at large scales, QM is obviously dominated at small scales! We think that the everlasting battle for exploring and understanding the Universe and for privileging a consistent perception of reality is therefore awaiting a consolidator rather than a conqueror. This should be capable of either unifying the two different benchmarks or at least bringing one closer to the other! The latter describes the consolidating approach, which is based on extending QM to relativistic energies and gravitational fields and generalizing Riemann to discretized Finsler geometry. Some properties of the additional geometric structures and curvatures which are disclosed by the proposed quantization and apparently overlooked in Einstein's GR, are analytically and numerically characterized. The analytical analyses introduce dynamics with possible nonlinear connections. We conclude that i) additional geometric structures and curvatures are intrinsic essential and ii) the spacetime at the quantum scale is no longer smooth or continuous. A maximal proper force is predicted, which maximally gravitationally accelerates a test particle and importantly manifests the quantum geometric nature of the generalized spacetime curvature.

A review of influence of drag-reducing agents on curved pipe flows is presented in this work. In addition, this review outlined proposed mechanism, friction factor and fluid flux models for drag-reducing agents in curved pipe flows. Our finding reveals that drag reduction by additives in curved pipes is quite significant but generally lower than the corresponding drag reduction in straight pipes. It decreases with increase in curvature ratio and more pronounced in the transition and turbulent flow regimes. Drag reduction strongly depends on the polymers and surfactants’ concentrations as well as the bubble fraction of micro-bubbles. It is also reported that drag reduction in curved pipes depends on temperature and existence of dissolved salts in the fluids. Maximum drag reduction asymptote differed between straight and curved pipes and between polymer and surfactant. No definite conclusion could be drawn as regards drag reduction for two-phase flow in curved pipes due to the limited studies in this area. Many questions such as the mechanism of drag reduction in curved pipes and how drag-reducing agents interact with secondary flows still remained unanswered. Hence, some research gaps have been identified with recommendations for areas of future researches.

When generalized noncommutative Heisenberg algebra accommodating impacts of finite gravitational fields as specified by loop quantum gravity, doubly–special relativity, and string theory, for instance, is thoughtfully applied to the eight-dimensional manifold (M8), the generalization of the Riemannian manifold becomes imminent. By constructing the deformed affine connections on a four-dimensional Riemannian manifold, we have determined the minimal length deformation of Riemann curvature tensor and its contractions, the Ricci curvature tensor, and Ricci scalar. Consequently, we have been able to construct the deformed Einstein tensor. As in Einstein’s classical theory of general relativity, we have proved that the covariant derivative of the deformed Einstein tensor vanishes, as well. We conclude that the minimal length correction suggests a correction to the spacetime curvature which is manifest in the additional curvature terms of the corrected Riemann tensor and its contractions. Accordingly, the spacetime curvature endows additional curvature and geometrical structure.

Difficult-to-cut materials are being increasingly used in many industries because of their superior properties, including high corrosion resistance, heat resistance and specific strength. However, these same properties make the materials difficult to machine using conventional machining techniques. Laser-assisted milling (LAM) is one of the effective methods for machining difficult-to-cut materials. In laser-assisted milling machining occur after the workpiece is locally preheated using a laser heat source. Laser assisted machining has been studied by many researchers on flat workpiece or micro end-milling. However, there is no research on the curved shape using laser assisted milling. This study investigated the use of laser assisted milling to machine a three-dimensional curved shape workpiece based on NURBS. A machining experiment was performed on Inconel 718 using different tool paths (ramping, contouring) under various machining conditions. Finite elements analysis was conducted to determine the depth of cut. Cutting force, specific cutting energy and surface roughness characteristics were measured, analyzed and compared for conventional and LAM machining. LAM significantly improved these machining characteristics, compared to conventional machining. There results can be applied to the laser-assisted machining of various three-dimensional shapes.

