From P/E Ratio to Fuzzy Infinite Spreadsheet—Mathematically Rigorous Derivations of the Zeroth and the First Order Solutions of Rate of Return

The P/E Ratio (Price/Earning) is one of the most popular concepts in stock analysis, yet its exact interpretation is lacking. Most stock investors know the P/E Ratio as a financial indicator with the useful characteristics of being relatively time-invariant. In this paper, a rigorous mathematical derivation of the P/E Ratio is presented. The derivation shows that, in addition to its assumptions, the P/E Ratio can be considered the zeroth order solution to the rate of return on investment. The commonly used concept of the Capitalization Rate (Cap Rate = Net Income / Price) in real estate investment analysis can also be similarly derived as the zeroth order solution of the rate of return on real estate investment. This paper also derives the first order solution to the rate of return (Return = Dividend/Price + Growth) with its assumptions. Both the zeroth and the first order solutions are derived from the exact future accounting equation (Cash Return = Sum of Cash Flow + Cash from Resale). The exact equation has been used in the derivation of the exact solution of the rate of return. Empirically, as an illustration of an actual case, the rates of return are 3%, 73%, and 115% for a stock with 70% growth rate for, respectively, the zeroth order, the first order, and the exact solution to the rate of return; the stock doubled its price in 2004. This paper concludes that the zero-th, the first order, and the exact solution of the rate of return all can be derived mathematically from the same exact equation, which, thus, forms a rigorous mathematical foundation for investment analysis, and that the low order solutions have the very practical use in providing the analytically calculated initial conditions for the iterative numerical calculation for the exact solution. The solution of value belongs to recently classified Culture Level Quotient CLQ = 10 and is in the process of being updated by fuzzy logic with its range of tolerance for predicting market crashes to advance to CLQ = 2.

An Analytical Approach to Obtaining the Stress Field of a Cylindrical Shell with a Circular Hole under Tension

The problem of a cylindrical shell with a circular hole under uniaxial tension is considered. The main obstacle of solving this problem is the necessity to find such coefficients in the expansion of the solution into a sum of basis functions, for which this solution satisfies the boundary conditions. The study of the classical works led to understanding that none of the so far proposed approaches can be considered successfully, and the results of these approaches differ, so it is not clear, which results can be used as a basis. In the present paper, a new analytical approach to studying this issue is proposed. It allows expanding the range of applicability of the solution and gives the opportunity for the analytical study of the stress state. The idea consists in expanding each of the basis functions in a Fourier series by dividing the variables, which allows obtaining explicitly an infinite system of algebraic equations for finding coefficients. One of the important steps of this research is that the authors were able to prove which exact equation is a linear combination of the others and exclude, which made it possible to compose a reduced system for finding unknown coefficients. The proposed approach, in contrast to most classical works, does not impose mathematical restrictions on the values of the main parameter characterizing the cylindrical shell. The existing restrictions are of mechanical nature, as larger cutouts require another model. Moreover, the numerical results obtained by the new method are presented in a fairly complete manner and they are compared with the results of the classical works.

On Linear Instability of Atmospheric Quasi-hydrostatic Equations in Response to Small Shortwave Perturbations

A set of 3-dimensional atmospheric-dynamics equations with quasi-hydrostatic approximation is proposed and justified with the practical goal to optimize atmospheric modelling at scales ranging from meso meteorology to global climate. Sound waves are filtered by applying the quasi-hydrostatic approximation. In the closed system of hydro/thermodynamic equations, the inertial forces are negligibly small compared to gravity forces, and the asymptotically exact equation for vertical velocity is obtained. Investigation of the stability of solutions to this system in response to small shortwave perturbations has shown that solutions have the property of shortwave instability. There are situations when the increment of the perturbation amplitude tends to infinity, corresponding to absolute instability. It means that the Cauchy problem for such equations may be ill-posed. Its formulation can become conditionally correct if solutions are sought in a limited class of sufficiently smooth functions whose Fourier harmonics tend to zero reasonably quickly when the wavelengths of the perturbations approach zero. Thus, the numerical scheme for the quasi-hydrostatic equations using the finite-difference method requires an adequately selected pseudo-viscosity to eliminate the instability caused by perturbations with wavelengths of the order of the grid size. The result is useful for choosing appropriate vertical and horizontal grid sizes for modelling to avoid shortwave instability associated with the property of the system of equations. Implementation of pseudo-viscosities helps to smoothen or suppress the perturbations that occur during modelling.

