In this paper, we consider the forgetful functor from the category LDcpoof local dcpos (respectively, Dcpo of dcpos) to the category Pos of posets (respectively, LDcpo of local dcpos), and study the existence of its left and right ADJOINTs. Moreover, we give the concrete forms of free and cofreeS-ldcpos over a local dcpo, where S is a local dcpo monoid. The main results are: (1) The forgetful functorU: LDcpo ®Pos has a left ADJOINT, but does not have a right ADJOINT; (2) The inclusion functorI: Dcpo®LDcpohas a left ADJOINT, but does not have a right ADJOINT; (3) The forgetful functorU: LDcpo-S ®LDcpo has both left and right ADJOINTs; (4) If (S, 0, 1) is a good ldcpo-monoid, then the forgetful functor U: LDcpo-S ®Pos-S has a left ADJOINT.

In this research, the continuous ADJOINT method is applied to optimize an airfoil in subsonic and transonic flows. An Euler flow solver is used to analyze the inviscid compressible flow over airfoils in each design cycle. Two design problems appearing in aerodynamic shape optimization, namely inverse pressure design and drag minimization were investigated. In the first part, a test case was carried out to evaluate the performance of the ADJOINT method in inverse design problem. The results show that we can use the ADJOINT method as an efficient tool in inverse aerodynamic design problems. In the second part, the constrained optimization was investigated in a drag minimization problem. The investigated samples show that a small variation of airfoil geometry has caused considerable decrease in the drag coefficient. To evaluate the performance of the ADJOINT method in design problems with numerous design variables and also to evaluate the effects of the adoption of the design vector on the optimization results, the constrained drag minimization was performed using two different design vectors. The results shows that the mechanism and the value of drag reduction are affected by the type of design vector. Also, computational cost of the ADJOINT method are independent of the number of design variables.

We will consider multiplication operators on Hilbert spaces of analyticfunctions on a domain $\omega \subset \cmark $. We determine the commutants of certainmultiplication operators with ADJOINTs in a Cowen-Douglas class operator.

Let D be the open unit disc in the complex plane C. A sandwich weighted composition operator S,' takes an analytic map f on the open unit disc D to the map (: f 0o')0, where ' is an analytic map of D into itself and is an analytic map on D. In this paper, we compute the ADJOINT of a sandwich weighted composition operator S,' on weighted Hardy spaces.

Joint inversion of different geophysical data is used to identify different geological structures and estimate some physical parameters. This study is aimed to do joint inversion of gravity and ADJOINT-state first arrival tomography by using the Gardner relation. Gardner's relation is an equation that relates seismic velocity to density.  In the gravity method, forward modeling aims to compute the gravity response at the surface due to a density distribution in the subsurface (Boulanger & Chouteau, 2001). RCG method introduced by Zhdanov (2015) has been used for inversion of gravity. In the forward modeling of the seismic method, the eikonal equation is used to model the first arrival traveltime, which approximates the high-frequency wave equation. The eikonal equation is nonlinear. There are different methods to solve it, such as Raytracing, Fast Marching, Fast Sweeping, and Finite Difference. This solving method is usually based on a beam, grid, or graph. This study examines the fast sweeping method (FSM) based on rectangular grids by Zhao (2005) has presented. Estimating model parameters from measured data generally minimizes the misfit function in geophysics. A classic way to solve a minimum problem is to determine the minimum of a series of linear problems sequentially. This formulation requires Frechet derivatives (Jacobin matrices), which need heavy and lengthy calculations. If minimization appears as a nonlinear optimization problem, only the misfit function gradient is required. In this study, the ADJOINT-state method is used to calculate the gradient. In the 1970s, the ADJOINT-state method was developed to efficiently calculate the gradient. It is now a well-known numerical community method for calculating the gradient of a function misfit. It can be used when this function misfit depends on the model parameters through state variables.  State variables are solutions to the forward problem (Plessix, 2006).  For example, state equations can be beam equations in the tomography problem that is the subject of this study. State variables are spatial coordinates and slowness vectors that describe beam paths. The ADJOINT-state method is effective because only one additional linear system has to be solved. This study performs the joint inversion of data based on the sequential inversion method using the petrophysical constraint, Gardner relation. The inversion process performs in several loops. In each loop, the individual seismic inversion is first performed in several iterations. The velocity model obtained using the Gardner relation is converted into a density model. The obtained density model is the initial model for individual inversion of gravity in the desired iterations. Finally, the obtained gravity model is converted to a velocity model using the inverse Gardner relation, and the above process is repeated to achieve the desired result. The results of the above inversion can be used to image shallow subsurface models and to limit the location of anomalies in hydrocarbon exploration in layered sediment environments such as basalt and salt dome modeling.

This paper will generalize $b$-frames, a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product, we will use a new product called the $b$-dual product, which is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames and $K$-$b$-frames and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will express the reconstruction formula of the elements of the Hilbert space. We will also study the stability and preservation of both $b$-frames and $K$-$b$-frames and to do so, we will give the equivalent of the ADJOINT operator according to the $b$-dual product.

In this paper, the effect of different kind of constraints on two and three dimensional ADJOINT-based shape optimization of aerodynamic design in inviscid and turbulent flows has been investigated. To assess the accuracy and efficiency of the ADJOINT-based approach, a complete set of optimization problem is investigated: from none to fully aerodynamic and geometrical constraints. With the ADJOINT method, the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding ADJOINT equations only once for each cost function, regardless of the number of design parameters. Drag forces was chosen to be the objective function. Hicks– Henne shape functions and free form deformation parameters are adopted for the surface geometry perturbations, for two and three dimensional geometries, respectivelyThe drag minimization results show that the ADJOINT equation converges well and with specifying the suitable constraints, the designed shape approaches to the most optimized level without the loss of performance.