Point Equation Research Articles

Plasmon properties of 4-bilayer graphene structures: Inhomogeneous background dielectric effects

In this paper, we investigate the plasmon properties of a 4-BLG structure on an inhomogeneous background dielectric within the random-phase approximation. Firstly, we determine collective excitations from the zeros of the dynamical dielectric function and then calculate respective broadening functions. Computations demonstrate that there are four solutions to the zero point equation of the dynamical dielectric function, corresponding to four phases of collective excitations in the system. Three of them, acoustical modes, are lower than that in BLG at the same parameters while the optical mode has a higher frequency at a given wave vector. Secondly, we present that both the inhomogeneity of background dielectric and the inter-layer separation strongly decrease the plasmonic frequency and respective broadening functions. Moreover, the inhomogeneity of the environment strengthens the effects of separation on plasmon properties. Lastly, the increase in doping density in BLG layers decreases strongly both plasmon frequency and its broadening functions. We observe that due to the pronounced contributions to plasmon properties of 4-BLG systems of the inhomogeneity, this factor should not be neglected in calculations.

Modeling the dynamics of information propagation in the temporal and spatial environment

In this paper, we try to establish a non-smooth susceptible–infected–recovered (SIR) rumor propagation model based on time and space dimensions. First of all, we prove the existence and uniqueness of the solution. Secondly, we divide the system into two parts and discuss the existence of equilibrium points for each of them. For the left part, we define R 0 to study the relationship between R 0 and the existence of equilibrium points. For the right part, we classify many different cases by discussing the coefficients of the equilibrium point equation. Then, on this basis, we perform a bifurcation analysis of the non-spatial system and find conditions that lead to the existence of saddle-node bifurcation. Further, we consider the effect of diffusion. We specifically analyze the stability of equilibrium points. In addition, we analyze the Turing instability and Hopf bifurcation occurring at some equilibrium points. According to the Lyapunov number, we also determine the direction of the bifurcation. When I = I c , we discuss conditions for the existence of discontinuous Hopf bifurcation. Finally, through numerical simulations and combined with the practical meaning of the parameters, we prove the correctness of the previous theoretical theorem.

Kinematic analysis and process optimization of root-cutting systems in field harvesting of garlic based on computer simulation technology.

Root cutting is an important process in garlic field harvesting but is the weakest link in the full mechanization of garlic production. To improve the current situation of technological backwardness and poor operational quality of mechanized garlic root-cutting in the main garlic-producing regions of China, this study combined the physical characteristics and agronomic requirements of garlic plants, and proposed an innovative floating root-cutting technology for garlic combine harvesters that enables the top alignment of bulb, adaptive profiling floating of cutter, and embedded cutting of roots. Through the kinematic analysis of the floating cutting process, the coordinate equations of the initial contact point of the bulb, the mathematical model of the floating displacement of the cutting component. Using computer simulation techniques, the dynamic simulation study of the floating cutting process was carried out in the rigid-flexible coupling numerical simulation model of root-cutting mechanism and garlic plant. The influence law of garlic conveying speed, extension spring preload force and stiffness on the floating displacement of the cutting component and the angular velocity of swing arm reset and its formation causes were analyzed by a single-factor simulation test. The key operating parameters of the root-cutting mechanism were optimized through the computerized virtual orthogonal test and fuzzy comprehensive evaluation. The significance of the factors affecting the floating cutting performance decreased in the following order: extension spring preload force, garlic conveying speed and extension spring stiffness. The optimal parameter combination of the root cutting mechanism obtained from the optimization were as follow: extension spring preload force was 16 N, garlic conveying speed was 0.8 m/s, and extension spring stiffness was 215 N/m. Tests conducted with the optimal parameter combination yielded a root excision rate of 92.72%, which meets the requirements of Chinese garlic field harvesting quality. This study provides computer simulation optimization methods for the optimal design of the root-cutting mechanism, and also provides technical and equipment support for the full mechanization of garlic production in China.

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

On the potential function of the colored Jones polynomial with arbitrary colors

We consider the potential function of the colored Jones polynomial for a link with arbitrary colors and obtain the cone-manifold structure for the link complement. In addition, we establish a relationship between a saddle point equation and hyperbolicity of the link complement. This provides evidence for the Chen-Yang conjecture on the link complement.

Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture

Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed $x$. This result is applied to infinite-dimensional, semi-linear control systems and their reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case $p=1$.

Critical point equation within the framework of various contact metric manifolds

The aim of the present paper is to study the critical point equation (shortly CPE) conjecture within the framework of various contact metric manifolds. First we establish that Kenmotsu manifold satisfying the CPE either becomes an Einstein manifold or the derivative of potential function along characteristic vector field satisfy a certain relation on the distribution of η. Next we study CPE on (κ, μ)'-almost Kenmotsu manifold and obtain that the manifold is Einstein. Later in case of 3-dimensional trans-Sasakian manifold, we get that either the manifold becomes α-Sasakian or it becomes Einstein. Finally we give examples of 3-dimensional trans-Sasakian manifold and (κ ,μ)'-almost Kenmotsu manifold to verify our outcomes.

