Elliptical Geometry Research Articles

Self-products of rationally elliptic spaces and inequalities between the ranks of homotopy and homology groups

We survey recent results on inequalities between the ranks of homotopy and cohomology groups (resp., graded components of mixed Hodge structures on these groups) for rationally elliptic spaces (resp., quasi-projective varieties which are rationally elliptic). We also discuss a refinement of these results describing a new invariant of rationally elliptic spaces allowing to compare the ranks of homotopy and homology groups. Its definition is based of the study of the range of parameter r such that rP(t)<Q(t)r for all t≥ε, where (P(t),Q(t)) is a pair of arbitrary polynomials with non-negative integer coefficients. We show that this range is related to the classical Lambert W-function W(z).

Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry.

Harmonic response analysis of elliptically curved thin plates

In this study, harmonic response analysis of isotropic elliptically curved thin plate structures has been conducted. The structure has been excited by a harmonic load, whose maximum magnitude is 100 N. The structure has been considered under fixed from both straight edges boundary conditions. The effect of the elliptical geometry on the harmonic response of the structure in terms of the critical frequency region, phase angle, stress, and displacement has been examined. For this purpose, the vertex to co-vertex ratio has been variated from 3 to 4 by 0.1 intervals. All analyses have been performed via ANSYS Workbench by employing the Mode Superposition Method. The results indicated that the elliptical geometry has a significant impact on the harmonic response of elliptically curved thin plate structures.

New distributed beamforming techniques based on optimized elliptical arc geometry for back lobe cancellation of linear antenna arrays

The presence of array back lobe is a major source of interference, power loss, and degrades the signal-to-interference ratio (SIR) in wireless communication systems. In this paper, new distributed beamforming techniques based on an optimized elliptical arc geometry (EAG) are proposed for back lobe cancellation of linear antenna arrays (LAA). The optimized EAG allows for flexible control of the elements' locations and initial transmission phases of their radiations, allowing them to add constructively towards the main beam direction and add destructively towards the back lobe direction. Firstly, two EAG based beamforming techniques are introduced using either a single ellipse-EAG (SE-EAG) or double ellipses-EAGs (DE-EAG). In these techniques, the back lobes are minimized by optimizing both the minor axes of the elliptical arcs and the elements' angular distributions using the particle swarm optimization (PSO) technique. While the elements' transmission energies are kept the same as that of the original LAA. For further back lobe minimization, the double ellipses-EAG with optimized energies (DE-EAG-WOE) beamforming technique is introduced, where both elements' transmission energies and positions are optimized. To verify the superiority and the possibility of practical validation of the proposed techniques, the proposed distributed array configurations based on the DE-EAG-WOE are realized using the CST Microwave Studio software package using dipole elements.

The Distribution of the Thermal Field in an Elliptical Electric Conductor Coated with Insulation

The paper determines the stationary thermal field in an elliptical cross-section electric conductor coated with insulation. Heat is generated by the flow of alternating current (AC) through the conducting core, and then dissipated from the insulation surface via convection and radiation. The authors have developed an original method for hybrid (analytical–numerical) modeling of a field. This method has been used to solve the relevant boundary problem of Poisson’s equation. While the eigenfunctions of the Laplace operator were determined analytically, the coefficients of the eigenfunctions were calculated by iteratively solving an appropriate system of algebraic equations. The proposed method enables the analysis of systems with an elliptical geometry and a heterogeneous layered structure (e.g., air, aluminum alloy, PCV), and does not require area discretization (grid). The developed analytical–numerical (AN) method has been positively verified using finite elements (FEs). The determined thermal field is illustrated graphically. The obtained solution has a good physical interpretation.

A Note on Gershgorin Disks in the Elliptic Plane

In this study, we derive Gershgorin discs of elliptic complex matrices in the elliptic plane. Also, we investigate the location of the zeros of an elliptic complex valued polynomial with the help of Gershgorin discs of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic complex numbers are a generalized form of complex and so real numbers. Thus, the obtained results extend, generalize and complement some known Gershgorin discs results from the literature.

Effective action in elliptic and hyperbolic spacetimes

The full one-loop effective action is computed for both hyperbolic and elliptic spaces.

