New full relativistic escape velocity and new Hubble related equation for the universe

The escape velocity derived from general relativity coincides with the Newtonian one. However, the Newtonian escape velocity can only be a good approximation when v ≪ c is sufficient to break free of the gravitational field of a massive body, as it ignores higher-order terms of the relativistic kinetic energy Taylor series expansion. Consequently, it does not work for a gravitational body with a radius at which v is close to c such as a black hole. To address this problem, we revisit the concept of relativistic mass, abandoned by Einstein, and derive what we call a full relativistic escape velocity. This approach leads to a new escape radius, where ve = c equal to a half of the Schwarzschild radius. Furthermore, we show that one can derive the Friedmann equation for a critical universe from the escape velocity formula from general relativity theory. We also derive a new equation for a flat universe based on our full relativistic escape velocity formula. Our alternative to the Friedmann formula predicts exactly twice the mass density in our (critical) universe as the Friedmann equation after it is calibrated to the observed cosmological redshift. Our full relativistic escape velocity formula also appears more consistent with the uniqueness of the Planck mass (particle) than the general relativity theory: whereas the general relativity theory predicts an escape velocity above c for the Planck mass at a radius equal to the Planck length, our model predicts an escape velocity c in this case.

The Fixed Angle Scattering Problem with a First-Order Perturbation

We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by 2n measurements up to a natural gauge. We also show that one can recover the full first-order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and Salo (SIAM J Math Anal 52(6):5467–5499, 2020) and (Inverse Probl 36(3):035005, 2020) to Hamiltonians with first-order perturbations, and it is based on wave equation methods and Carleman estimates.

Tolerance, Cultural Diversity and Economic Growth: Evidence from Dynamic Panel Data Analysis

This study aims to examine the impact of social tolerance of cultural diversity, and the ability to speak widely spoken languages, on economic performance. Based on the literature, the evidence is still controversial and unclear. Therefore, the study used panel data relating to (99) non-English speaking economies during the time period between 2009 and 2017. Following the augmented Solow model approach, the related equation was expanded, in this study, to include (besides human capital) social tolerance, the English language (as a lingua franca) and the level of openness. The model was estimated using the two-step system GMM approach. The results show that social tolerance of diversity and English language competence have a positive, but insignificant impact on the economy. Regarding policy implications, government and decision-makers can avoid the costs deriving from cultural diversity by adopting democratic and effective institutions that aim to achieve cultural justice and recognition, which, in turn, enhance the level of tolerance, innovation and productivity in the economy. Moreover, to ease intercultural communication within heterogeneous communities, it is necessary to invest in enhancing the quality of second language education which is necessary to make society more tolerant and the country more open to the global economy.

A COMPLEX ORDER MODEL OF ATRIAL ELECTRICAL PROPAGATION FROM FRACTAL POROUS CELL MEMBRANE

Cardiac tissue is characterized by structural and cellular heterogeneities that play an important role in the cardiac conduction system. Under persistent atrial fibrillation (persAF), electrical and structural remodeling occur simultaneously. The classical mathematical models of cardiac electrophysiological showed remarkable progress during recent years. Among those models, it is of relevance the standard diffusion mathematical equation, that considers the myocardium as a continuum. However, the modeling of structural properties and their influence on electrical propagation still reveal several limitations. In this paper, a model of cardiac electrical propagation is proposed based on complex order derivatives. By assuming that the myocardium has an underlying fractal process, the complex order dynamics emerges as an important modeling option. In this perspective, the real part of the order corresponds to the fractal dimension, while the imaginary part represents the log-periodic corrections of the fractal dimension. Indeed, the imaginary part in the derivative implies characteristic scales within the cardiac tissue. The analytical and numerical procedures for solving the related equation are presented. The sinus rhythm and persAF conditions are implemented using the Courtemanche formalism. The electrophysiological properties are measured and analyzed on different scales of observation. The results indicate that the complex order modulates the electrophysiology of the atrial system, through the variation of its real and imaginary parts. The combined effect of the two components yields a broad range of electrophysiological conditions. Therefore, the proposed model can be a useful tool for modeling electrical and structural properties during cardiac conduction.

