/library/oar/handle/123456789/404 Wed, 05 Nov 2025 18:22:22 GMT 2025-11-05T18:22:22Z /library/oar/handle/123456789/140818 Title: Approximating photon trajectories in spherically symmetric spacetimes Authors: Sultana, Joseph Abstract: In this paper we use the Homotopy analysis method to obtain an analytic approximation for the entire photon trajectory in the Schwarzschild spacetime. This is usually expressed exactly in terms of an elliptic integral. We compare our approximation with other formulae found in the literature, which were specifically obtained for the Schwarzschild solution. Unlike some of these formulae, our approximation can be applied and maintains a good accuracy for emission point close to the event horizon and also for emission angles close to and greater than π/2. We show that our method can easily be applied to other spherically symmetric solutions such as the Reissner-Nordström solution. Such an approximation would be useful when accurate determination of the light trajectories around compact objects is required without the need to revert to time consuming numerical integration of elliptic integrals. Mon, 01 Jan 2024 00:00:00 GMT /library/oar/handle/123456789/140818 2024-01-01T00:00:00Z /library/oar/handle/123456789/139154 Title: Reconstruction of shear force in atomic force microscopy from measured displacement of the cone-shaped cantilever tip Authors: Hasanov, Alemdar; Kawano, Alexandre; Baysal, Onur Abstract: In this paper, a dynamic model of reconstruction of the shear force g(t) in the Atomic Force Microscopy (AFM) cantilever tip-sample interaction is proposed. The interaction of the cone-shaped cantilever tip with the surface of the specimen (sample) is modeled by the damped Euler-Bernoulli beam equation ρ A ( x ) u t t + μ ( x ) u t + ( r ( x ) u x x + κ ( x ) u x x t ) x x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) , subject to the following initial, u(x,0)=0, u t ( x , 0 ) = 0 and boundary, u(0,t)=0, u x ( 0 , t ) = 0 , ( r ( x ) u x x ( x , t ) + κ ( x ) u x x t ) x = ℓ = M ( t ) , ( − ( r ( x ) u x x + κ ( x ) u x x t ) x ) x = ℓ = g ( t ) conditions, where M ( t ) : = 2 h cos ⁡ θ   g ( t ) / π is the momentum generated by the transverse shear force g(t). For the reconstruction of g(t) the measured displacement ν ( t ) : = u ( ℓ , t ) is used as an additional data. The least square functional J ( F ) = 1 2 ∥ u ( ℓ , ⋅ ) − ν ∥ L 2 ( 0 , T ) 2 is introduced and an explicit gradient formula for the Fr\'echet derivative through the solution of the adjoint problem is derived. This allows to construct a gradient based numerical algorithm for the reconstructions of the shear force from noise free as well as from random noisy measured output ν ( t ) . Computational experiments show that the proposed algorithm is very fast and robust. This allows to develop a numerical "gadget" for computational experiments of generic AFMs. Mon, 01 Jan 2024 00:00:00 GMT /library/oar/handle/123456789/139154 2024-01-01T00:00:00Z /library/oar/handle/123456789/139000 Title: Turnaround radius for charged particles in the Reissner-Nordström deSitter spacetime Authors: German, Ethan James; Sultana, Joseph Abstract: We investigate the turnaround radius of the Reissner–Nordström deSitter Spacetime and how the turnaround radius changes if a test particle carries charge. We also consider the Martínez–Troncoso–Zanelli (MTZ) solution of conformally coupled gravity and investigate how the turnaround radius changes for a scalar test charge. In both scalar and electric interaction cases we find that the Turnaround Radius depends on the particle’s energy. Mon, 01 Jan 2024 00:00:00 GMT /library/oar/handle/123456789/139000 2024-01-01T00:00:00Z /library/oar/handle/123456789/138980 Title: A sharp upper bound for the harmonious total chromatic number of graphs and multigraphs Authors: Abreu, Marién; Gauci, John Baptist; Mattiolo, Davide; Mazzuoccolo, Giuseppe; Romaniello, Federico; Rubio-Montiel, Christian; Traetta, Tommaso Abstract: A proper total colouring of a graph G is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edgeswith their colours, then no pair of colours appears twice. The smallest number of colours for it to exist is called the harmonious total chromatic number of G, denoted by ht(G). Here, we give a general upper bound for ht(G) in terms of the order n of G. Our two main results are obvious consequences of the computation of the harmonious total chromatic number of the complete graph Kn and of the complete multigraph λKn, where λ is the number of edges joining each pair of vertices of Kn. In particular, Araujo-Pardo et al. have recently shown that 3/2 n ≤ ht(Kn)≤ 5/3 n + θ(1). In this paper, we prove that ht(Kn) = ⌈3/2 n⌉ except for ht(K₁) = 1 and ht(K₄) = 7; therefore, ht(G)≤ ⌈3/2 n⌉, for every graph G on n > 4 vertices. Finally, we extend such a result to the harmonious total chromatic number of the complete multigraph λKn and as a consequence show that ht(G) ≤ (λ-1)(2⌈n/2⌉-1)+⌈3n/2⌉ for n > 4, where G is a multigraph such that λ is the maximum number of edges between any two vertices. Mon, 01 Jan 2024 00:00:00 GMT /library/oar/handle/123456789/138980 2024-01-01T00:00:00Z