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Copyright of Lusitania Sacra is the property of Universidade Catolica Portuguesa, Centro de Estudos de Historia Religiosa and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let $\mathbb{F}_{q}$ denote the finite field of $q$ elements, where $q$ is a prime power. In this paper, we study the twisted Hessian curves denoted $H_{a,d}(\mathbb{F}_{q}[e])$ over the ring $\mathbb{F}_{q}[e]$, where $e^{3}=e^2$ and $(a,d)\in (\mathbb{F}_{q}[e])^{2}$. More precisely, we study some arithmetical properties of this ring and using the Twisted Hessian equation, we define the twisted Hessian curves $H_{a,d}(\mathbb{F}_{q}[e])$. This work study the twisted Hessian curve helped us to define two twisted Hessian over the finite field $\mathbb{F}_{q}$. We end this paper by giving the classification of the elements in $H_{a,d}(\mathbb{F}_{q}[e])$.

In this paper, a combination of boundary value problems for the nonhomogeneous analytic equations have been studied. The aim of this paper is to find a solution for the Schwarz-(Dirichlet-Neumann) problem and obtain its solvability conditions.

This paper presents several sufficient conditions for the existence of at least three classical solutions of a boundary value problem for a fourth-order difference equation. Fourth-order boundary value problems act as models for the bending or deforming of elastic beams. In different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others, the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. Our technical approach is based on variational methods. An example is included in the paper. Numerical computations of the example confirm our theoretical results.

With the current struggles of the world nowadays with several epidemics, modeling the dynamics of disease outbreaks has become much more important than any time before. In this context, the present paper aims at studying a stochastic hepatitis B virus epidemic model with treatment cure rate. Our model consists of three epidemic compartments describing the interaction between the susceptible, the infected and the recovered individuals; an SIR model where the infected individuals transmit the infection to the susceptible ones with a transmission rate perturbed by white noise. Our paper begins by establishing that our hepatitis B stochastic model has unique global solution. It moves then to giving sufficient conditions for the stochastic extinction and persistence of the hepatitis B disease. Finally, our paper provides some numerical results to support the analytical study, showing numerically that the treatment cure rate facilitates the extinction of the hepatitis B disease among the population.

In this paper, we are interested to prove a general fixed point theorem for a mapping in fuzzy b-metric spaces. The results in this paper generalize the Banach fixed point theorem in fuzzy b-metric spaces. To show the significance of our result an application is presented to establish the existence of a solution for an integral equation.

This paper aims to identify and analyze the theme of Financial Education (EF) in academic research in the latest editions of ENEM, CIBEM and SIPEM. The methodological approach is qualitative of the bibliographical. We believe that the SBEM electronic database enabled a bibliographic survey of academic production. The results pointed to the need to promote discussions about EF in the school environment, recognizing its contribution to the integral and critical formation of the subject, its articulation with social, political, economic, environmental and cultural problems, making education a progressively active social force. They also highlighted the urgent need to implement continuing education programs that enable the planning and implementation of more assertive teaching practices in this area.

Since the publication of the Base Nacional Comum Curricular (National Common Curricular Base), Financial Education has become a topic to be developed in the classroom, mainly related to the mathematics curriculum. In this sense, teachers are in charge of addressing it. In view of this, in this paper we aim to discuss understandings of Financial Education aimed at Basic Education and produced by mathematics undergraduates, mobilized in two training meetings. Based on Critical Mathematics Education and the qualitative research paradigm, we identified five topics using content analysis procedures. The results of the analysis of these topics indicate that the future teachers have knowledge of School Financial Education, although they relate the subject to economic aspects and Financial Mathematics.

