In this paper we will try to introduce a good smoothness notion for a functor. We consider properties and conditions from geometry and algebraic geometry which we expect a smooth functor should have.

This paper presents an analytical solution for stability analysis of thick rectangular functionally graded plates with porosity subjected to in-plane loadings using the higher-ORDER shear and normal deformation plate theory, for the FIRST time. The plate material and its porosity are assumed to vary along the thickness direction. Also, three types of porosity pattern along the thickness are considered. Since the plate structure is not generally symmetry to the mid-plane it is assumed that the in-plane loads are applied to its neutral plane to remove the bending-stretching coupling. Stability equations are derived and then analytically solved for rectangular plates with simple supports using Legendre orthonormal polynomials and Navier’s method to determine the critical buckling load. The results are then compared with estimates made using higher-ORDER shear deformation (HSDT) and classical plate theories (CPT) available in the literature for FG non-porous plates. It is shown that compared to the HSDT, the HOSNDT yields smaller values for the plate critical buckling load and the effect of HSNDT is more important as the plate thickness increases. In addition, it is demonstrated that compared to the uniaxial load, the effect of HSNDT is greater as the plate is subjected to a biaxial compression load. Finally, the effects of the porosity distribution, porosity, power-law index, loading condition, and thickness ratio are studied in detail using HOSNDT. The results show that the porosity effect is greater in smaller values of the power law index parameter.

One of the nice properties of the FIRST-ORDER logic is the compactness of satisfiability. It states that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in fuzzy logics will pose various kinds of compactness in these logics.In this article, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in H\'{a}jek Basic logic, some new results are given around this issue.It will be shown that there are topologies on $[0,1]$ and $[0,1]^2$ for which the interpretation of all logical connectives of the Basic logic is continuous. Furthermore, a topology on FIRST-ORDER structures will be introduced for any similarity relation as well. Then by the same ideas as in continuous logic, the results around the compactness of satisfiability will be extended for Basic logic.

The aim of this paper is to analyze the undamped second ORDER descriptor systems to show the possibility of transform an undamped second ORDER descriptor system to a pure FIRST ORDER descriptor system, on different ways, while stating some of its benefits. Meanwhile we will introduce pure FIRST ORDER systems and undamped second ORDER systems and indicate that under some assumptions every undamped second ORDER descriptor system can be changed to a pure FIRST ORDER one.

In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrodinger equation (NLS) by considering the modulus of the solutions in the (x, t) plan in the cases of ORDERs 3 and 4. For this, we use a representation of solutions to NLS equation as a quotient of two determinants by an exponential depending ont. This formulation gives in the case of the ORDER 3 and 4, solutions with, respectively 4 and 6 parameters. With this method, beside Peregrine breathers, we construct all characteristic patterns for the modulus of solutions, like triangular configurations, ring and others.

Today, sensors and actuators have a special place in the science and industry world. Therefore, the electromechanical analysis of piezoelectric materials is one of the topics of interest for researchers. To that effect, besides understanding the behavior of piezoelectric structures by optimizing this behavior, they could facilitate designing and manufacturing structures. In this study, using the FIRST-ORDER shear deformation theory and the FIRST-ORDER electrical potential theory and applying the energy method, the governing equations are extracted for thick piezoelectric cylinders subjected to mechanical and electrical loading at their internal and external surfaces under various boundary conditions in two cylinder heads. Then an analytical solution is presented for the governing equations. Using this solution, the results of the electromechanical behavior of the cylinders under different boundary conditions in two cylinder heads are estimated and evaluated. Then these results are compared with the finite element method’s results. The analytical solution is not a series solution, and therefore there would be no need to investigate the convergence. Furthermore, it is of less computational volume. The results obtained from the analytical method and the finite element method have a good agreement, showing that the presented analytical solution can be used with higher accuracy.

In this paper, FIRSTly, it is shown that the standard account of the concept ‘practice’ in the Wittgenstein’s philosophy is a therapeutic one. According to this account, the normative standard of the rightness and wrongness of the use of words cannot be put into words. This account can be seen as the negative aspect of the whole idea of normativity. Secondly, I argue that therapy alone is not adequate. In ORDER to give a plausible account of the whole idea of normativity, the positive aspect of the concept ‘practice’ has to be presented. Thirdly, I suggest that distinguishing between the FIRST ORDER and the second ORDER account of the concept ‘practice’ might be a way of explaining the positive aspect. According to the FIRST ORDER account, the constituents of the concept ‘practice’ have to be given. The second ORDER account sketches out the idea that practice goes all the way down. Our confrontation with things in the world is based upon doing rather than theorising at the very basic level. Overt activities ultimately depend on mental activities.

