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Solid Earth tide represents the response of solid Earth to the lunar (solar) gravitational force. The yielding solid Earth due to the force has been thought to be a prolate ellipsoid since the time of Lord Kelvin, yet the ellipsoid’s geometry such as major semi-axis’s length, minor semi-axis’s length, and flattening remains unresolved. Additionally, the tidal displacement of reference point is conventionally resolved through a combination of expanded potential equations and given Earth model. Here we present a geometric model in which both the ellipsoid’s geometry and the tidal displacement of reference point can be resolved through a rotating ellipse with respect to the Moon (Sun). We test the geometric model using 23-year gravity data from 22 superconducting gravimeter (SG) stations and compare it with the current model recommended by the IERS (International Earth Rotation System) conventions (2010), the average Root Mean Square ( RMS ) deviation of the gravity change yielded by the geometric model against observation is 6.47 µGal (equivalent to 2.07 cm), while that yielded by the current model is 30.77 µGal (equivalent to 9.85 cm). The geometric model will greatly contribute to many application fields such as geodesy, geophysics, astronomy, and oceanography.

The four-dimensional ellipsoid of an anisotropic hyper-structure corresponds to the four-dimensional sphere of an isotropic hyper-structure. In three dimensions, both theories for spherical and ellipsoidal harmonics have been developed by Laplace and Lamé, respectively. Nevertheless, in four dimensions, only the theory of hyper-spherical harmonics is hitherto known. This void in the literature is expected to be filled up by the present work. In fact, it is well known that the spectral decomposition of the Laplace equation in three-dimensional ellipsoidal geometry leads to the Lamé equation. This Lamé equation governs each one of the spectral functions corresponding to the three ellipsoidal coordinates, which, however, live in non-overlapping intervals. The analysis of the Lamé equation leads to four classes of Lamé functions, giving a total of 2n + 1 functions of degree n. In four dimensions, a much more elaborate procedure leads to similar results for the hyper-ellipsoidal structure. Actually, we demonstrate here that there are eight classes of the spectral hyper-Lamé equation and we provide a complete analysis for each one of them. The number of hyper-Lamé functions of degree n is (n + 1)2; that is, n2 more functions than the three-dimensional case. However, the main difficulty in the four-dimensional analysis concerns the evaluation of the three separation constants appearing during the separation process. One of them can be extracted from the corresponding theory of the hyper-sphero-conal system, but the other two constants are obtained via a much more complicated procedure than the three-dimensional case. In fact, the solution process exhibits specific nonlinearities of polynomial type, itemized for every class and every degree. An example of this procedure is demonstrated in detail in order to make the process clear. Finally, the hyper-ellipsoidal harmonics are given as the product of four identical hyper-Lamé functions, each one defined in its own domain, which are explicitly calculated and tabulated for every degree less than five.

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigenfunctions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour’s ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

As originally introduced, the Einstein, Podolsky and Rosen (EPR) phenomenon was the ability of one party (Alice) to steer, by her choice between two measurement settings, the quantum system of another party (Bob) into two distinct ensembles of pure states. As later formalized as a quantum information task, EPR-steering can be shown even when the distinct ensembles comprise mixed states, provided they are pure enough and different enough. Consider the scenario where Alice and Bob each have a qubit and Alice performs dichotomic projective measurements. In this case, the states in the ensembles to which she can steer form the surface of an ellipsoid in Bob’s Bloch ball. Further, let the steering ellipsoid have nonzero volume (as it must if the qubits are entangled). It has previously been shown that if Alice’s first measurement setting yields an ensemble comprising two pure states, then this, plus any one other measurement setting, will demonstrate EPR-steering. Here we consider what one can say if the ensemble from Alice’s first setting contains only one pure state , occurring with probability p p . Using projective geometry, we derive the necessary and sufficient condition analytically for Alice to be able to demonstrate EPR-steering of Bob’s state using this and some second setting, when the two ensembles from these lie in a given plane. Based on this, we show that, for a given , if p p is high enough [ ] then any distinct second setting by Alice is sufficient to demonstrate EPR-steering. Similarly, we derive a such that is necessary for Alice to demonstrate EPR-steering using only the first setting and some other setting. Moreover, the expressions we derive are tight; for spherical steering ellipsoids, the bounds coincide: .

Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman-Kundu-Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulerʼs inequality for the circumradius and inradius of a triangle.

The integrated hydrobulging of stainless-steel prolate ellipsoids from preforms with two thicknesses was investigated. The produced ellipsoids were closed with two 16 mm thick closures and had nominal semiminor and semimajor axes of 89 and 125 mm, respectively. The ellipsoidal preforms comprised eight conical segments inscribed inside the target perfect ellipsoid. The four end and middle segments of the preforms had nominal thicknesses of 0.67 and 0.83 mm, respectively. The hydrobulging of these preforms was explored analytically and numerically and was compared with that of prolate ellipsoids with constant thickness. Two nominally identical ellipsoidal preforms were fabricated, measured, and hydrobulged to confirm the theoretical predictions. The results indicated that varying the preform thicknesses is an efficient method of overcoming insufficient hydrobulging of the ends of prolate ellipsoids in other methods. Moreover, hydrobulging instability can be effectively monitored by measuring geometric dimensions, such as axial height.

The emergence of 3D Gaussian splatting (3DGS) has greatly accelerated rendering in novel view synthesis. Unlike neural implicit representations like neural radiance fields (NeRFs) that represent a 3D scene with position and viewpoint-conditioned neural networks, 3D Gaussian splatting utilizes a set of Gaussian ellipsoids to model the scene so that efficient rendering can be accomplished by rasterizing Gaussian ellipsoids into images. Apart from fast rendering, the explicit representation of 3D Gaussian splatting also facilitates downstream tasks like dynamic reconstruction, geometry editing, and physical simulation. Considering the rapid changes and growing number of works in this field, we present a literature review of recent 3D Gaussian splatting methods, which can be roughly classified by functionality into 3D reconstruction, 3D editing, and other downstream applications. Traditional point-based rendering methods and the rendering formulation of 3D Gaussian splatting are also covered to aid understanding of this technique. This survey aims to help beginners to quickly get started in this field and to provide experienced researchers with a comprehensive overview, aiming to stimulate future development of the 3D Gaussian splatting representation.

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the virtues of the ellipse and her higher-dimensional cousins for both these purposes in a variety of contexts, including linear models, multivariate linear models and mixed-effect models. We emphasize the strong relationships among statistical methods, matrix-algebraic solutions and geometry that can often be easily understood in terms of ellipses.

A Hilbert plane is a model of the hyperbolic plane if and only if at least one circle is a quadric. Further, a Hilbert plane is a model of the hyperbolic plane if and only if at least two confocal ellipses are quadrics, or at least one ellipse with numerical eccentricity greater than 1/3 is a quadric.

We establish novel Hermite-Hadamard-type inequalities for the product of two strongly -convex functions defined on balls and ellipsoids in multidimensional Euclidean spaces. Additionally, we investigate mappings associated with these inequalities and explore their applications. Our results generalize several existing findings in the literature.