Physical activity provides numerous benefits to physical and mental health, even affecting learning. Despite this, one of the main modern problems is represented by sedentariness. Children and adolescents spend most of their time at school. Therefore, a valid teaching methodology to benefit from the positive effects of physical activity and reduce sedentary moments could be interdisciplinary learning, which integrates physical education into various curricular subjects. Specifically, this paper proposes a systematic literature review of some protocols (n = 11) to investigate whether an elementary school’s integrated physical education and mathematics curriculum can improve learning in the logic-cognitive domain. From all the studies reviewed, the strong importance of interdisciplinary teaching, linking mathematics to physical education, was confirmed to reduce sedentary moments and enable a better attitude toward this discipline, breaking down the many stereotypes that often paint it as a complex and abstract subject.

Copyright of Educação Matemática Pesquisa is the property of Pontificia Universidade Catolica de Sao Paulo, Programa de Estudos Pos-Grad. em Educacao Matematica and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we investigate the classical identities of the repunit sequence with integer indices in light of the properties of Horadan-type sequences. We highlight particularly the Tagiuri-Vajda Identity and Gelin-Cesàro Identity. Additionally, we prove that no repunit is a perfect power, either even or odd. Finally, we address a divisibility criterion for the terms of repunit rn by a prime p and its powers.

In this paper, we study the Cauchy problem for the Fractional Porous Medium Equation in Rn for n ≥ 2. By using the contraction mapping method, Littlewood-Paley theory and Fourier analysis, we get, when 1 < β ≤ 2, the local solution v ∈ XT := LT ∞(FBp,r (2 − 2m −β + n/p' )(Rn))∩ LTρ1(FBp,r s1(Rn))∩ LTρ2(FBp,r s2 ( Rn)) with 1 ≤ p < ∞, 1 ≤ r ≤ ∞, and the solution becomes global when the initial data is small in critical Fourier-Besov spaces FBp,r (2 − 2m −β + n/p' )(Rn) . In addition, We establish a blowup criterion for the solutions. Furthermore, the global existence of solutions with small initial data in FB∞,1 (2 − 2m −β + n )(Rn) is also established. In the limit case β = 1, we prove global well-posedness for small initial data in critical Fourier-Besov spaces FBp,1 (2 − 2m + n/p' )(Rn) with 1 ≤ p < ∞ and FB∞,1 (2 − 2m + n )(Rn), respectively.

The paper deals with the following Robin problem $$ \left\lbrace \begin{aligned} - \mathcal{M} \left( \int _{\Omega} \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)} dx + \int _{\partial \Omega } \frac{a(x)}{p(x)} \vert \nabla u \vert ^{p(x)} d \sigma \right) \mathop{\rm div} (\vert \nabla u \vert ^{p(x)-2} \nabla u) &= \lambda h(x,u) \ \ \text{ in } \Omega,\\ \vert \nabla u \vert ^{p(x)-2} \frac{\partial u}{\partial \nu} + a(x) \vert u \vert ^{p(x)-2} u &=0 \quad \quad \quad \ \text{ on } \partial \Omega . \end{aligned} \right. $$ The goal is to determine the precise positive interval of $\lambda $ for which the problem admits at least two nontrivial solutions via variational approach for the above problem without assuming the Ambrosetti-Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the fountain theoreom with Cerami condition.

Quite recently, a new minimal structure m^\star_H has been introduced in [ ] by using a minimal structure m and a hereditary class H. In this paper, we introduce and investigate the notion of H-scattedness in a hereditary minimal space (X, m, H).

This paper focuses on the time fractional wave problem with the use of a new fractional derivative in Colombeau algebra. Using Banach's fixed point theorem and Laplace transforms, we give and prove the integral solution of the problem. In Colombeau's algebra, The existence and uniqueness of the solution are demonstrated using the Gronwall lemma.

The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds

This paper concerns another approach for solving the matrix differential equations of the second order $X''(t)=AX'(t)+BX(t)$, where $A$, $B$ are square matrices of order ${d\times d}$. Such approach is based on some properties of matrix square root and the linear difference equation. We establish various new results and explicit formulas for the solutions of this type of matrix differential equations. Moreover, because of the non-commutativity condition between $A$ and $B$, the matrix fundamental system will play an important role. Finally, our results and their robustness, are validated by providing some examples and applications.

