On A New Formulation of Micro-phenomena: Basic Principles, Stationary Fields And Beyond

In a series of essays, beginning with this article, we are going to develop a new formulation of micro-phenomena based on the principles of reality and causality. The new theory provides with us a new depiction of micro-phenomena assuming an unified concept of information, matter and energy. So, we suppose that in a definite micro-physical context (including other interacting particles), each particle is enfolded by a probability field whose existence is contingent upon the existence of the particle, but it can locally affect the physical status of the particle in a context-dependent manner. The dynamics of the whole particle-field system obeys deterministic equations in a manner that when the particle is subjected to a conservative force, the field also experiences a conservative complex force which its form is determined by the dynamics of particle. So, the field is endowed with a given amount of energy, but its value is contingent upon the physical conditions the particle is subjected to. Based on the energy balance of the particle and its associated field, we argue why the field has a probabilistic objective nature. In such a way, the basic elements of this new formulation, its application for some stationary states and its nonlinear generalization for conservative systems are discussed here.Comment: 35 pages, 5 figures, 3 appendice

Excited states of spherium

We report analytic solutions of a recently discovered quasi-exactly solvable model consisting of two electrons, interacting {\em via} a Coulomb potential, but restricted to remain on the surface of a $\mathcal{D}$-dimensional sphere. Polynomial solutions are found for the ground state, and for some higher ($L\le3$) states. Kato cusp conditions and interdimensional degeneracies are discussed.Comment: 6 pages, 2 figures, to appear in Mol. Phy

Wave reflection and transmission in multiply stented blood vessels

Closed circulatory systems display an exquisite balance between vascular elasticity and viscous fluid effects, to induce pulse-smoothing and avoid resonance during the cardiac cycle. Stents in the arterial tree alter this balance through stiffening and because a periodic structure is introduced, capable of interacting with the fluid in a complex way. While the former feature has been investigated, the latter received no attention so far. But periodic structures are the building blocks of metamaterials, known for their ‘non-natural’ behaviour. Thus, the investigation of a stent's periodic microstructure dynamical interactions is crucial to assess possible pathological responses. A one-dimensional fluid–structure interaction model, simple enough to allow an analytical solution for situations of interest involving one or two interacting stents, is introduced. It is determined: (i) whether or not frequency bands exist in which reflected blood pulses are highly increased and (ii) if these bands are close to the characteristic frequencies of arteries and finally, (iii) if the internal structure of the stent can sensibly affect arterial blood dynamics. It is shown that, while the periodic structure of an isolated stent can induce anomalous reflection only in pathological conditions, the presence of two interacting stents is more critical, and high reflection can occur at frequencies not far from the physiological values.This work was supported by the ERC advanced grant ERC-2013-ADG-340561-INSTABILITIES (2014– 2019). T.K.P. acknowledges support from the aforementioned programme for the period from 10 May 2016 to 13 September 2016. A.B.M. acknowledges the support from the aforementioned ERC Advanced Grant during his Visiting Professorship at Trento University in 2016