The exact determination of radiative exchanges between solids and surfaces has been a long sought-for question in heat transfer science. Being the canonical equation that rules such phenomena, a fourfold integral, it is extremely difficult to obtain an accurate solution like a formula or abacus. Over the last thirty years, the author has tried to integrate the canonical expression by sundry procedures and they have published two books and a dozen of articles on the matter, recently by virtue of computational geometry and graphic algorithms as a new way to solve the finite-difference problems that arise on complex geometries. In architectural engineering curved radiant emitters are customary since antiquity, especially in domes and vaults and their oculus, However, a consistent procedure to handle them was not readily available. The principles that are described hereby based on Cabeza-Lainez’ first principle for spherical fragments offer a complete panorama on the manner in which surface sources related or contained in spheres can be interpreted and accounted for without resorting to integration. The main advance is that a variety of unexplained problems of radiative heat transfer, applicable to aerospace engineering, meteorological, architectural and medical sciences can be sorted out as exactly as quickly.

Nanometer-curved surfaces are abundant in biological systems as well as in nano-sized technologies. Properly functionalized polymer-grafted nanoparticles (PGNs) adhere to surfaces with different geometries and curvatures. This work explores some of the energetic and mechanical characteristics of the adhesion of PGNs to surfaces with positive, negative and zero curvatures using Coarse-Grained Molecular Dynamics (CGMD) simulations. Our calculated free energies of binding of the PGN to the curved and flat surfaces as a function of separation distance show that curvature of the surfaces critically impacts the adhesion strength. We find that the flat surface is the most adhesive, and the concave surface is the least adhesive surface. This somewhat counterintuitive finding suggests that while a bare nanoparticle is more likely to adhere to a positively curved surface than a flat surface, grafting polymer chains to the nanoparticle surface inverts this behavior. Moreover, we studied the rheological behavior of PGN upon separation from the flat and curved surfaces under external pulling force. The results presented herein can be exploited in drug delivery and self-assembly applications.

The fundamental challenge for the structural design of complicated shapes and lightweight structures such as Tensile Membrane Structures (TMS) is a reliable estimation of lateral loading such as wind load. Wind pressure coefficients can’t be correctly calculated by standards for a large range of intricate forms of structures. Computational Fluid Dynamics (CFD), a strong computational technique for evaluating wind pressure distribution on complicated geometry, can be used to determine wind-related parameters. The average wind related-distributions for the tensile membrane surface are validated with a few contributions from prior studies, which are important to validate with benchmarks in experimental wind tunnel experiments. For the double-curved surface, the results show good agreement between experimental tests and CFD simulations. In this study, CFD simulation was used to calculate the mean surface pressure coefficient (Cp) for generic shapes and tensile membrane forms using the steady Reynolds-Averaged Navier-Stokes (RANS) technique. Wind force coefficients and the effect of wind directions are also explored across a wide range of classical and tensile forms. The purpose of the current research is to look at the effect of wind assault angle on the Cp distribution for tensile membrane structures and general shapes.

The magnetic field of the earth plays an important role in the ecosystem, and the magnetic field of celestial bodies is also important in the formation of cosmic large-scale structures, but the origin and evolution of the celestial magnetic field is still an unresolved mystery. Many hypotheses to explain the origin have been proposed, but there are some insurmountable difficulties for each one. At present, the theory widely accepted in scientific society is the dynamo model, it says that, the movement of magnetofluid inside celestial bodies, which can overcome the Ohmic dissipative effect and generate persistent weak electric current and macroscopic magnetic field. However, this model needs an initial seed magnetic field, and the true values of many physical parameters inside the celestial body are difficult to obtain, and there is no stable solution to the large range of fluid motion. These are all difficulties for the dynamo model. Furthermore, it is difficult for the dynamo to explain the correlation between the dipole magnetic field and angular momentum of a celestial body. In this paper, by calculating the interaction between spin of particles and gravity of celestial body according to Clifford algebra, we find that a rotational celestial body provides a field $\Omega^\alpha$ for spins, which is similar to the magnetic field of a dipole, and the spins of charged particles within the celestial body are arranged along the flux line of $\Omega^\alpha$, then a macroscopic magnetic field is induced. The calculation shows that the strength of $\Omega^\alpha$ is proportional to the angular momentum of the celestial body, which explains the correlation between the magnetic intensity and angular momentum. The results of this paper suggest that further study of the effects of internal variables such as density, velocity, pressure and temperature of a celestial body on $\Omega^\alpha$ may provide some new insights into the origin and evolution of celestial magnetic field.