In this paper, we consider matrix maps over a zero-symmetric right nearring $N$ with 1. We define the notions of essential ideal, superfluous ideal, generalized essential ideal of a matrix nearring and prove results which exhibit the interplay between these ideals and the corresponding ideals of the base nearring $N$. We discuss the combinatorial properties such as connectivity, diameter, completeness of a graph (denoted by $\mathcal{L}_{g}(H)$) defined on generalized essential ideals of a finitely generated module $H$ over $N$. We prove a characterization for $\mathcal{L}_{g}(H)$ to be complete. We also prove $\mathcal{L}_{g}(H)$ has diameter at-most 2 and obtain related properties with suitable illustrations.

Let A and B be two algebras. A linear mapping 𝜟:A → B is called a strongly generalized derivation of order n, if there exist the families {𝐸_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐹_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐺_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} 𝑎𝑛𝑑 {𝐻_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} of linear mappings which satisfy 𝛥(𝑎𝑏) = Σ [𝐸_𝑘(𝑎) 𝐹_𝑘(𝑏) + 𝐺_𝑘(𝑎)𝐻_𝑘(𝑏)] 𝑛𝑘 =1 for all 𝑎, 𝑏 ⋲ 𝐴. The main purpose of this paper is to study the image of such derivations. Our main result on the image of strongly generalized derivations of order one reads as follows: Let A be a unital, commutative Banach algebra and let 𝛥: 𝐴 → 𝐴 be a continuous strongly generalized derivation of order one; that is, there exist the linear mappings 𝐸, 𝐹, 𝐺, 𝐻: 𝐴 → 𝐴 satisfying 𝐷(𝑎𝑏) = 𝐸(𝑎) 𝐹(𝑏) + 𝐺(𝑎) 𝐻(𝑏) for all 𝑎, 𝑏 ⋲ 𝐴. Let 𝐸, 𝐹, 𝐺 and 𝐻 be continuous linear mappings. We prove that, under certain conditions, 𝐻 (𝐴), 𝐸(𝐴) 𝑎𝑛𝑑 𝛥(𝐴) are contained in the Jacobson radical of A. This result generalizes Singer-Wermer theorem about the image of continuous derivations on commutative Banach algebras.

In this paper we are interested to the existence and multiplicity of sign changing radial solutions of problem of elliptic equations $\Delta U(x)+\varphi(|x|)f(U)=0$ with a nonlinear boundary conditions on exterior of the unite ball centered at the origin in $\mathbb{R}^{N}$ such that $ u(x) \rightarrow 0$ as $ |x|\to \infty $, with any given number of zeros where the nonlinearity $ f(u) $ is odd, superlinear for $ u $ lager enough and $ f0$ on $(\beta,\infty) $. The function $\varphi>0$ is $ C^{1} $ on $ [R,\infty) $ where $ 02(N-1) $ and $ N>2 $ for large $ |x| $.

In this paper, an optimal trajectory of the rocket angle with a variable mass will be calculated by considering the aerodynamic forces, the acceleration of gravity and moves with a non-rectilinear motion from a initial state to a final state with a known altitude. The aim is to optimize the lateral oset of the rocket. For this, we formulate an optimal control problem where the rocket angle is the control. In order to solve the problem, let applied Shooting method's based on the Pontryagin's maximum principle, and study the precision and a duration time. Finally, we validate the results by using MATLAB software.

In this paper, we investigated non strong hyperconnected generalized topological spaces. Ekici \cite{Eki11} and Devi \cite{Ren12} have provided the results of hyperconnectedness for strong generalized topological spaces. We generalized these results for arbitrary generalized topological spaces. Through the notion of hyperconnectedness of arbitrary generalized topological spaces, we constructed an example which fails Hausdorff characterization of topological spaces \lq\lq A first countable spaces is Hausdorff if and only if every convergent sequence has unique limit\rq\rq. This example also serves the purpose of constructing Anti Hausdorff Fr$\acute{e}$chet space in which every convergent sequence has unique limit required by Novak in \cite{Nov39}.