Optical soliton perturbation with Kudryashov’s generalized law of refractive index and generalized nonlocal laws by improved modified extended tanh method

In this work, the improved modified extended tanh scheme is implemented to extract exact travelling wave solutions for perturbed nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index and dual form of generalized nonlocal nonlinearity. Various types of solutions are extracted such as bright solitons, singular solitons, dark solitons, singular periodic wave solutions, periodic wave solutions, Jacobi elliptic functions, plane wave and hyperbolic wave solutions. Moreover, 3D and contour plots of some solutions are illustrated to show the physical nature of obtained solutions.

Effect of orientation of elliptic tube on the total melting time of latent thermal energy storage systems

Encapsulation of Phase Change Materials (PCM) for energy storage, thermal comfort and many other energy applications is receiving much attention due to the fact that material, physical characteristics and geometry of the container can affect drastically the thermal performance of the PCM. Phase change materials have usually low thermal conductivity which impairs their thermal charging and discharging characteristics. Different geometries were investigated including rectangular, cylindrical and spherical with and without extended surfaces to investigate the heat charge processes. Cylindrical geometries of circular sections were intensively investigated while cylinders and tubes with elliptic and elongated cross section received less attention, although they may have better thermal performance for thermal storage. The present numerical investigation is aimed at contributing to better understand the effects of the elliptic geometry and how the different geometrical and operational parameters can affect the thermal performance of the enclosed PCM. The present investigation reports the results of a numerical study on elliptic cylinders containing PCM under melting conditions. The 2D inward melting problem is modeled by using a CFD code. The numerical model is based upon the enthalpy-porosity method along with the finite control volume techniques. The numerical predictions are validated against available experimental results. The inward melting process is analyzed for two orientations of the elliptic enclosures. Due to the flow field effect namely the Rayleigh-Bénard convection, the numerical results showed that the horizontal elliptic enclosure have higher melting rate and hence lower total melting time compared to those of the vertical elliptic enclosure.

Geometric representation of the irregular cross-section of old timber elements: Comparison of different approaches for mechanical characterisation

The assessment and the mechanical evaluation of the wooden elements in historical buildings are prerequisites to guarantee their conservation and safety at the same time. The availability of reliable reference values for the mechanical properties of old timber is crucial and they must refer to a well-defined cross-section. This could be a huge problem when the structural elements are irregular, as frequently happens with historical timber.A sample of 24 wooden elements dismantled from the roof of two buildings located in central Italy dated to the late nineteenth century and with very irregular cross-sections was used to investigate the geometry that best represents the actual cross-section for the mechanical characterization. Rectangular, elliptical and octagonal geometry patterns were defined and compared.Bending strength and stiffness were underestimated of 23% on average when calculated on rectangular cross-section and overestimated of 30% on average when the ellipse was used as a reference. The representation by an irregular octagon proved to be an excellent schematization of the actual cross-section, with an average overestimation of just 1%.For the tested elements, the value of 0.66 (2/3 of the side) of mean wane was identified as the threshold above which the elliptical representation should be preferred to the octagonal one. The data also suggested a sufficient representativeness of the rectangle for an average wane of less than 0.2 (1/5 of the side). The last conclusions, however, need further confirmation on a wider sample.

An engineering method to design daggerboards and rudders for small racing yachts

The variety of shapes and sizes among daggerboards and rudders of small racing yachts indicates the lack of a comparative design method and that some aspects may be neglected in the designs currently proposed. In particular, a literature survey showed that: (i) the conventional design approaches are exclusively based on upwind conditions, disregarding any downwind contribution to the race, (ii) the detrimental effects related to the typically low Reynolds numbers involved are marginally considered, and (iii) there are ambiguous indications about sweep angle effects on foil performance. In this paper, a design method based on a Velocity Prediction Program compares different appendage planforms for a yacht, identifying the ones minimizing the overall racing time, sum of upwind and downwind contributions. The method is applied to a 4.6 m skiff and the performance of the daggerboards analysed are compared with xflr5 simulations, validated against experimental tests. The results quantify the time gains of longer foils and the similar performance of elliptic and rectangular geometries with equal drafts but different aspect ratios. Swept foils are investigated as well, showing lower lift curve slopes compared to straight geometries for the high aspect ratios typically used on small yachts.

Homology versus homotopy in rational fibrations

Motivated by prominent problems like the Hilali conjecture, Yamaguchi– Yokura recently proposed certain estimates on the relations of the dimensions of rational homotopy and rational cohomology groups of fibre, base and total spaces in a fibration of rationally elliptic spaces. In this article we prove these estimates in the category of formal elliptic spaces and, in general, whenever the total space in addition has positive Euler characteristic or has the rational homotopy type of a homogeneous manifold (respectively of a known example) of positive sectional curvature. Additionally, we provide general estimates approximating the conjectured ones. Moreover, we suggest to study families of rationally elliptic spaces under certain asymptotics, and we discuss the conjectured estimates from this perspective for twostage spaces.