Contact stress analysis of external non-variable multi-modulus spur gear pairs

The conditions of correct gear engagement of a pair of involute gears and the principle of geometric relationship of involute tooth profile are derived to determine the pitch point of externally meshing non-variable multi-modulus spur gear pairs into the double tooth engagement region. The Hertzian contact stress model is used as the basis to derive the contact stress calculation equations for pitch point and other characteristic contact points, and the variation of the contact stress distribution on the contact line with the module ratio is analyzed. The results demonstrate that when the number of teeth of the gear pairs is determined, the larger the value of the module difference, the easier it is for the pitch point of the gear pairs to be in the double tooth meshing zone. Except for the double tooth meshing out point, the contact stress decreases with the increase of module ratio, and the maximum point of contact stress is at the upper boundary point of the single tooth meshing zone.

On the and universality classes

We recently determined the exact fixed point equations and the spaces of solutions of the two-dimensional and models using scale invariant scattering theory. Here, we discuss subtleties hidden in some solutions and related to the difference between ferromagnetic and antiferromagnetic interaction.

Realization of Bounce in a Modified Gravity Framework and Information Theoretic Approach to the Bouncing Point

In this work, we report a study on bouncing cosmology with modified generalized Chaplygin Gas (mgCG) in a bulk viscosity framework. Reconstruction schemes were demonstrated in Einstein and modified f(T) gravity framework under the purview of viscous cosmological settings. We also took non-viscous cases into account. We studied the equation of state (EoS) parameter under various circumstances and judged the stability of the models through the sign of the squared speed of sound. We observed the mgCG behaving like avoidance of Big Rip in the presence of bulk viscosity at the turnaround point and in non-viscous cases, a phantom-like behavior appears. The turnaround point equation of state parameter crosses the phantom boundary, violating NEC. The role of the mgCG’s model parameters was also investigated before and after the bounce. A Hubble flow dynamics was carried out and, it was revealed that mgCG is capable of realizing an inflationary phase as well as an exit from inflation. An f(T) gravitational paradigm was also considered, where the mgCG density was reconstructed in the presence of bulk viscosity. The role of the parameters associated with the bouncing scale factor, describing how fast the bounce takes place, was also studied in this framework. Finally, the reconstructed mgCG turned out to be stable against small perturbations irrespective of the presence of bulk viscosity and modified gravity scenario. Finally, the reconstruction scheme was assessed using statistical analysis, Shannon entropy.

Dynamical analysis of a [formula omitted] epidemic-like propagation model with non-smooth control

The continuous acceleration of information communication makes the spread of rumors constantly develop and change. Analysis of the dynamics of rumor propagation allows for its better control. In this paper, we study a rumor transmission model with a nonlinear transmission rate and a non-smooth threshold transmission function, based on the classical SI infectious disease model. In this model, we first prove the existence of the model solutions, and then obtain the number of non-negative equilibrium points. By dividing the model into spatially homogeneous and inhomogeneous parts with continuous and discontinuous, we discuss the characteristic equations of the equilibrium points and obtain the existence conditions for Saddle–node bifurcation, Turing bifurcation and Hopf bifurcation under the corresponding conditions. Finally, the feasibility of the theoretical calculation results is further verified by using the simulation results.

Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

In Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022), an estimate for skew-symmetric 2-tensors was claimed. Soon after, this estimate has been exploited to claim powerful classification results: Most notably, it has been employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive scalar curvature (Xu and Ye in Invent Math 33(2):64, 2022) and in connection with the Besse conjecture (Yun and Hwang in Critical point equation on three-dimensional manifolds and the Besse conjecture). In the present note, we point out an issue in the argument proposed in Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022) and we provide a counterexample to the estimate.

Best Proximity Point Theorem for α ̃–ψ ̃-Contractive Type Mapping in Fuzzy Normed Space

The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems offer an approximate solution to the fixed point equation . It is used to solve the problem in order to come up with a good approximation. This paper's main purpose is to introduce new types of proximal contraction for nonself mappings in fuzzy normed space and then proved the best proximity point theorem for these mappings. At first, the definition of fuzzy normed space is given. Then the notions of the best proximity point and - proximal admissible in the context of fuzzy normed space are presented. The notion of α ̃–ψ ̃- proximal contractive mapping is introduced. After that, the best proximity point theorem for such type of mapping in a fuzzy normed space is state and prove. In addition, the idea of α ̃–ϕ ̃-proximal contractive mapping is presented in a fuzzy normed space and under specific conditions, the best proximity point theorem for such type of mappings is proved. Furthermore, some examples are offered to show the results' usefulness.

Certain Paracontact Metrics Satisfying the Critical Point Equation

The aim of this paper is to study theCPE (Critical Point Equation) on some paracontact metric manifolds.First, we prove that if a para-Sasakian metric satisfies the CPE,then it is Einstein with constant scalar curvature -2n(2n+1). Next,we prove that if $(\kappa,\mu)$-paracontact metric satisfies theCPE, then it is locally isometric to the product of a flat$(n+1)$-dimensional manifold and $n$-dimensional manifold ofnegative constant curvature$-4$.

