This article studies electrostatic fields and potentials in the presence of conductors and point charges under the framework of solving Laplace’s equation with specifi ed boundary conditions. The results demonstrate that many problems posed and solved in elementary electrostatics through various heuristics such as the method of images, can be more rigorously treated under the solution framework of Laplace’s equation.

Recently, Li et al [1] investigated the nonlinear Langevin equations existence including two fractional orders with infinite-point boundary conditions. It has shown that the outcomes given it does count on the solution form for which it has boundary values. It means that their solution needs more to be in the final step. In current work, we will have discussion about the existence and uniqueness for the same boundary conditions by investigating the closely explicit solution form that has no boundary values. Numerical example is supported to have more vision to compare the between new and previous results. It turns out that our results are superior.

The study aims to examine the mediating role of job crafting between inclusive leadership and innovative work behavior. The data were collected from 314 workers employed in China's small and medium-sized industries. The data collection was done through survey design. The data analysis was done through structural equation modeling using Spss 26 and Mplus 8. Inclusive leadership was found to be related to job crafting and innovative work behavior of the employees. Job crafting was found to be mediating between inclusive leadership and innovative work behavior. The study delineated the link mechanism between inclusive leadership and innovative work behavior. Studying inclusive leadership in the context of Chinese culture is a powerful complement to inclusive leadership theory. This paper provides the managers of SMEs with significant managerial insights into how inclusive leadership can effectively motivate employees’ innovative work behaviors.

Notion of median in one dimension is a foundational element in nonparametric statistics. It has been extended to multi-dimensional cases both in location and in regression via notions of data depth. Regression depth (RD) and projection regression depth (PRD) represent the two most promising notions in regression. Carrizosa depth DC is another depth notion in regression. Depth induced regression medians (maximum depth estimators) serve as robust alternatives to the classical least squares estimator. The uniqueness of regression medians is indispensable in the discussion of their properties and the asymptotics (consistency and limiting distribution) of sample regression medians. Are the regression medians induced from RD, PRD, and DC unique? Answering this question is the main goal of this article. It is found that only the regression median induced from PRD possesses the desired uniqueness property. The conventional remedy measure for non-uniqueness, taking average of all medians, might yield an estimator that no longer possesses the maximum depth in both RD and DC cases. These and other findings indicate that the PRD and its induced median are highly favorable among their leading competitors.

Land engineering is a specific new academic discipline in China. Although the undergraduate major of land engineering was officially approved and established lately since 2017, the birth of land engineering as an academic discipline dates back 40 years ago. It has passed through four development stages: the incubation stage in 1978-1985, the initial stage in 1986-1997, the growth stage in 1998-2011, and the expanding stage from 2012 to present. However, land engineering as an academic discipline remains immature and seriously lags behind practice. There are still no unified academic community and broad academic consensus. After a historical overview of the four development stages, this study gave a strategic consideration to five key questions. We argue that the study object of the discipline is land engineering activity, which is defined as the artificial transformation of a land complex combined by various natural and human elements. The uniqueness of the discipline is rooted in its ability to study the comprehensive and integrated reorganization or rebuilding of various elements of land as a complex, with the theory of land complex reconstruction being the core theory. The discipline of land engineering is based on land pure science and land technology, and is one basis of land management. It consists of two modules (rural land engineering and urban land engineering), five secondary disciplines of each module (land development, land rearrangement, land improvement, land protection, and land remediation), and more than 30 research directions. Various technologies are only instrumental but not essential components of land engineering as an academic discipline.

\abstract{The existence of vorticies is well known in fluid mechanics and particularly in turbulence theory. In the context of the millennium problem of the Clay Institute on the existence (or not) of Global solutions in time of the Periodic Navier Stokes equations (PNS) results are now proposed to confirm that Global solutions do not exist in the general case of PNS defined on $\mathbb{T}^3$. This corresponds to part ($D$) proposition in the Navier Stokes problem as posted by C.Fefferman. Recent work has shown that for the Euler equations defined on the 3-Torus there exists a H\"older continuous solution for the third component of velocity $v_3$ at the center of each cell or cube of the 3-Torus Lattice. It has also been determined that as the kinematic viscosity changes from the corresponding value at $\nu=0$ to the fully viscous case for the Periodic Navier Stokes equations (PNS) at the value $\nu=1$ the solution reaches a peak ($\frac{\partial v_3}{\partial y_3} = 0$ ) in $y_3$. (third component of $\vec{r}=(y_1,y_2,y_3)$ in Cartesian co-ordinates) Here in the viscous case there exists a dipole which is not centered at the center of the cell of the Lattice. This immediately implies that since the dipole by definition has an equal in magnitude positive and negative peak in the third component of velocity, then the dipole Riemann cut-off surface is covered by a closed surface which is the sphere $\|\vec{r}\|=1$ and where a given cell of dimensions $[-1,1]^3$ is circumscribed on a sphere of radius $1$. For such a closed surface containing a dipole it necessarily follows that the flux at the surface of the sphere of $v_3$ wrt to surface normal $\vec{n}$ is zero including at $y_i, i=1\dots 3$ is zero at the points where the surface of sphere touches the cube walls. At the finite time singularity on the sphere a rotation boundary condition is deduced. Here $\frac{\partial v_3}{\partial \vec{n}}= \vec{n}\cdot \nabla v_3$. Also a boundary condition on the sphere shows rotation of sphere. Finally as a result, on the sphere a solution for $v_3$ is obtained which is proven to be H\"older continuous and it is proven that it is possible to extend H\"older continuity on the sphere uniquely to all of the interior of the ball.}