On M-theory on real toric fibrations

Borrowing ideas from elliptic complex geometry, we approach M-theory compactifications on real fibrations. Precisely, we explore real toric equations rather than complex ones exploited in F-theory and related dual models. These geometries have been built by moving real toric manifolds over real bases. Using topological changing behaviors, we unveil certain data associated with gauge sectors relying on affine lie symmetries.

Panoramic Imaging Assessment of Different Bladder Phantoms – An Evaluation Study

ObjectiveTo evaluate “panoramic image stitching” for cystoscopy, a novel technique to augment a urologist's field of view transoperatively in real-time during a cystoscopic “keyhole” procedure, 3-D bladder phantoms provide a suitable setting. Thus, the objective is the evaluation of different 3-D printed bladder phantoms with respect to their ability to be used for extended experiments of panoramic cystoscopy. Materials and MethodsFive bladder phantoms with different geometries, surface textures, stiffness and materials were assessed with rigid 0- and 30-degree lenses and a video-cystoscope regarding suitability for image stitching. For panoramic image generation, we use an established real-time stitching approach successfully applied to retrospective cystoscopy image sequences from real clinical cases. For evaluation of the experiments two quality criteria were defined, namely ‘completeness’ (describing the internal area of the phantom that can be stitched), and ‘extension’ (how far does an acquired panorama extend beyond the assumed ‘equator’ of the phantom). ResultsPanoramas of all phantom and endoscope combinations were computed. Using landmarks (south pole, north pole, equator) in the phantoms, maximum extension of the panoramas was assessed. The computed panoramas yield maximum extensions between 270o (0-degree cystoscope) and 330o (video cystoscope). Deformable phantoms yield larger panoramas than the rigid models. ConclusionIn can be concluded that the stitching process works quite well with all evaluated phantoms. Nevertheless, a novel phantom would be needed, combing a deformable, elliptical geometry and real-type vascular texture in the inside, that can be filled with water.

Cavitation and charge separation in laser-produced copper and carbon plasma in transverse magnetic field

In the present work, we report the dynamics and geometrical features of the plasma plume formed by the laser ablation of copper and graphite (carbon) targets in the presence of different transverse magnetic field. This work emphasizes on the effect of atomic mass of the plume species on the diamagnetic behaviour and geometrical aspect of the expanding plasma plume in the magnetic field. The time-resolved analysis of the simultaneously captured two directional images in orthogonal to the expansion axis is carried out for the comparative study of projected three-dimensional structure of copper and carbon plasma plume. In the presence of magnetic field, sharp differences are observed between the copper and carbon plasma plumes in terms of formation of diamagnetic cavity and structure formation. An elliptical cavity-like structure is observed in case of copper plasma plume which attains the sharp conical shape with increasing the time delay or magnetic field strength. On the other hand, splitted carbon plasma plume appears as a Y-shape structure in the presence of magnetic field where the cavity-like structure is not observed for the considered time and magnetic field. Based on the modified energy balance relation for the elliptic cylindrical geometry, we have also simulated the dynamics of the plume which is in close agreement with observed plasma expansion in diamagnetic and non-diamagnetic regions.

Completeness of Pledger’s modal logics of one-sorted projective and elliptic planes

Ken Pledger devised a one-sorted approach to the incidence relation of plane geometries, using structures that also support models of propositional modal logic. He introduced a modal system 12g that is valid in one-sorted projective planes, proved that it has finitely many non-equivalent modalities, and identified all possible modality patterns of its extensions. One of these extensions 8f is valid in elliptic planes. These results were presented in his 1980 doctoral dissertation, which is reprinted in this issue of the Australasian Journal of Logic. &#x0D; Here we show that 12g and 8f are strongly complete for validity in their intended one-sorted geometrical interpretations, and have the finite model property. The proofs apply standard technology of modal logic (canonical models, filtrations) together with a step-by-step procedure introduced by Yde Venema for constructing two-sorted projective planes.