Incidence of Severe Acute Respiratory Syndrome Coronavirus-2 infection among previously infected or vaccinated employees

Introduction: The protective effect of previous infection versus vaccination is poorly studied. Among a clinical laboratory that has been conducting routine workforce screening since the beginning of the pandemic, we aimed to assess the relative risk of Severe Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2) infection among individuals who were SARS-CoV-2 naive, previously infected, or vaccinated. Methods: Using an electronic laboratory information system, employees were divided into three groups: (1) SARS-CoV-2 naive and unvaccinated, (2) previous SARS-CoV-2 infection, and (3) vaccinated. Person-days were measured from the date of the employee first test and truncated at the end of the observation period. SARS-CoV-2 infection was defined as two positive SARS-CoV-2 PCR tests in a 30-day period. Individuals with fewer than 14 days of follow up were excluded. Incidence estimates and the 95% confidence intervals were calculated using the Poisson Exact equation. The incidence rate ratio (IRR) was used as a measure of association between groups. Analyses were performed on StataSE (StataCorp, College Station, TX). Results: We identified 4313, 254 and 739 employee records for groups 1, 2, and 3, respectively. The median age of employees was 29.0 years (interquartile range: 23.6, 39.9). During the observation period, 254, 0, and 4 infections were identified among groups 1, 2, and 3, respectively. Group 1 had an incidence of 25.9 per 100 person-years (95% CI: 22.8-29.3). Group 2 had an incidence of 0 per 100 person-years (95% CI: 0-5.0). Group 3 had an incidence of 1.6 per 100 person-years (95% CI: 0.04-4.2). The IRR of reinfection among those with previous infection compared to SARS-CoV-2 naive was 0 (95% CI: 0-0.19). The IRR of those vaccinated compared to SARS-CoV-2 naive was 0.06 (95% CI: 0.02-0.16). The IRR of those vaccinated compared to prior SARS-CoV-2 was 0 (95% CI: 0-4.98). Conclusion: Previous SARS-CoV-2 infection and vaccination for SARS-CoV-2 were associated with decreased risk for infection or re-infection with SARS-CoV-2 in a routinely screened workforce. The was no difference in the infection incidence between vaccinated individuals and individuals with previous infection. Further research is needed to determine whether our results are consistent with the emergence of new SARS-CoV-2 variants.

On the Generation of Harmonics by the Non-Linear Buckling of an Elastic Beam

The Euler–Bernoulli theory of beams is usually presented in two forms: (i) in the linear case of a small slope using Cartesian coordinates along and normal to the straight undeflected position; and (ii) in the non-linear case of a large slope using curvilinear coordinates along the deflected position, namely, the arc length and angle of inclination. The present paper starts with the exact equation in a third form, that is, (iii) using Cartesian coordinates along and normal to the undeflected position like (i), but allowing exactly the non-linear effects of a large slope like (ii). This third form of the equation of the elastica shows that the exact non-linear shape is a superposition of linear harmonics; thus, the non-linear effects of a large slope are equivalent to the generation of harmonics of a linear solution for a small slope. In conclusion, it is shown that: (i) the critical buckling load is the same in the linear and non-linear cases because it is determined by the fundamental mode; (ii) the buckled shape of the elastica is different in the linear and non-linear cases because non-linearity adds harmonics to the fundamental mode. The non-linear shape of the elastica, for cases when powers of the slope cannot be neglected, is illustrated for the first four buckling modes of cantilever, pinned, and clamped beams with different lengths and amplitudes.

NONLINEAR DIFFUSION EQUATION WITH A PERTURBED CONVECTIONTERM: POTENTIAL SYMMETRIES WITH RESPECT TO THE SECOND CONSERVATION LAW

We consider the nonlinear diffusion equation with a perturbed convection term. The potential symmetries for the exact equation with respect to the second conservation law are classified. It is found that these exist only in the linear case. It is further shown that no nontrivial approximate potential symmetries of order one exists for the perturbed equation with respect to the other conservation law.