The main object of this work is to establish the existence and uniqueness of a solution to the 3D Navier-Stokes (NS) system for an incompressible fluid with viscosity. The nonlinearity of the NS system, as well as the need to estimate velocity and pressure for every value of the viscosity parameter [1] make them challenging to solve. In this regard, in the present work, we study the Navier-Stokes system, which describe the flow of a viscous incompressible fluid and the solution was obtained for velocity and pressure in an analytical form. In addition, the found pressure distribution law, which is described by a Poisson type equation and plays a fundamental role in the theory of Navier-Stokes systems in constructing analytic smooth (conditionally smooth) solutions.

In this paper we study solutions and drift homogenization for a class of viscous lake equations by using the method of semigroups of bounded operators. Suppose that the initial value i.e.,for some Hölder continuous function onwith smooth function value satisfying and Then the initial value problem (2) for viscous lake equations has a unique smooth local strong solution. Using this result we study the drift homogenization for three-dimensional stationary Stokes equation in the usual sense

Conditions under which compositions of component systems form a well-defined system-of-systems here are formulated at a fundamental level. Statement of what defines a well-defined composition and sufficient conditions guaranteeing such a result offers insight into exemplars that can be found in special cases such as differential equation and discrete event systems. For any given global state of a composition, two requirements can be stated informally as: 1) the system can leave this state, i.e., there is at least one trajectory defined that starts from the state, and 2) the trajectory evolves over time without getting stuck at a point in time. Considered for every global state, these conditions determine whether the resultant is a well-defined system and if so, whether it is non-deterministic or deterministic. We formulate these questions within the framework of iterative specifications for mathematical system models that are shown to be behaviorally equivalent to the Discrete Event System Specification (DEVS) formalism. This formalization supports definitions and proofs of the afore-mentioned conditions. Implications are drawn at the fundamental level of existence where the emergence of a system from an assemblage of components can be characterized. We focus on systems with feedback coupling where existence and uniqueness of solutions is problematic.

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context the numerical test is illustrated by two examples where we find meaningful numerical results.

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop mathematical apparatus of studying inhomogeneous boundary value problems under consideration. It is based on using of a weak solution of the boundary value problem and construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce original problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

We presented a fractional order model to capture the transmission dynamics of Anthrax infection with a nonlinear force of infection and its long-term impacts. Analysis by the standard Fabrizio Caputo approach established the model is well-posed and investigation on the stability of the model, Anthrax-free equilibrium as connected to the basic reproduction number R0

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo type fractional stochastic delay differential systems (FSDDSs) with Poisson jumps by utilizing delayed perturbation of Mittag-Leffler function. Moreover, by using Burkholder-Davis-Gundy's inequality, Doob's martingale inequality and Holder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

In this study, we construct a new Atangana-Baleanu fractional model for whooping cough disease to predict future dynamics of the disease, as well as to suggest strategies to eliminate the disease in an optimal way. We prove that the proposed model has a unique solution which is positive and bounded. To measure contagiousness of the disease, we determine the reproduction number R0 and use it to examine the local and global stability at equilibrium points that have symmetry. Through sensitivity analysis, we determine parameters of the model that are most sensitive to R0. The ultimate aim of this research is to analyze different disease prevention approaches in order to find the most suitable one. For this, we include the vaccination and quarantine compartments in the proposed model and formulate an optimal control problem to assess the effect of vaccination and quarantine rates on disease control in three distinct scenarios. Firstly, we study the impact of vaccination strategy and conclude findings with presentation of graphical results. Secondly, we examine the impact of quarantine strategy on whooping cough infection with possible elimination from the society. Lastly, we implement vaccination and quarantine strategies together to visualize their combine effect on infection control. In addition to the study of an optimal control problem, we examine the effect of fractional order on disease dynamics as well as the impact of constant vaccination and quarantine rates on disease transmission and control. We determine that the optimal control strategy with the three controls is more effective in reducing the spread of whooping cough infection. Implementation of Toufik-Atangana type numercial scheme both for the state and adjoint equations is another contribution of this article.

Meixner's historical remark in 1969 "... it can be shown that the concept of entropy in the absence of equilibrium is in fact not only questionable but that it cannot even be defined...." is investigated from today's insight. Several statements --such as the three laws of phenomenological thermodynamics, the embedding theorem and the adiabatical uniqueness-- are used to get rid of non-equilibrium entropy as a primitive concept. In this framework, Clausius inequality of open systems can be derived by use of the defining inequalities which establish the non-equilibrium quantities contact temperature and non-equilibrium molar entropy which allow to describe the interaction between the Schottky system and its controlling equilibrium environment.