We give in this paper some useful calculus results related to the s-H$\ddot{o}$lder subdifferential of extended-real valued functions defined on arbitrary real Banach spaces.

This paper discusses relative uniform convergence of triple sequence of functions that are defined on a compact domain. Another central idea that is discussed is the regular relative uniform convergence and the Cauchy relative uniform convergence of triple sequence of functions. The idea that a continuous triple sequence defined on a compact domain is relative uniform convergent if and only if it is relative uniform Cauchy has been discussed and established. Subsequently, a glimpse into Cesaro summability of triple sequences and a theorem regarding triple Cesaro summability of bounded relative uniform triple sequence of functions have been introduced.

Throught this paper,we study the cauchy problem for the conformable fractional Navier-Stokes Equations (FNSE) with finite delay external forces, contains some hereditary features, on a bounded domain. We prove that there exist a unique local mild solutions for the initial datum, by using semigroup theory, conformable fractional calculus and Banach contraction theorem.With more conditions on delay external forces we establish the globality and continuation of the mild solutions.

In this paper, we propose and study a Bilevel quasimonotone Variational Inequality Problem (BVIP) in the frame work of Hilbert space. We introduce a new modified inertial iterative technique with self-adaptive step size for approximating a solution of the BVIP. In addition, we established a strong convergence result of the proposed iterative technique with an adaptive step-size conditions without prior knowledge of Lipschitz’s constant of the cost operators as well as the strongly monotonicity coefficient under some standard mild assumptions. Finally, we provide some numerical experiments to demonstrate the efficiency of our proposed methods in comparison with some recently announced results in the literature.

In this paper, we investigate stochastic optimal intervention control of mean-field nonlinear random Poisson-jump-system with related noisy process. We derive the necessary conditions of optimality for partially observed optimal intervention control problems of mean-field type. The coefficients depend on the state of the solution process as well as of its probability distribution and the control variable. The proof of our main result is obtained by applying L-derivatives in the sense of Lions. In our control model, there are two models of jumps for the state process, the inaccessible ones which come from the random Poission process and the predictable ones which come from the intervention control. Finally, we apply our result to study conditional mean-variance portfolio selection problem with interventions, where the foreign exchange interventions are intended to contain excessive fluctuations in foreign exchange rates and to stabilize them.

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

In this paper we investigate existence and non-existence of solutions for a Dirichlet boundary value problem involving the $(p_1(k), p_2(k))$-Laplacian operator when variational methods are applied to obtain the results.

In this paper it is shown that, if a spacetime contains a Cauchy surface then the causally convex hulls of compact sets are compact. The converse is not true in general, however if the spacetime is causal then the compactness of causally convex hulls of compact sets of the spacetime ascertain the existence of a Cauchy surface. Also, the convex hull operator is denoted and some results regarding it has been explored.

Let L = K(α) be an extension of a number field K where α satisfies the monic irreducible polynomial P(X) = Xp −a ∈ R[X] of prime degree p and such that a is pth power free in R := OK (the ring of integers of K). The purpose of this paper is to give an explicit formula for the ideal discriminant DL/K of L over K involving only the prime ideals dividing the principal ideals aR and pR. As an illustration, we compute the discriminant DL/K of a family of septic and quintic pure fields over quadratic fields. Hence a slightly simpler computation of discriminant DL/K is obtained.

In the present paper, we study $p(x)-$biharmonic problem involving $q(x)-$Hardy type potential with no-flux boundary condition. By using the mountain pass type theorem and Ekeland variatoinal principle, we obtain at least two nontrivial weak solutions.

In this paper, a study of generalized quasi-conformal curvature tensor has been made on the four dimensional spacetime of general relativity. Some results related to the application of such spacetime in the general relativity are obtained. Perfect fluid and dust fluid cosmological models have also been studied.