THEORETICAL FOUNDATIONS OF THE POINT CALCULUS AS A MATHEMATICAL APPARATUS FOR GEOMETRIC AND COMPUTER MODELING

The purpose of the article is to acquaint the reader with the new mathematical apparatus – point calculus, which was developed for solving engineering and scientific problems by the leading scientists of the Melitopol School of Applied Geometry and continues its active development in the works of Russian and Ukrainian scientists-geometers. The article presents the fundamental definitions and terms, basic methods, metrics, fundamental theorems of point calculus, shows the principles of parametrization of a straight line, plane and three-dimensional space in point calculus. Possibilities of point calculus include the definition of continuous and discrete geometric models of objects, processes and phenomena in the form of point equations and computational algorithms based on them. The advantage of the new mathematical apparatus is the representation of geometric objects in the form of a set of projections on the axes of the global coordinate system, which makes it possible to determine the geometric models of objects, processes and phenomena in spaces of any dimension. The use of point calculus in computer graphics and virtual reality, solid modeling of isotropic and anisotropic bodies and computer modeling of objects consisting of nanoparticles is seen as promising. Therefore, the main direction of promising research is seen as the further expansion of theoretical and applied point calculus tools, and its popularization by solving a wide range of important engineering and scientific problems of an applied nature.

Systemic-Risk and Evolutionary Stable Strategies in a Financial Network

We consider a financial network represented at any time instance by a random liability graph which evolves over time. The agents connect through credit instruments borrowed from each other or through direct lending, and these create the liability edges. These random edges are modified (locally) by the agents over time, as they learn from their experiences and (possibly imperfect) observations. The settlement of the liabilities of various agents at the end of the contract period (at any time instance) can be expressed as solutions of random fixed point equations. Our first step is to derive the solutions of these equations (asymptotically and one for each time instance), using a recent result on random fixed point equations. The agents, at any time instance, adapt one of the two available strategies, risky or less risky investments, with an aim to maximize their returns. We aim to study the emerging strategies of such replicator dynamics that drives the financial network. We theoretically reduce the analysis of the complex system to that of an appropriate ordinary differential equation (ODE). Using the attractors of the resulting ODE we show that the replicator dynamics converges to one of the two pure evolutionary stable strategies (all risky or all less risky agents); one can have mixed limit only when the observations are imperfect. We verify our theoretical findings using exhaustive Monte Carlo simulations. The dynamics avoid the emergence of the systemic-risk regime (where majority default). However, if all the agents blindly adapt risky strategy it can lead to systemic risk regime.

Existence of best proximity points for the sum of two operators

In this paper, we work on the existence of approximate solution for the fixed point equation Rx + Qx = x, when R is a weak contractive mapping and Q is a continuous self-map which is a new topic related to the background literature. Till now, no results were found to find the existence of approximate solution for such a kind of mappings in literature to the best of our knowledge. Here we prove the existence of best proximity point for the sum of a weak contractive non self map and a continuous self map. Also, we provide an example to support our main theorem. And also, we prove the existence of best proximity point for the sum of Geraghty-contraction and continuous map.

Performance Bounds of Model Predictive Control for Unconstrained and Constrained Linear Quadratic Problems and Beyond*

We study unconstrained and constrained linear quadratic problems and investigate the suboptimality of the model predictive control (MPC) method applied to such problems. Considering MPC as an approximate scheme for solving the related fixed point equations, we derive performance bounds for the closed-loop system under MPC. Our analysis, as well as numerical examples, suggests new ways of choosing the terminal cost and terminal constraints, which are not related to the solution of the Riccati equation of the original problem. The resulting method can have a larger feasible region, and cause hardly any loss of performance in terms of the closed-loop cost over an infinite horizon.

Fixed point equations for superlinear operators with strong upper or strong lower solutions and applications

<abstract><p>It is well known that sublinear operators and superlinear operators are two classes of important nonlinear operators in nonlinear analysis and dynamical systems. Since sublinear operators have only weak nonlinearity, this advantage makes it easy to deal with them. However, superlinear operators have strong nonlinearity, and there are only a few results about them. In this paper, the convergence of Picard iteration for the superlinear operator $ A $ is obtained based on the conditions that the fixed point equation $ Ax = x $ has a strong upper solution and a lower solution (or alternatively, an upper solution and a strong lower solution). Besides, the uniqueness of the fixed point of strongly increasing operators as well as the global attractivity of strongly monotone dynamical systems are also discussed. In addition, the main results are applied to monotone dynamics of superlinear operators and nonlinear integral equations. The method used in our work develops the traditional method of upper and lower solutions. Since a strong upper (upper) solution and a lower (strong lower) solution are easily checked, the obtained results are effective and practicable in the study of nonlinear equations and dynamical systems. The main novelty is that this paper provides new fixed point results for increasing superlinear operators and the obtained results are applied to strongly monotone systems to investigate their global attractivity.</p></abstract>

Critical points in coupled Potts models and correlated percolation

We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a q-state Potts model coupled to an r-state Potts model in two dimensions. For integer values of q and r the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin–Teller case (). Since our results extend to continuous values of the number of states, we can access the limit r → 1 corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin–Kasteleyn clusters by analytical continuation.