Prime n solutions of the Brocard’s Problem

In this paper on the [1]“Brocard’s Problem” , I have worked on case when n is prime and n divides m-1. Necessary conditions on m are given in Theorem and Corollaries.I used necessary and sufficient condition of primes. Assuming that n is prime and divides m-1, I applied Inverse Laplace Transform on the obtained equation and got a polynomial function which is easier to deal with. I worked with zero of the polynomial function and got lower bound of p which was not useful as p tends to infinity, but solving quartic equation which I have given at the end could give significant upper, lower bounds of p.What would happen to those upper, lower bounds if p tends to infinity?

Permitted speed decision of single-unit trucks with emergency braking maneuver on horizontal curves under rainy weather

Under adverse weather conditions, visibility and the available pavement friction are reduced. The improper selection of speed on curved road sections leads to an unreasonable distribution of longitudinal and lateral friction, which is likely to cause rear-end collisions and lateral instability accidents. This study considers the combined braking and turning maneuvers to obtain the permitted vehicle speed under rainy conditions. First, a braking distance computation model was established by simplifying the relationship curve between brake pedal force, vehicle braking deceleration, and braking time. Different from the visibility commonly used in the meteorological field, this paper defines "driver’s sight distance based on real road scenarios" as a threshold to measure the longitudinal safety of the vehicle. Furthermore, the lateral friction and rollover margin is defined to characterize the vehicle’s lateral stability. The corresponding relationship between rainfall intensity-water film thickness-road friction is established to better predict the safe speed based on the information issued by the weather station. It should be noted that since the road friction factor of the wet pavement not only determined the safe vehicle speed but also be determined by the vehicle speed, so we adopt Ferrari’s method to solve the quartic equation about permitted vehicle speed. Finally, the braking and turning maneuvers are considered comprehensively based on the principle of friction ellipse. The results of the TruckSim simulation show that for a single-unit truck, running at the computed permitted speed, both lateral and longitudinal stability meet the requirements. The proposed permitted vehicle speed model on horizontal curves can provide driving guidance for drivers on curves under rainy weather or as a decision-making basis for road managers.

A conversation on the quartic equation of the secular determinant of methylenecyclopropene

The Hückel method (HM) is based on quantum mechanics and it is used for calculating the energies of molecular orbitals of π electrons in conjugated systems. The HM involves the setting up of the secular determinant which is expanded to obtain a polynomial which is to be solved. In general, the polynomial is one which may be factorized. However, in May 2020, students brought to my attention that the secular determinant of methylenecyclopropene could not be factorized completely. As a result of this, we used a combination of online tools, technology and visualization to calculate the roots of the secular determinant. This write-up, in a playwriting format, describes the conversation between the facilitator and the students.

Plane-wave propagation in media with electromagnetic anisotropy

The propagation of electromagnetic waves in a medium with electrical and magnetic anisotropy is a subject that is not usually handled in conventional optics and electromagnetism books. During this work, we try to give a pedagogical approach to the subject, using tools that are accessible to an average physics student. In this article, we obtain the Fresnel relation in a media with electromagnetic anisotropy, which corresponds to a quartic equation in the refraction index, assuming only that the principal axes of the electric and magnetic tensors coincide. Additionally, we find the geometric location related to the different situations the discriminant of the quartic equation provides. In order to illustrate our findings, we determine the refractive index together with the plane wave equations for certain values of the parameters that meet the established conditions. The target readers of the paper are students pursuing physics at the intermediate undergraduate level.

Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes

Presented is a very detailed two-tier analysis of the location of the real roots of the general quartic equation $$x^4 + a x^3 + b x^2 + c x + d = 0$$ x 4 + a x 3 + b x 2 + c x + d = 0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. The presented classification of the roots of the quartic equation is particularly useful in situations in which the equation stems from a model the coefficients of which are (functions of) the model parameters and solving cubic equations, let alone using the explicit quartic formulæ , is a daunting task. The only benefit in such cases would be to gain insight into the location of the roots and the proposed method provides this. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the “sub-quartic” $$x^4 + ax^3 + bx^2$$ x 4 + a x 3 + b x 2 with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.

Quadratic-stretch elasticity

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed.