A relationship between the functional Schr\"odinger representation and the precanonical quantization of a nonlinear scalar field theory is extended to arbitrary curved space-times. The canonical functional derivative Schr\"odinger equation is derived from the manifestly covariant precanonical Schr\"odinger equation in a singular limiting case when the ultraviolet parameter $\varkappa$ introduced by precanonical quantization is identified with the invariant delta-function at equal spatial points. In the same limiting case, the Schr\"odinger wave functional is expressed as the trace of the multidimensional product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration. Thus the standard QFT in curved space-time in functional Schr\"odinger representation emerges from the precanonical formulation of quantum fields as a singular limiting case.

Steel reinforcement in concrete has the tendency to corrode and this process can lead to structural damage. FRP reinforcement represents a viable alternative for structures exposed to aggressive environments and has many possible applications where superior corrosion resistance properties are required. The use of FRP rebars as internal reinforcements for concrete, however, is limited to specific structural elements and does not yet extend to the whole structure. The reasons for this relate to the limited availability of curved or shaped reinforcing elements on the market and their reduced structural performance. Various studies, in fact, have shown that the mechanical performance of bent portions of composite bars is reduced significantly under a multiaxial combination of stresses and that the tensile strength can be as low as 25% of the maximum tensile strength that can be developed in the straight part. In a significant number of cases, the current design recommendations for concrete structures reinforced with FRP, however, were found to overestimate the bend capacity of FRP rebar. This paper presents the state-of-the art review of the research works on the strength degradation in curved FRP composites and highlighted the performance of exiting predictive models for the bend capacity of FRP reinforcement. Recent practical predictive model based on the Tsai-Hill failure criteria by considering the material at marcromechanical level is also discussed and highlighted. The review also identifies the challenges and highlights the future directions of research to explore the use of shaped FRP composites in civil engineering applications and the trends for future research in this area.

The inability to delineate a unified physical ontology that accounts simultaneously for the laws of special relativity and the results of quantum experiments has been a defining problem in physics for more than 100 years. This analysis addresses the problem by positing a novel mixed ontology composed of a “discrete” 4D spacetime and a physical, ultra-high dimensional (3 x N) “Planck Space.” Together, Planck Space and the three spatial dimensions of 4D spacetime form a tightly integrated ((3 x N) + 3) hyperspace (the “Dual Ontology”).Critically, the Dual Ontology’s structure replaces 1) the continuous, differentiable manifold of 4D spacetime with a discrete, background-independent 4D spacetime that is intrinsically curved and 2) mathematical spaces with a physical (3 x N) Planck Space. Moreover, the Dual Ontology is structurally and dynamically predicated on the one-to-one mapping and identity between the discrete spatial units that simultaneously form 4D spacetime and Planck Space. The one-to-one mapping and identity of the discrete spatial units support an integrated quantum dynamics based upon the dynamic evolution of single and N-body quantum states in 4D spacetime in full compliance with the laws of special relativity and the instantaneous collapse of all quantum states in an ontic Planck Space, where special and general relativity and more generally, 4D spacetime’s laws of physics, do not apply.