In this paper we continue as in \cite{Rezguietal} to exploit the modified variants of Bessel function in the framework of $q$-theory to construct wavelet operators. A generalized $q$-Bessel type function has been introduced leading to an associated mother wavelet which in turns has induced a continuous wavelet transform. Finally, Plancherel/Parceval type relations have been proved. Such variants of wavelets permit in some sense to approximate solutions of ODEs and PDEs by transforming them to recurrent sequences.

In this present paper, we will envisaged the existence and uniqueness of solutions for the following boundary value problem for a nonlinear fractional differential equation involving with $\psi$-Caputo fractional derivative. Our results are proved under Banach contraction principle and Krasnoselkii's fixed point theorem.

The main objective of this research paper is to embed $\Phi$-Caputo fractional derivative in the Colombeau algebra of generalized functions and we investigate the existence and uniqueness of the Cauchy problem involving $\Phi$-Caputo fractional derivatives in the extended Colombeau algebras. Finally, we give an example of how the ideas presented in the document can be applied.

Let $K$ be a pure number field with $\alpha$ a complex root of a monic irreducible polynomial $F(x) = x^{120}-m \in \mathbb{Z}[x]$ with $ m \neq \pm 1 $. In this paper, we study the monogenity of $K$. More precisely, we prove that if $m$ is square-free, $m \not \equiv 1\md{4}$, $m \not \equiv \pm 1 \md{9} $, and $\ol{m}\not \in \{ \mp 1, 7, 18 \} \md {25}$, then $K$ is monogenic. On the other hand, if $m \equiv 1\md{4}$, $m \equiv 1 \md{9} $, or $m \equiv 1 \md{25}$, then $K$ is not monogenic. Our results are illustrated by some computational examples.

Let $M$ be an unital left module over a ring $R$ with unity. We define an undirected (\textit{nil graphs}) for the module $M$ as a graph whose vertex set is $M^*=M-{0}$ and any two distinct vertices $x$ and $y$, in these graphs, are adjacent if and only if there exist $r \in R$ such that $r^{2} (x+y) = 0$ and $r(x+y)\neq 0$. In this paper, we study the graph's adjacency, diameter, radius, and eulerian and hamiltonian properties. We also defined another nil graph $\Gamma^{*}_{N} (M)$, in which we reduced the vertex set to $N(M^*)$, set of all non-zero nil elements of the module, and keep the adjacency relation same as that of $\Gamma_{N} (M)$. We investigate the adjacency, diameter, radius, eulerian and hamiltonian properties of the graph $\Gamma^{*}_{N} (\mathbb{Z}_{p^n})$ and compare these properties among both the graphs.

One of an important problems in finite groups theory is, characterizable of groups by specific property. In this paper, we prove that groups $B_4 (q)$, where $3< q$ be prime number and $ \frac{q^4+1}{2}$ is a prime number, can be uniquely determined by the largest elements order and the order of group.

In this paper, we introduce the concept of graded S−comultiplication modules. Several results concerning graded S−comultiplication modules are proved. We show that N is a graded S−second submodule of a graded S−comultiplication R−module M if and only if Ann_R(N) is a graded S−prime ideal of R and there exists x ∈ S such that xN ⊆x- for every x- ∈ S.

In this paper, we consider a Laguerre differential operator on half line by accomplishing harmonic analysis tools with respect to the operator. We study some definitions and properties of Laguerre continuous wavelet transform. We also explore generalized Laguerre Fourier transform and convolution product on half line associated with the Laguerre differential operator. Also a new continuous wavelet transform associated with Laguerre function is constructed and investigated.

In this paper, we study the limit cycles of a perturbed differential in $\mathbb{R} ^2$, given by \begin{equation*} \left\{ \begin{array}{ccl} \overset{.}{x} &=& y ,\\ \overset{.}{y} &=& -x-\epsilon (1+\sin^n (\theta) \cos^m (\theta))H(x,y), \end{array} \right. \end{equation*} where $\epsilon$ is a small parameter, $m$ and $n$ are non-negative integers, $\tan(\theta)=y/x$, and $P(x,y)$ is a real polynomial of degree $n \geq 1$. Using Averaging theory of first order we provide an upper bound for the maximum number of limit cycles. Also, we provide some examples to confirm and illustrate our results.

