Search Results (29)

Intrinsic optical bi-stability in dense two-level systems is developed for the bad cavity limit where electromagnetic modes are adiabatically eliminated. Each atom interacts via dipole–dipole forces with its nearby spatial distribution of atoms. The theory is developed into two parts, corresponding to the short sample, with dimensions shorter than the wavelength, and the long sample. In both cases, the local field corrections modify the Maxwell–Bloch equations, so that cubic or quartic equations are obtained for the inversion of population as a function of the external light intensity, thus leading to intrinsic bi-stability. The effects of noise sources on intrinsic bi-stability were treated, and I found that while the observability of bi-stability was not obtained experimentally for a simple two-level system, there were many observations of bi-stability obtained through the ‘up-conversion’ of rare earth excited crystals. I show the differences between these two systems. Full article

An innovative high-order dimensionality reduction approach, which integrates a condensed finite-difference scheme with proper orthogonal decomposition techniques, has been explored for solving diffusion equations. The difference scheme with forth order accurate in both space and time is introduced through the idea of interpolation approximation. The quartic spline function and $(2,2)$ Padé approximation were utilized in space and time discretization, respectively. The stability and convergence were proven. Moreover, the dimensionality reduction formulas were derived using the proper orthogonal decomposition (POD) method, which is based on the matrix representation of the compact finite-difference scheme. The bases of the POD method were established by cumulative contribution rate of the eigenvalues of snapshot matrix that is different from the traditional ways in which the bases were established by the first eigenvalues. The method of cumulative contribution rate can optimize the degree of freedom. The error analysis of the reduced bases high-order POD finite-difference scheme was provided. Numerical experiments are conducted to validate the soundness and dependability of the reduced-order algorithm. The comparisons between the $(2,2)$ finite-difference method, the traditional POD method, and reduced dimensional method with cumulative contribution rate were discussed. Full article

In this work, we define the multi-radical quadratic and multi-radical quartic mappings as two systems of radical functional equations and then represent them as two equations. Then, we establish some results concerning the stability of multi-radical quadratic and multi-radical quartic mappings by applying a fixed-point based on Brzdȩk. As a direct consequence, we prove the Rassias and Hyers stability of the mentioned mappings. Full article

We present a mechanistic mathematical model of the basal state for type 2 diabetes mellitus (T2DM) in an analytical form and illustrate its use for in silico basal-state and dynamic studies. At the core of the basal-state model is a quartic equation that expresses the basal plasma glucose concentration solely in terms of model parameters. This analytical model avoids a computationally intensive numerical solver and is illustrated by an investigation of how glucose-utilization parameters impact basal glucose, insulin, insulin-dependent utilization, and hepatic extraction, leveraging median parameter values of early-stage T2DM. Furthermore, the presented basal-state model ensures accurate execution of the corresponding dynamic model, which contains basal quantities within its derivative functions; erroneous, unintended dynamics in plasma glucose, insulin, and C-peptide are illustrated using an incorrect basal glucose value. The presented basal model enables efficient and accurate basal-state and dynamic studies, facilitating the understanding of T2DM pathophysiology and the development of T2DM diagnosis, treatment, and management strategies. Full article

In this work, we investigate the transition from regular dynamics to chaotic behavior in a one-dimensional quartic anharmonic classical oscillator driven by a time-dependent external square-wave force. Owing to energy conservation, the motion of an undriven quartic anharmonic oscillator is regular, periodic, and stable. For a driven quartic anharmonic oscillator, the equations of motion cannot be solved analytically due to the presence of an anharmonic term in the potential energy function. Using the fourth-order Runge–Kutta method to numerically solve the equations of motion for the driven quartic anharmonic oscillator, we find that the oscillator motion under the influence of a sufficiently small driving force remains regular, while by gradually increasing the driving force, a series of nonlinear resonances can occur, grow, overlap, and ultimately disappear due to the emergence of chaos. Full article

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function. Full article

This study considers the specific case of a flat, minimally coupled to gravity, quintessence cosmology with a dark energy quartic polynomial potential that has the same mathematical form as the Higgs potential. Previous work on this case determined that the scalar field is given by a simple expression of the Lambert W function in terms of the easily observable scale factor. This expression provides analytic equations for the evolution of cosmological dark energy parameters as a function of the scale factor for all points on the Lambert W function principal branch. The Lambert W function is zero at a scale factor of zero that marks the big bang. The evolutionary equations beyond the big bang describe a canonical universe that is similar to $\Lambda$CDM, making it an excellent dynamical template to compare with observational data. The portion of the W function principal before the big bang extends to the infinite pre-bang past. It describes a noncanonical universe with an initially very low mass density that contracts by rolling down the dark energy potential to a singularity, big bang, at the scale factor zero point. This provides a natural origin for the big bang. It also raises the possibility that the universe existed before the big bang and is far older, and that it was once far larger than its current size. The recent increasing interest in the possibility of a dynamical universe instead of $\Lambda$CDM makes the exploration of the nature of such universes particularly relevant. Full article

In this work the matrix exponential function is solved analytically for the special orthogonal groups $SO\left(n\right)$ up to $n=9$. The number of occurring k-th matrix powers gets limited to $0\le k\le n-1$ by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of $SO\left(3\right)$, a quadratic equation needs to be solved for $n=4,5$, a cubic equation for $n=6,7$, and a quartic equation for $n=8,9$. As an interesting subgroup of $SO\left(7\right)$, the exceptional Lie group $G_2$ of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, $\xi =1$. The traces of the $SO\left(n\right)$-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities. Full article

In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures. Full article

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials, respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes. Full article

In this work, we investigate the refined stability of the additive, quartic, and quintic functional equations in modular spaces with and without the ${\Delta }_2$-condition using the direct method (Hyers method). We also examine Ulam stability in 2-Banach space using the direct method. Additionally, using a suitable counterexample, we eventually demonstrate that the stability of these equations fails in a certain case. Full article

In this article, we defined the generalized intuitionistic P-pseudo fuzzy 2-normed spaces and investigated the Hyers stability of m-mappings in this space. The m-mappings are interesting functional equations; these functional equations are additive for m = 1, quadratic for m = 2, cubic for m = 3, and quartic for m = 4. We have investigated the stability of four types of functional equations in generalized intuitionistic P-pseudo fuzzy 2-normed spaces by the fixed point method. Full article

This paper implements the trial equation approach to retrieve cubic–quartic optical solitons in fiber Bragg gratings with the aid of the trial equation methodology. Five forms of nonlinear refractive index structures are considered. They are the Kerr law, the parabolic law, the polynomial law, the quadratic–cubic law, and the parabolic nonlocal law. Dark and singular soliton solutions are recovered along with Jacobi’s elliptic functions with an appropriate modulus of ellipticity. Full article

In this work, we introduce a new type of generalised quartic functional equation and obtain the general solution. We then investigate the stability results by using the Hyers method in modular space for quartic functional equations without using the Fatou property, without using the ${\Delta }_b$-condition and without using both the ${\Delta }_b$-condition and the Fatou property. Moreover, we investigate the stability results for this functional equation with the help of a fixed-point technique involving the idea of the Fatou property in modular spaces. Furthermore, a suitable counter example is also demonstrated to prove the non-stability of a singular case. Full article