Numerical solutions of random mean square Fisher-KPP models with advection

[EN] This paper deals with the construction of numerical stable solutions of random mean square Fisher-Kolmogorov-Petrosky-Piskunov (Fisher-KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-PCasabán Bartual, MC.; Company Rossi, R.; Jódar Sánchez, LA. (2020). Numerical solutions of random mean square Fisher-KPP models with advection. Mathematical Methods in the Applied Sciences. 43(14):8015-8031. https://doi.org/10.1002/mma.5942S801580314314FISHER, R. A. (1937). THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES. Annals of Eugenics, 7(4), 355-369. doi:10.1111/j.1469-1809.1937.tb02153.xBengfort, M., Malchow, H., & Hilker, F. M. (2016). The Fokker–Planck law of diffusion and pattern formation in heterogeneous environments. Journal of Mathematical Biology, 73(3), 683-704. doi:10.1007/s00285-016-0966-8Okubo, A., & Levin, S. A. (2001). Diffusion and Ecological Problems: Modern Perspectives. Interdisciplinary Applied Mathematics. doi:10.1007/978-1-4757-4978-6SKELLAM, J. G. (1951). RANDOM DISPERSAL IN THEORETICAL POPULATIONS. Biometrika, 38(1-2), 196-218. doi:10.1093/biomet/38.1-2.196Aronson, D. G., & Weinberger, H. F. (1975). Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial Differential Equations and Related Topics, 5-49. doi:10.1007/bfb0070595Aronson, D. ., & Weinberger, H. . (1978). Multidimensional nonlinear diffusion arising in population genetics. Advances in Mathematics, 30(1), 33-76. doi:10.1016/0001-8708(78)90130-5Weinberger, H. F. (2002). On spreading speeds and traveling waves for growth and migration models in a periodic habitat. Journal of Mathematical Biology, 45(6), 511-548. doi:10.1007/s00285-002-0169-3Weinberger, H. F., Lewis, M. A., & Li, B. (2007). Anomalous spreading speeds of cooperative recursion systems. Journal of Mathematical Biology, 55(2), 207-222. doi:10.1007/s00285-007-0078-6Liang, X., & Zhao, X.-Q. (2006). Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Communications on Pure and Applied Mathematics, 60(1), 1-40. doi:10.1002/cpa.20154E. Fitzgibbon, W., Parrott, M. E., & Webb, G. (1995). Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete & Continuous Dynamical Systems - A, 1(1), 35-57. doi:10.3934/dcds.1995.1.35Kinezaki, N., Kawasaki, K., & Shigesada, N. (2006). Spatial dynamics of invasion in sinusoidally varying environments. Population Ecology, 48(4), 263-270. doi:10.1007/s10144-006-0263-2Jin, Y., Hilker, F. M., Steffler, P. M., & Lewis, M. A. (2014). Seasonal Invasion Dynamics in a Spatially Heterogeneous River with Fluctuating Flows. Bulletin of Mathematical Biology, 76(7), 1522-1565. doi:10.1007/s11538-014-9957-3Faou, E. (2009). Analysis of splitting methods for reaction-diffusion problems using stochastic calculus. Mathematics of Computation, 78(267), 1467-1483. doi:10.1090/s0025-5718-08-02185-6Doering, C. R., Mueller, C., & Smereka, P. (2003). Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality. Physica A: Statistical Mechanics and its Applications, 325(1-2), 243-259. doi:10.1016/s0378-4371(03)00203-6Siekmann, I., Bengfort, M., & Malchow, H. (2017). Coexistence of competitors mediated by nonlinear noise. The European Physical Journal Special Topics, 226(9), 2157-2170. doi:10.1140/epjst/e2017-70038-6McKean, H. P. (1975). Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov. Communications on Pure and Applied Mathematics, 28(3), 323-331. doi:10.1002/cpa.3160280302Berestycki, H., & Nadin, G. (2012). Spreading speeds for one-dimensional monostable reaction-diffusion equations. Journal of Mathematical Physics, 53(11), 115619. doi:10.1063/1.4764932Cortés, J. C., Jódar, L., Villafuerte, L., & Villanueva, R. J. (2007). Computing mean square approximations of random diffusion models with source term. Mathematics and Computers in Simulation, 76(1-3), 44-48. doi:10.1016/j.matcom.2007.01.020Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling, 40(21-22), 9362-9377. doi:10.1016/j.apm.2016.06.017Sarmin, E. N., & Chudov, L. A. (1963). On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method. USSR Computational Mathematics and Mathematical Physics, 3(6), 1537-1543. doi:10.1016/0041-5553(63)90256-8Sanz-Serna, J. M., & Verwer, J. G. (1989). Convergence analysis of one-step schemes in the method of lines. Applied Mathematics and Computation, 31, 183-196. doi:10.1016/0096-3003(89)90118-5Calvo, M. P., de Frutos, J., & Novo, J. (2001). Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations. Applied Numerical Mathematics, 37(4), 535-549. doi:10.1016/s0168-9274(00)00061-1Cox, S. M., & Matthews, P. C. (2002). Exponential Time Differencing for Stiff Systems. Journal of Computational Physics, 176(2), 430-455. doi:10.1006/jcph.2002.6995De la Hoz, F., & Vadillo, F. (2016). Numerical simulations of time-dependent partial differential equations. Journal of Computational and Applied Mathematics, 295, 175-184. doi:10.1016/j.cam.2014.10.006Company, R., Egorova, V. N., & Jódar, L. (2018). Conditional full stability of positivity-preserving finite difference scheme for diffusion–advection-reaction models. Journal of Computational and Applied Mathematics, 341, 157-168. doi:10.1016/j.cam.2018.02.031Kaczorek, T. 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Fluid flow through porous media using distinct element based numerical method

Many analytical and numerical methods have been developed to describe and analyse fluid flow through the reservoir’s porous media. The medium considered by most of these models is continuum based homogeneous media. But if the formation is not homogenous or if there is some discontinuity in the formation, most of these models become very complex and their solutions lose their accuracy, especially when the shape or reservoir geometry and boundary conditions are complex. In this paper, distinct element method (DEM) is used to simulate fluid flow in porous media. The DEM method is independent of the initial and boundary conditions, as well as reservoir geometry and discontinuity. The DEM based model proposed in this study is appeared to be unique in nature with capability to be used for any reservoir with higher degrees of complexity associated with the shape and geometry of its porous media, conditions of fluid flow, as well as initial and boundary conditions. This model has first been developed by Itasca Consulting Company and is further improved in this paper. Since the release of the model by Itasca, it has not been validated for fluid flow application in porous media, especially in case of petroleum reservoir. In this paper, two scenarios of linear and radial fluid flow in a finite reservoir are considered. Analytical models for these two cases are developed to set a benchmark for the comparison of simulation data. It is demonstrated that the simulation results are in good agreement with analytical results. Another major improvement in the model is using the servo controlled walls instead of particles to introduce tectonic stresses on the formation to simulate more realistic situations. The proposed model is then used to analyse fluid flow and pressure behaviour for hydraulically induced fractured and naturally fractured reservoir to justify the potential application of the model

Profile Prediction and Fabrication of Wet-Etched Gold Nanostructures for Localized Surface Plasmon Resonance

Dispersed nanosphere lithography can be employed to fabricate gold nanostructures for localized surface plasmon resonance, in which the gold film evaporated on the nanospheres is anisotropically dry etched to obtain gold nanostructures. This paper reports that by wet etching of the gold film, various kinds of gold nanostructures can be fabricated in a cost-effective way. The shape of the nanostructures is predicted by profile simulation, and the localized surface plasmon resonance spectrum is observed to be shifting its extinction peak with the etching time