Surface model of the human red blood cell simulating changes in membrane curvature under strain

We present mathematical simulations of shapes of red blood cells (RBCs) and their cytoskeleton when they are subjected to linear strain. The cell surface is described by a previously reported quartic equation in three dimensional (3D) Cartesian space. Using recently available functions in Mathematica to triangularize the surfaces we computed four types of curvature of the membrane. We also mapped changes in mesh-triangle area and curvatures as the RBCs were distorted. The highly deformable red blood cell (erythrocyte; RBC) responds to mechanically imposed shape changes with enhanced glycolytic flux and cation transport. Such morphological changes are produced experimentally by suspending the cells in a gelatin gel, which is then elongated or compressed in a custom apparatus inside an NMR spectrometer. A key observation is the extent to which the maximum and minimum Principal Curvatures are localized symmetrically in patches at the poles or equators and distributed in rings around the main axis of the strained RBC. Changes on the nanometre to micro-meter scale of curvature, suggest activation of only a subset of the intrinsic mechanosensitive cation channels, Piezo1, during experiments carried out with controlled distortions, which persist for many hours. This finding is relevant to a proposal for non-uniform distribution of Piezo1 molecules around the RBC membrane. However, if the curvature that gates Piezo1 is at a very fine length scale, then membrane tension will determine local curvature; so, curvatures as computed here (in contrast to much finer surface irregularities) may not influence Piezo1 activity. Nevertheless, our analytical methods can be extended address these new mechanistic proposals. The geometrical reorganization of the simulated cytoskeleton informs ideas about the mechanism of concerted metabolic and cation-flux responses of the RBC to mechanically imposed shape changes.

COMPARISON OF PREDICTED FAILURE AREA AROUND THE BOREHOLES IN THE STRIKE-SLIP FAULTING STRESS REGIME WITH HOEK-BROWN AND FAIRHURST GENERALIZED CRITERIA

Breakout is a shear failure due to compression that forms around the borehole due to stress concentration. In this paper, the breakout theory model is investigated by combining the equilibrium elasticity equations of stress around the borehole with two Hoek-Brown and Fairhurst generalized fracture criteria, both of which are based on the Griffith criterion. This theory model provides an explicit equation for the breakout failure width, but the depth of failure is obtained by solving a quartic equation. According to the results and in general, in situ stresses and rock strength characteristics are effective in developing the breakout failure area, As the ratio of in-situ stresses increases, the breakout area becomes deeper and wider. Because in the shear zone, the failure envelope of the Fairhurst criterion is lower than the Hoek-Brown failure criterion, the Fairhurst criterion provides more depth for breakout than the Hoek-Brown criterion. However, due to the same compressive strength of the rock in these two criteria, the same failure width for breakout is obtained from these two criteria. Also, the results obtained for the depth of failure from the theoretical model based on the Fairhurst criterion are in good agreement with the laboratory results on Westerly granite.

Surface model of the human red blood cell simulating changes in membrane curvature under strain

The highly deformable red blood cell (erythrocyte; RBC) responds to mechanically imposed shape changes with enhanced glycolytic flux and cation transport. Such morphological changes are produced experimentally by suspending the cells in a gelatin gel, which is then elongated or compressed in a special apparatus inside an NMR spectrometer. However, direct mathematical predictions of the shapes of the morphed cells have not been reported before. We used recently available functions in Mathematica to triangularize and then compute four types of curvature. The RBCs were described by a previously presented quartic equation in three dimensional (3D) Cartesian space. A key finding was the extent to which the maximum and minimum Principal Curvatures were localized symmetrically in patches at the poles or equators and distributed in rings around the main axis of the strained RBC. The simulations, on the nano-metre to micro-meter scale of curvature, suggest activation of only a subset of the intrinsic mechanosensitive cation channels, Piezo1, during experiments carried out with controlled distortions that persist for many hours. This view is consistent with a recent proposal for non-uniform distribution of Piezo1 molecules around the RBC membrane. On the other hand, if the curvature that gates Piezo1 is at a much finer length scale, then membrane tension will determine local curvature and micron scale curvature as described here will be less likely to influence Piezo1 activity. The geometrical reorganization of the simulated cytoskeleton helps understanding of the concerted metabolic and cation-flux responses of the RBC to mechanically imposed shape changes.

Location-Price Game in a Dual-Circle Market with Different Demand Levels

This paper researches a location-price game in a dual-circle market system, where two circular markets are interconnected with different demand levels. Based on the Bertrand and Salop models, a double intersecting circle model is established for a dual-circle market system in which two players (firms) develop a spatial game under price competition. By a two-stage (location-then-price) structure and backward induction approach, the existence of price and location equilibrium outcomes is obtained for the location game. Furthermore, by Ferrari method for quartic equation, the location equilibrium is presented by algebraic expression, which directly reflects the relationship between the equilibrium position and the proportion factor of demand levels. Finally, an algorithm is designed to simulate the game process of two players in the dual-circle market and simulation results show that two players almost reach the equilibrium positions obtained by theory, wherever their initial positions are.