Journal of Mathematical Extension
Volume:16 Issue: 9, Jan 2022

In this paper, we defined the notions of (∈, ∈)-fuzzy implicative deductive systems and (∈, ∈ ∨ q)-fuzzy implicative deductive systems of hoops and studied some traits and tried to define some definitions that are equivalent to them. Thus by using the notion of (∈, ∈)- fuzzy deductive system of hoop, we defined a new congruence relation on hoop and show that the algebraic structure that is made by it is a Brouwerian semilattice, Heyting algebra and Wajesberg hoop.

We say that an isometric immersion hypersurface $ x:M^n\rightarrow\mathbb{E}^{n+1}$  is ofnull $L_k$-2-type  if  $x =x_1+x_2$, $ x_1, x_2:M^n\rightarrow\mathbb{E}^{n+1}$ are smooth maps and $L_k x_1 =0, ~ L_k x_2 =\lambda x_2$,  $\lambda$ is non-zero real number,  $L_k$ is the linearized operator ofthe $(k + 1)$th mean curvature of the hypersurface, i.e., $L_k( f ) =\text{tr} (P_k \circ \text{Hessian} f )$ for$f \in C^\infty(M)$, where $P_k$ is the $k$th Newton transformation,  $L_k x = (L_k  x_1, \ldots , L_k x_{n+1}), ~x = (x_1, \ldots, x_{n+1})$. In this article,  we classify $\delta (2)$-idealEuclidean hypersurfaces of  null $L_1$-2-type.

To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing as compared to simple models. This paper considers mixture of Topp-Leone distributions under classical and Bayesian perspective based on complete sample. The new distribution which exhibits decreasing and upside down bathtub shaped density while the distribution has the ability to model lifetime data with decreasing, increasing and upside down bathtub shaped failure rates. We derive several properties of the new distribution such as moments, moment generating function, conditional moment, mean deviation, Bonferroni and Lorenz curves and the order statistics of the proposed distribution. Moreover, we estimate the parameters of the model by using frequentist and Bayesian approaches. For Bayesian analysis, five loss functions, namely the squared error loss function (SELF), weighted squared error loss function (WSELF), modified squared error loss function (MSELF), precautionary loss function (PLF), and K-loss function (KLF) and uniform as well as gamma priors are considered to obtain the Bayes estimators and posterior risk of the unknown parameters of the model. Furthermore, credible intervals (CIs) and highest posterior density (HPD) intervals are also obtained. Monte Carlo simulation study is done to access the behavior of these estimators. For the illustrative purposes, a real-life application of the proposed distribution to a tensile strength data set is provided

Controlled frames in Hilbert spaces have been introducedby Balazs, Antoine and Grybos to improve the numerical output of inrelation to algorithms for inverting the frame operator. In this paper,we introduce some new concepts and show results on controlled fusionframes for Hilbert spaces. It is shown that controlled fusion framesare a generalization of fusion frames giving a generalized way to obtainnumerical advantage in the sense of preconditioning to check the fusionframe condition. For this end, we introduce the notion of Q-duality forControlled fusion frames. Also, we survey the robustness of controlledfusion frames under some perturbations.

‎This contribution ‎‎deals with the mathematical modeling of R-norm entropy and R-norm divergence in quantum logics. ‎‎W‎e extend some results concerning the R-norm entropy and conditional R-norm entropy given in (Inf. Control 45, 1980), ‎‎‎‎‎to ‎the‎ quantum logics.‎ Firstly, the concepts of ‎‎R-norm entropy ‎and‎ ‎conditional R-norm entropy in quantum logics are introduced. ‎We ‎prove‎ ‎the concavity property for the notion of R-norm entropy in quantum logics ‎and we ‎show‎‎ that this entropy measure does not have the property of sub-additivity in a true sense. ‎It ‎is ‎prove‎n that ‎‎‎the monotonicity ‎property ‎for ‎the suggested type of ‎conditional ‎version ‎of ‎R-norm ‎entropy, holds. Furthermore, we introduce the concept of R-norm divergence of states in quantum logics and we derive basic properties of this quantity. ‎In particular‎, a relationship between the R-norm divergence and the R-norm entropy of partitions is provided.‎‎‎

In this paper, we present the notion of weakly compact topological fuzzes and give some characterizations of them. In particular, a characterization of weakly compactness are given by the closedness of the projection fuzz maps. Also, we study some properties of proper fuzz maps as an important class of closed fuzz maps.

‎In this paper‎, ‎we formulate and study the duality problems in Wolfe type for the mathematical programs with vanishing constraints in nonsmooth case‎, ‎whereas Mishra \emph{et al.} (Ann Oper Res 243(1):249–272‎, ‎2016) investigated it in smooth case‎. ‎Also‎, ‎we derive the weak‎, ‎strong‎, ‎strict converse duality results for the problems with Lipschitzian data utilizing Clarke subdifferential‎.

In the current article, we introduce and investigate two new families of strongly Ozaki $\lambda$-pseudo bi-close-to-convex functions in the open unit disk. We determine upper bounds for the second and third coefficients of functions belonging to these new subclasses. Also, we point out several certain special cases for our results.

In this study, pseudo almost automorphic(PAA) solutions ofa Lienard-type system with multiple delays are considered. By applyingthe main features of PAA, Banach xed point theorem and some dier-ential inequalities, sucient conditions for the existence and uniquenessof such solutions are obtained . Since the PAA is more general than thealmost periodicity(AP) and pseudo almost periodicity(PAP), this workis a new and complementary compared to previous studies. In addition,an example is given to show the correctness of the created conditions.

The main purpose of this paper is to investigate the effect of Φ-derivatives on the commutativity of rings and algebras. Let R be a 2-torsion free prime ring, d:R → R be a Φ-derivation such that Φ is an epimorphism and d Φi = Φd = d. If [Φ(a), Φ(x)]d(y) = d(x) [y,a] for all x,y,a ⋲ R, then R is commutative or d is zero. Another result in this regard reads as follows. Let (A, *) be a unital, involutive algebra, and let ψ : A ⨯ A → A be a *-two variable Φ-derivation such that ψ(e,a_{0}) = e for some a_{0} ⋲ A, where e is the unity of A. If {a ⋲ A : ψ(a, a_0) = 0} = {0}, then A is commutative. Some other related results are also discussed