Some Interrelations between Geometry and Modal Logic

This is a reprinting of Ken Pledger’s PhD thesis, submitted to the University of Warsaw in 1980 with the degree awarded in 1981. It develops a one-sorted approach to the theory of plane geometry, based on the idea that the usually two-sorted theory “can be made one-sorted by keeping careful account of whether the incidence relation is iterated an even or odd number of times”.The one-sorted structures can also serve as Kripke frames for modal logics, and the thesis defines and studies two such logics that are validated by projective planes and elliptic planes respectively. It raises questions of logical completeness for these systems that are addressed in the first article of this journal issue.

Volume of the whale shark and their mechanism of vertical feeding

The present study provides a noninvasive method to estimate the body volume of sharks (Elasmobranchii, Selachii) using a computational geometric model. This method allows the volume of sharks to be estimated from lateral and ventral photographs assuming an elliptical body cross-sectional geometry. A comparison of the estimated and actual body volumes of several shark species showed that the estimation error was < 0.5%. The accuracy of the model decreased if photographs that were inclined to the orthogonal plane were used, although this error was on average < 2.3% if the inclination angle was 10° or less.Applying this model to captive whale sharks (Rhincodon typus) that were 8.0 and 8.8 m in total length revealed that their body volumes were 3.5 and 4.5 m3, respectively. These estimates allowed for the quantitative evaluation of our hypothesis, that the whale shark uses suctioned air for buoyancy control during vertical feeding—a behavior unique to this species among elasmobranchs. The volume estimates of the captive whale sharks, together with the density estimates from their liver proportions, revealed that the air occupying a part of oro-pharyngeal and branchial cavities can help the whale sharks to keep their body floating. This hypothesis may explain how the whale shark sometimes stays at the water surface without fin motion during vertical feeding, even though their body density is greater than that of seawater.

Design consideration of compact cardiac TOF-PET systems: a simulation study

Myocardial perfusion imaging (MPI) with PET plays a vital role in the management of coronary artery disease. High sensitivity systems can contribute to maximizing the potential value of PET MPI; therefore, we have proposed two novel detector arrangements, an elliptical geometry and a D-shape geometry, that are more sensitive and more compact than a conventional large-bore cylindrical geometry. Here we investigate two items: the benefits of the proposed geometries for cardiac imaging; and the effects of scatter components on cardiac PET image quality. Using the Geant4 toolkit, we modeled four time-of-flight (TOF) PET systems: an 80 cm diameter cylinder, a 40 cm diameter cylinder, a compact ellipse, and a compact D-shape. Spatial resolution and sensitivity were measured using point sources. Noise equivalent count rate and image quality were examined using an anthropomorphic digital chest phantom. The proposed geometries showed higher sensitivity and better count rate characteristics with a fewer number of detectors than the conventional large-bore cylindrical geometry. In addition, we found that the increased intensity of the scatter components was a big factor affecting the contrast in defect regions for such a compact geometry. It is important to address the issue of the increased intensity of the scatter components to develop a high-performance compact cardiac TOF PET system.

An improved ring test to assess cracking resistance of concrete under restrained shrinkage

Self-restraint caused by differential shrinkage along the radius of a concrete ring has great driving effect on the restrained cracking in the traditional ring test. In order to estimate cracking potential of concrete under restrained shrinkage, i.e. steel ring restraint effect, an improved circular/elliptical ring test was proposed by pre-setting a 6 mm long real crack at the inner circumference of the concrete specimen. Meanwhile, a numerical study based on fracture theory was conducted to investigate the cracking mechanism of the improved ring test. By introducing the fictitious crack model, the crack initiation age and propagation process in the ring specimen were simulated using the initial fracture toughness-based cracking criterion. The driving forces of the shrinkage cracking, provided by self-restraint and steel ring restraint, were analyzed and the influence of the pre-set crack length on the cracking age was discussed. The results indicated that, different from the traditional ring test, the shrinkage crack firstly initiated from the pre-set crack and then unstably propagated toward the outer circumference of the concrete in the improved method. In the crack propagation process, the self-restraint provided cracking insistence rather than driving force on the shrinkage cracking due to the bending effect along the radial direction caused by non-uniform moisture distribution in concrete. Therefore, the driving force of the shrinkage cracking was totally provided by the steel ring restraint and would overcome the cracking resistances from concrete material and non-uniform shrinkage. Furthermore, the advantage of the elliptical geometry in increasing the restraining level was still available for the improved ring test with a pre-set crack, which could make the shrinkage cracking occur in advance and enhance the steel ring restraint.