In this paper, we intend to prove that the modulus $\mathcal{A}-$lacunary statistical convergence of fractional difference double sequences and modulus lacunary fractional matrix of four-dimensions taken over the space of modulus $\mathcal{A}-$lacunary fractional difference uniformly integrable real sequences are equivalent. We represent another version of the Brudno-Mazur Orlicz bounded consistency theorem by using modulus function, lacunary sequence, and fractional difference operator. We show that the four-dimensional $RH-$ regular matrices $\mathcal{A}$ and $\mathcal{B}$ are modulus lacunary fractional difference consistent over the multipliers space of modulus fractional difference $\mathcal{A}-$summable sequences and an algebra $Z.$

In the present paper, we prove the algebraic independence of a finite number of real continued fractions which have partial quotients that increase rapidly. We then use a general Liouville criteria to justiy the algebraic independence of limits of some real series. We note that these results extend some work of Bundschuh and we use a new and simple method.

In this paper we introduce the notion of generalized permuting tri-derivations, g-derivations, and g-tri-derivations on lattices, and we study and generalize some properties discussed in [14] and [29]. We also give some properties characterizing the g-derivations, the g-tri-derivations, and the generalized permuting tri-derivations and their trace.

In this paper, we review the works of the authors ([10], [23]) etc. and establish two theorems on degree of convergence of a function g and conjugate of a function ˜g in generalized Zygmund (Z(λ) r , r ≥ 1) space using Hausdorff-Matrix (∧T) operator of its Fourier series and conjugate Fourier series respectively based on the findings of the review. Our results generalize several earlier results. Some important corollaries are also deduced from our main theorems.

In this paper we consider a mathematical model describing a quasistatic frictional contact problem between a deformable body and an obstacle, say a foundation. We assume that the behavior of the material is described by a linear electro-elastic constitutive law with long memory. The contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. Moreover, we consider a slip dependent coefficient of friction. We derive a variational formulation for the model, in the form of a coupled system for the displacements and the electric potential. Under a less assumption on the coecient of friction, we prove the existence result of weak solutions of the model. We can show the uniqueness of solution by adding another condition (3.10). The proofs are based on arguments of time-dependent variational inequalities, dierential equations and Banach fixed point theorem.

Due to the recent increasing interest in hyperbolic-cotangent types of scalar-or two-dimensional systems of difference equations and treatment of some particular states. This paper presents a natural extension of the p-dimensional of four-systems of this generalized type and treats general states. Which is an extension of Stevic's work (J. Inequal. Appl., 2021, 184 (2021)). We also show these systems are solvable by using appropriate variable transformations and obtaining systems of homogeneous linear difference equations with constant coefficients. Some numerical examples of these systems are presented.

In this paper, boundedness results for the continuous quaternion wavelet transform on Besov, $BMO$ and Hardy $H^{p}$ spaces are established. Furthermore, the continuous quaternion wavelet transform is also studied on the weighted Besov, $BMO_k$ and $H^{p}_{k}$ spaces associated with a tempered weighted function.

In this paper, we study geometry of $(LCS)_{n}$-manifold focusing on some conditions of $E$-Bochner curvature tensor. First, we describe an $E$-Bochner pseudo-symmetric $(LCS)_{n}$-manifold is never reduces to $E$-Bochner semi-symmetric manifold under the condition ($(\alpha^{2}-\rho)\neq0$). Next, we characterize certain results of $(LCS)_{n}$-manifold satisfying $B^{e}(U,V)\xi=0$, $B^{e}(\xi,V)\cdot B^{e}=0$ and $B^{e}(\xi,V)\cdot S=0$.