ABSTRACT Using a high safety factor in construction without precise calculations imposes high costs on the execution project. Using a low safety factor exposes the project to a high risk of failure. Therefore, determining the impact of uncertain parameters greatly aids in designing with the least possible risk. The contribution of each parameter to uncertainty can be identified through sensitivity analysis. This paper considers the impact of the uncertainty of three parameters—elastic modulus, pipe thickness, and pipe diameter—on the buckling of a marine pipeline, as these three parameters significantly influence the geometry of the pipeline. Two analyses, FIRST ORDER Second Moment (FORM) and Second ORDER Reliability ORDER (SORM), are employed for the sensitivity analysis of the pipeline parameters. The FIRST ORDER Second Moment analysis provides four importance vectors, while the Second ORDER Reliability ORDER analysis yields only one importance index. The results indicate that the pipe diameter has a greater impact compared to the other two parameters. The smallest difference is observed between the elastic modulus and pipe thickness, suggesting that these two parameters have a similar level of sensitivity. The reliability index values in FORM and FOSM are distinct yet close to each other, with FOSM yielding a lower value.   INTRODUCTION Submarine pipelines are subjected to various forces of different natures during installation and operation, such as hoop stress, axial force, and so on. The high temperature and pressure gradient between the well and the terminal (platform or refinery) in gas transmission lines generates significant axial force in the pipeline. If the mentioned forces exceed a certain threshold, despite the friction of the seabed and the weight of the submerged pipeline, they can cause large DEFORMATIONS in the pipeline or, in other words, lead to pipeline buckling. The DNV-F-110 standard (DET NORSKE VERITAS) classifies this type of buckling as global buckling, which can be categorized into horizontal and vertical buckling based on the deformation mode (Shabani et al., 2017). Global buckling is a common and likely phenomenon that can occur during the operation of a pipeline. This phenomenon can lead to defects such as failure, bending, and so on in the pipeline (Karampour and Albermani, 2014). As accessible oil fields are depleted, the need for drilling at greater depths has increased, resulting in higher temperatures and pressures due to the transportation of hydrocarbon products. Consequently, the likelihood of global buckling occurring in deep waters has increased. Due to the lower critical temperature for lateral buckling compared to the critical temperature for vertical buckling, non-buried pipelines often face lateral buckling (Karampour et al., 2013). The presence of an initial vertical defect in a non-buried submarine pipeline makes the occurrence of vertical buckling possible. MATERIALS AND METHODS The marine pipeline studied in this research has a diameter of 0.816 meters, a thickness of 0.0242 meters, and a length of 40 meters. The Poisson's ratio of the pipeline is considered to be 0.3, and the thermal expansion coefficient is taken as 1.5 × 10^-6. The residual stress from the installation of the pipeline is not considered in this study due to its insignificance compared to other forces affecting the axial force. Horizontal compressive forces are considered with a high conservative coefficient. RESULTS Three parameters E, t, and D are geometric parameters of the underwater pipeline subjected to buckling. Among the three parameters mentioned, parameters of pipeline thickness and diameter are resistance variables, and elasticity modulus is the load resistance variable. The results show that the pipeline diameter is the most important among the mentioned parameters and the modulus of elasticity is the least important. In the analysis of FORM and FOSM, there are reliability indicators for both of them, which are the basic part of reliability evaluation by the two mentioned analyses. The reliability index values in FORM and FOSM are distinct but close values, which are lower in FOSM. According to Figure (2), the pipeline diameter is the most important variable of the defined limit state function. As it is clear, the smallest difference is between E and t, in other words, these two parameters are close to each other in terms of sensitivity, with the difference that one is a load variable and the other is a resistance variable. Also, this point can be concluded that the values elated to the importance index κ have a big difference with other importance indices and the rest of the importance indices have close values. DISCUSSION AND CONCLUSION Among the three mentioned parameters, the thickness and diameter of the pipeline are variables affecting resistance, while the modulus of elasticity is a variable affecting load resistance. The diameter of the pipeline holds the greatest importance among the mentioned parameters, while the modulus of elasticity is of lesser significance. The reliability index values in FORM and FOSM are distinct yet close to each other, with FOSM yielding a lower value. The approximate equality of the reliability index in the two analyses can be expressed by the difference in the importance vectors in the two analyses, and as the importance of the variables decreases, the difference becomes more pronounced. The values related to the importance index Kappa show a significant difference compared to other importance indices, while the remaining importance indices have values that are close to each other.