Journal of Mathematical Extension
Volume:13 Issue: 2, Spring 2019

Let $L$ be a complete lattice. Let $R$ be a commutative ring, $M$ an $R$-module and $\nu$ an $L$-submodule of $M$. $\nu$ is called a classical prime $L$-submodule of $M$ if for any $L$-fuzzy points $a_r, b_s\in L^R$ and $x_t\in L^M$ ($a, b\in R$, $x\in M$ and $r, s, t\in L$), $a_rb_sx_t\in \nu$ implies that either $a_rx_t\in \nu$ or $b_sx_t\in \nu$. Assume that $\nu$ is an $L$-submodule of $mmu\in L(M)$. We say that $\nu$ is a $2$-absorbing $L$-submodule of $\mu$ if for any $L$-fuzzy points $a_r, b_s\in L^R$ and $x_t\in L^M$ ($a, b\in R$, $x\in M$ and $r, s, t\in L$), $a_rb_sx_t\in \nu$ implies that $a_rb_s\mu\subseteq \nu$ or $a_rx_t\in \nu$ or $b_sx_t\in \nu$. In this case every prime $L$-submodule of $M$ is a classical prime $L$-submodule, and every classical prime $L$-submodule is a $2$-absorbing $L$-submodule. In this paper we give some basic results concerning these classes of $L$-submodules. Finally we topologize $L-Cl.Spec(M)$, the set of all classical prime $L$-submodules of $M$, with Zariski topology.

Let $R$ be a commutative ring and let $M$ be an $R$-module. In this paper, we introduce the dual notion of fuzzy prime (that is, fuzzy second) submodules of $M$ and explore some of the basic properties of this class of submodules. We say a non-zero fuzzy submodule $\mu$ of $M$ is fuzzy second if for each $r \in R$, we have $1_r.\mu = \mu$ or $1_r.\mu = 1_\theta$. It is shown that the fuzzy second submodules is a proper subclass of the fuzzy coprimary submodules.

In this paper, the problem of MHD ow and radiation heat transfer of nanouids against a at plate in porous medium with the eects of variable surface heat ux and rst-order chemicalreaction is investigated numerically. Three dierent types of nanoparticles, namely Cu, Al2O3 andAg are considered by using water as a base uid with Prandtl number Pr = 6:2. The governingpartial dierential equations can be written as a system of nonlinear ordinary dierential equationsover a semi-innite interval using a similarity transformation. A new eective collocation methodis proposed based on exponential Bernstein functions to simulate the solution of the resultingdierential systems. The advantage of this method is that it does not require truncating ortransforming the semi-innite domain of the problem to a nite domain. In addition, this methodreduces the solution of the problem to the solution of a system of algebraic equations. Graphicaland tabular results are presented to investigate the inuence of the solid volume fraction, types ofnanoparticles, radiation and suction/blowing, magnetic eld, permeability, Schmidt number andchemical reaction, on velocity, temperature and concentration proles. The obtained results ofthe current study are in excellent agreement with previous works.

Data envelopment analysis (DEA) is one of the best tools for evaluating units with multiple inputs and multiple outputs. In multiplier models of DEA sometimes data on inputs or outputs is available, and/or some assumptions are imposed to the model that result in some conditions on weights vectors, in addition to non-negative conditions of the weight vectors of u and v. These conditions are called weights restrictions. Applying weights restrictions on the multiplier model creates new variables in its corresponding DEA model. Thus, applying weights restrictions on the multiplier model leads to the development of technology model in the envelopment form. This makes the projection of an inefficient unit that is on the efficiency frontier of the developed technology not to be necessarily producible. Therefore, applying weights restrictions on multiplier models will enjoy this defect. To solve this problem weights restrictions are applied through the trade-off matrix that is a simultaneous change in inputs and outputs. Applying weights restrictions in DEA model helps us keep under control the significance of one output to other output or the significance of one input to other input and/or the significance of one output to one input.

In this paper,the notion of a non-commutative hypervaluation as a generalization of a non-commutative valuation is introducedand some results are proved in this respect.Also, we define the notion of a non-commutative hypervaluationover a Krasner hyperfield and some basic results and characterizations are obtained. Finally, we introduce the non-commutative discretehypervaluation of a hyperfield and investigate the important properties.

The purpose of the present paper is to introduce a class $\boldsymbol{D}% _{\lambda ,\delta }^{k,\alpha }C_{0}(\beta )$ of bi-concave functions defined by a differential operator. We find estimates on the Taylor-Maclaurin coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ for functions in this class. Several consequences of these results are also pointed out in the form of corollaries.

Using the concept of eta-convex functions as generalization of convex functions, we inquiry about the relation between minimization problem and Kuhn-Tucker problem with new settings and give sucient and necessary optimality condition. Also the relation between minimization problem and it's Mond-Weir dual problem in convex case is investigated.

In this paper, we classify translation surfaces of Type1 in the threedimensional semi- isotropic space {SI}^{3} under the condition Delta ^{J}x_{i}=\lambda _{i}x_{i}, where \Delta ^{J} denotes the Laplacian of the surface with respect to the first, the second and the third fundamental forms. We also give explicit forms of these surfaces.