Extending the Debye scattering equation for diffraction from a cylindrically averaged group of atoms: detecting molecular orientation at an interface

The Debye scattering equation is now over 100 years old and has been widely used to interpret diffraction patterns from randomly oriented groups of atoms. The present work develops and applies a related equation that calculates diffraction intensity from groups of atoms randomly oriented about a fixed axis, a scenario that occurs when molecules are oriented at an interface by the presentation of a binding motif as in antibody binding. Using an example biomolecule, the high level of sensitivity of the diffraction pattern to the orientation of the molecule and to the direction of the incident beam is shown. The use of the method is proposed not only for determining the orientation of molecules in biosensors and at membrane interfaces, but also for determining molecular conformation without the need for crystallization.

Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation

New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth-order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.

The effect of rock composition on muon tomography measurements

In recent years, the use of radiographic inspection with cosmic-ray muons has spread into multiple research and industrial fields. This technique is based on the high-penetration power of cosmogenic muons. Specifically, it allows the resolution of internal density structures of large-scale geological objects through precise measurements of the muon absorption rate. So far, in many previous works, this muon absorption rate has been considered to depend solely on the density of traversed material (under the assumption of a standard rock) but the variation in chemical composition has not been taken seriously into account. However, from our experience with muon tomography in Alpine environments, we find that this assumption causes a substantial bias in the muon flux calculation, particularly where the target consists of high {Z2∕A} rocks (like basalts and limestones) and where the material thickness exceeds 300 m. In this paper, we derive an energy loss equation for different minerals and we additionally derive a related equation for mineral assemblages that can be used for any rock type on which mineralogical data are available. Thus, for muon tomography experiments in which high {Z2∕A} rock thicknesses can be expected, it is advisable to plan an accompanying geological field campaign to determine a realistic rock model.

Numerical Modeling of Thermal-Dependent Creep Behavior of Soft Clays under One-Dimensional Condition

Creep is a common phenomenon for soft clays. The paper focuses on investigating the influence of temperature on the time-dependent stress-strain evolution. For this purpose, the temperature-dependent creep behavior for the soft clay has been investigated based on experimental observations. A thermally related equation is proposed to bridge the thermal creep coefficient with temperature. By incorporating the equation to a selected one-dimensional (1D) elastic viscoplastic (EVP) model, a thermal creep-based EVP model was developed which takes into account the influence of temperature on creep. Simulations of oedometer tests on reconstituted clay are made through coupled consolidation analysis. The bonding effect of the soil structure on compressive behavior for intact clay is studied. By incorporating the influence of the soil structure, the thermal creep EVP model is extended for intact clay. Experimental predictions for thermal creep oedometer tests are simulated at different temperatures and compared to that obtained from reconstituted clay. The results show that the influence of temperature on the creep behavior for intact clay is significant, and the model, this paper proposed, can successfully reproduce the thermal creep behavior of the soft clay under the 1D loading condition.

The effect of rock composition on muon tomography measurements

In recent years, the use of radiographic inspection with cosmic-ray muons has spread into multiple research and industrial fields. This technique is based on the high-penetration power of cosmogenic muons. Specifically, it allows the resolution of internal density structures of large scale, geological objects through precise measurement of the muon absorption rate. So far in many previous works, this muon absorption rate has been considered to depend solely on the density of traversed material (under the assumption of a standard rock) but the variation in chemical composition has not been taken seriously into account. However, from our experience with muon tomography in Alpine environments we find that this assumption causes a substantial bias on the muon flux calculation, particularly where the target consists of high {Z2/A} (like basalts) or low {Z2/A} (e.g. dolomites) rocks and where the material thickness exceeds 300 m. In this paper we derive an energy loss equation for different minerals and we additionally derive a related equation for mineral assemblages that can be used for any rock type on which mineralogical data is available. Thus, for muon tomography experiments in which high/low {Z2/A} rock thicknesses can be expected, it is advisable to plan an accompanying geological field campaign to determine a realistic rock model.

Oscillation Criteria and Spectrum of Self-Adjoint even Order Two-Term Differential Operators

In this paper we consider the oscillation and spectrum of a class of high order two-term differential operators. Using the oscillation and non-oscillation criteria of related equation obtained here, we describe some spectral information of differential operators. In particular, the conditions which guarantee that any self-adjoint extension of differential operators has spectrum discrete and bounded below are given.