Can a gravitational lens magnify gravity ? The answer to this question is the subject of this article. Light is deflected as it passes through curved spacetime caused by a mass distribution. This phenomenon, known as "gravitational lensing," has been extensively studied and is well understood. According to the general theory of relativity, the same deflection is expected for gravitational waves, because they also propagate at the speed of light. But this applies to gravitational fields as well. Thus, gravitational fields, which extend through curved spacetime, are also deflected correspondingly. This effect is referred to as "Gravitationally Lensed Gravity". It leads to the fact that the gravitational field strength originating from an object in the background of a gravitational lens is enhanced in the same way as its light intensity. If the background object is visible to an observer several times due to the gravitational lens, each of the images also exerts a corresponding gravitational attraction on her ! For the first time, a proof of the existence of this amazing effect is given. In a simple thought experiment the contraction of a spherically symmetric gravitational lens is discussed. The proof is essentially based on Birkhoff's theorem. According to this theorem, during the contraction additional null geodesics between the background object and the observer come into existence. To these null geodesics also contributions to the gravitational field are to be assigned. In the literature, similar descriptions can be found for the gravitational self-interaction of a particle in a curved spacetime background. The article is organized as follows: In the introduction, the historical origin of the idea of the "Gravitationally Lensed Gravity" is briefly outlined. In the second chapter, the subsequently required basic principles and essential features of general relativity are explained. In the third chapter I analyze different arrangements of masses and describe the observations to be expected. The arrangements show step by step the transition from the classical "Newtonian limit" in a flat spacetime background to the "Gravitationally Lensed Gravity" in a curved spacetime background. Due to its clarity, the simple scenario may also serves as a suitable test case for the mathematical methods of general relativity. The concluding outlook provides references to current questions in astrophysics, especially the question of the existence of Dark Matter. Here, the general theory of relativity offers much simpler explanations with the "Gravitationally Lensed Gravity".

The enormous diversity of hair fibers found in human population is striking, while the basic structure of human fibers is often the same! But why do some fibers curl? Curling is related to the machinery and processes that produce different keratin filament orientation in relation to the axis of hair growth, influenced by developmental and molecular control. It is established that a curly hair shaft is produced by a curved hair follicle that have an angulated dermal implantation, with a bulb bent in the shape of golf club, while the outer root sheath is dissymmetrical; have an outer side that is convex bulging and thin and a concave side in which the cells express alpha-smooth muscle protein reflecting internal mechanical force. Yet, the exact cause of hair curling is not known. All the previous studies were short of impacting curly hair treatment. My hypothesis is that the curling mechanical force is generated by a hypersensitive contracted Arrector Pili muscle (APM) of the hair follicle. In curly hair, each APM fiber is shortened; pulling its attachment spot upward towards the epidermis and obliquely or laterally; curling the outer root sheath to produce the curved follicle described above. This contraction has the influence at a time when the young hair root is being formed and keratin microfibrils are being oriented and packed. APM is a multi-unit muscle wherein each fiber contracts independently to create the enormous diversity of curly hair types. APM contraction can be modulated by oral administration of selective smooth muscle relaxing treatment that relax the curved follicle from within. Further studies are needed to confirm the role of APM in shaping curly hair follicle including AI studies, i.e., computer-aided target identification to demonstrate APM active site that can interact with herbal small molecules.

This article lays the foundation for Extended Relativity (ER) – a unified geometric approach to the electromagnetic and gravitational field and relativistic dynamics in them. We explain here the basic tenets of ER, including the influenced spacetime of an object and the Extended Principle of Inertia. These two new ideas enable a unified theory of these fields. We introduce local scaling functions for describing the geometry of a spacetime influenced by a field and construct a simple, universal local scaling function which unifies electromagnetism and gravity. We recover the full electromagnetic theory, including Maxwell’s equations. For gravitation, ER passes all of the tests of General Relativity. Despite the non-linearity of relativistic gravitation, we obtain a method of combining fields from multiple sources. Finally, we identify the differences in the relativistic treatment of the electromagnetic field by Special Relativity and of the gravitational field by General Relativity and indicate how these differences are resolved in ER.