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1. Gravitational Interaction in the Chimney Lattice Universe

   https://doi.org/10.3390/universe7040101  

   Eingorn, Maxim; McLaughlin, Andrew; Canay, Ezgi; Brilenkov, Maksym; Zhuk, Alexander (April 2021, Universe)

   null (Ed.)

   We investigate the influence of the chimney topology T×T×R of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision. 

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2. Gravitation in the Space with Chimney Topology

   https://doi.org/10.3390/ECU2021-09295  

   Eingorn, Maxim; McLaughlin, Andrew; Canay, Ezgi; Brilenkov, Maksym; Zhuk, Alexander (January 2021, Physical Sciences Forum)

   null (Ed.)

   Searching for possible indicators of spatial topology of the Universe in the Cosmic Microwave Background data, one recognizes a quite promising interpretation which suggests that the shape of the space manifests itself in the form of anomalies in the large angular scale observations, such as the quadrupole and octopole alignment. Motivated by the presumptive existence of such a tempting connection, we study the chimney topology, T×T×R, which belongs to the class of toroidal topologies with a preferred direction. The infinite axis in this case may be attributed to the preferred axis of the aforementioned quadrupole and octopole alignment. We investigate the gravitational aspects of such a configuration. Namely, we reveal the form of the gravitational potential sourced by point-like massive bodies. Starting from the perturbed Einstein equations, which ensure the proper demonstration of relativistic effects, one can derive the Helmholtz equation for the scalar perturbation (gravitational potential). Through distinct alternative methods, we present the physically meaningful nontrivial exact solutions of this equation. Our approach excludes any presumptions regarding the spatial distribution of gravitating sources. We show that the particular solution that appears in the form of summed Yukawa potentials is indeed very convenient for the use in numerical calculations, in the sense that it provides the desired accuracy with fewer terms in the series. 

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3. Exact and approximate energy sums in potential wells

   https://doi.org/10.1088/1751-8121/ab69a6  

   Berry, M. V.; Burke, Kieron (February 2020, Journal of Physics A: Mathematical and Theoretical)

   Abstract Sums of theNlowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regimeN  ≫  1. Euler-Maclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levelsN  +  1,N  +  2…. For the harmonic oscillator and the particle in a box, the formula is exact. For wells where the levels are known approximately (e.g. as a WKB series), with the higher levels being more accurate, the formula improves accuracy by avoiding the lower levels. For a linear potential, the formula gives the first Airy zero with an error of order 10⁻⁷. For the Pöschl–Teller potential, regularisation is not immediately applicable but the energy sum can be calculated exactly; its semiclassical approximation depends on howNand the well depth are linked. In more dimensions, the Euler–Maclaurin technique is applied to give an analytical formula for the energy sum for a free particle on a torus, using levels determined by the smoothed spectral staircase plus some oscillatory corrections from short periodic orbits. 

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4. Modified Representations for the Close Evaluation Problem

   https://doi.org/10.3390/mca26040069  

   Carvalho, Camille (December 2021, Mathematical and Computational Applications)

   When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem’s solution. Similar to Gauss’s law used to modify Laplace’s double-layer potential, we use modified representations of Laplace’s single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Some techniques have been developed in the context of the representation formula or using interpolation techniques. We provide alternative modified representations of the layer potentials directly (or when only one density is at stake). Several numerical examples illustrate the efficiency of the technique in two and three dimensions. 

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5. Electromagnetic–gravitational perturbations of Kerr–Newman black hole

   https://doi.org/10.1142/S0218271825400036  

   Giorgi, Elena (January 2025, International Journal of Modern Physics D)

   Black holes are important objects in our understanding of the universe, as they represent the extreme nature of General Relativity. The Kerr–Newman black hole is the most general asymptotically flat black hole solution and its stability properties have long been elusive due to the interaction between gravitational and electromagnetic radiations. We illustrate the main conjectures regarding the stability problem of known black hole solutions and present some recent theorems regarding the evolution of the Kerr–Newman black holes to coupled perturbations. 

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