Detection and monitoring for cancer and abnormal vasculature by photoacoustic signal characterization of structural morphology

Photoacoustic systems can produce high-resolution, high-contracts images of vascular structures. To reconstruct images at very high-resolution, signals must be collected from many transducer locations, which can be time consuming due to limitations in transducer array technology, In this thesis a method is presented to discriminate between normal and abnormal tissue based on the structural morphology of vasculature and permits data to be acquired quickly. To demonstrate that the approach may be useful for cancer detection, a special simulator that produces photoacoustic signal from 3D models of vascular tissue is developed. Validation of the simulator is performed against a derived exact equation for finite-length cylindrical photoacoustic sources and through FEM models. Results show that is possible to differentiate tissue classed even when it is not possible to resolve individual blood vessels. Performance of the algorithm remains strong as the number of transducer locations decreases and in the presence of noise.

Detection and monitoring for cancer and abnormal vasculature by photoacoustic signal characterization of structural morphology

Photoacoustic systems can produce high-resolution, high-contracts images of vascular structures. To reconstruct images at very high-resolution, signals must be collected from many transducer locations, which can be time consuming due to limitations in transducer array technology, In this thesis a method is presented to discriminate between normal and abnormal tissue based on the structural morphology of vasculature and permits data to be acquired quickly. To demonstrate that the approach may be useful for cancer detection, a special simulator that produces photoacoustic signal from 3D models of vascular tissue is developed. Validation of the simulator is performed against a derived exact equation for finite-length cylindrical photoacoustic sources and through FEM models. Results show that is possible to differentiate tissue classed even when it is not possible to resolve individual blood vessels. Performance of the algorithm remains strong as the number of transducer locations decreases and in the presence of noise.

Salvesen’s Method for Added Resistance Revisited

Almost 4 years after the appearance of Salvesen–Tuck–Faltinsen (STF) strip theory (Salvesen et al., 1970, “Ship Motions and Sea Loads,” Annual Meeting of the Society of Naval Architecture and Marine Engineers (SNAME), New York, Nov. 12–13), Salvesen in 1974 published his popular method for calculation of added resistance (Salvesen, 1974, “Second-Order Steady State Forces and Moments on Surface Ships in Oblique Regular Waves,” Vol. 22; Salvesen, 1978, “Added Resistance of Ships in Waves,” J. Hydronautics, 12(1), pp. 24–34). His method is based on an exact near-field formulation; however, he applied the long-wave and the weak-scatterer assumptions to present his approximate method using the integrated quantities (hydrodynamic and geometrical coefficients). Considering the available computational powers in the 1970s, both of these assumptions were absolutely justifiable. The intention of this paper is to disseminate the results of a recent study at the Technical University of Denmark, whereby the Salvesen’s formulation has been revisited and the added resistance is computed from the original exact equation without invoking the weak-scatterer or the long-wave assumptions. This is performed using the solutions of the radiation and the scattering problems, obtained by a low-order boundary element method and the two-dimensional free-surface Green function inside our in-house STF theory implementation (Bingham and Amini-Afshar, 2020, DTU_Strip Theory Solver). The weak-scatterer assumption is then removed through a direct calculation of the x-derivatives of the velocity potentials and the normal vectors along the body. Knowing the velocity potentials over each panel, the long-wave assumption is also avoided by a piece-wise analytical integration of sectional Kochin Function (Kochin, 1936, “On the Wave Resistance and Lift of Bodies Submerged in Fluid,” Transactions of the Conference on the Theory of Wave Resistance, Moscow.). The presented results for five ship geometries testify that the correct treatment of the original equation is achieved only after both of the above-mentioned assumptions are removed. Implemented in this manner, Salvesen’s method proves to be relatively more accurate and robust than has been generally perceived during all these years.

A High Resolution Simulation of a Single Shock-Accelerated Particle

Particle drag models, which capture macro viscous and pressure effects, have been developed over the years for various flow regimes to enable cost effective simulations of particle-laden flows. The relatively recent derivation by Maxey and Riley has provided an exact equation of motion for spherical particles in a flow field based on the continuum assumption. Many models that have been simplified from these equations have provided reasonable approximations; however, the sensitivity of particle-laden flows to particle drag requires a very accurate model to simulate. To develop such a model, a 2D axisymmetric Navier-Stokes direct numerical simulation of a single particle in a transient, shock-driven flow field was conducted using the hydrocode FLAG. FLAG's capability to run arbitrary Lagrangian-Eulerian hydrodynamics coupled with solid mechanic models makes it an ideal code to capture the physics of the flow field around and in the particle as it is shock-accelerated -- a challenging regime to study. The goal of this work is twofold: to provide a validation for FLAG's Navier-Stokes and heat diffusion solutions, and to provide a rationale for recent experimental particle drag measurements.