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In this paper, the competitive equilibrium of the Bertrand game is discussed with bounded rationality and the utility function. When the parameters changing and the number of firms increasing, the competitive equilibrium valve of Bertrand duopoly game is gradually being blurred. When the number of competitors is more than four, it is very difficult to derive the value of the equilibrium points. How to find the general competitive equilibrium points for Bertrand game, which is studied from spatial agglomeration with mean value theorem. A Bertrand duopoly game is proposed based on a demand function. Celestial bodies motion as method is introduced to handle the number and stability of competitive equilibrium points, and the stable points is symmetry. The results are supported by numerical computation and simulations.

In this paper, a three-parameter bounded unit distribution with a flexible hazard rate called the unit generalized log Burr XII (UGLBXII) distribution is derived. To show the importance of the proposed distribution, we establish some of its mathematical properties such as random number generator, ordinary moments, generalized TL moments, conditional moments, reliability and uncertainty measures. We characterize the UGLBXII distribution via innovative techniques. We also present the bivariate‐ and multivariate‐type distributions via Morgenstern (Mor) family and via Clayton family. Six estimation methods such as the maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling methods are adopted to estimate its unknown parameters. We perform simulation studies on the basis of the graphical results to see the performance of the above estimators. Two real data sets are considered to prove the empirical superiority of the proposed model.

This paper proposes a new definition of the nonlinear Fredholm integro-differential equation of the second kind with continuous kernel in two-dimensional (NT-DFIDE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions of NT-DFIDE are obtained by two powerful methods Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.

In this paper, we establish a new integral identity involving local fractional integral on Yang's fractal sets. Using this integral identity, some new generalized Hermite-Hadamard type inequalities whose function is monotonically increasing and generalized harmonically convex are obtained. Finally, we construct some generalized special means to explain the applications of these inequalities.

The conjugate gradient (CG) method is a method to solve unconstrained optimization problems. Moreover CG method can be applied in medical science, industry, neural network, and many others. In this paper a new three term CG method is proposed. The new CG formula is constructed based on DL and WYL CG formulas to be non-negative and inherits the properties of HS formula. The new modification satisfies the convergence properties and the sufficient descent property. The numerical results show that the new modification is more efficient than DL, WYL, and CG-Descent formulas. We use more than 200 functions from CUTEst library to compare the results between these methods in term of number of iterations, function evaluations, gradient evaluations, and CPU time.

In this paper, we investigate blow-up conditions for the semilinear generalized Tricomi equation with a general nonlinear memory term in $ \mathbb{R}^n $ by using suitable functionals and employing iteration procedures. Particularly, a new combined effect from the relaxation function and the time-dependent coefficient is found.

Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.

Among various improvements in fuzzy set theory, a progressive development has been in process to investigate fuzzy analogues of fixed point theorems of the classical fixed point results. In this direction, taking the ideas of $ \theta $-contractions as well as Feng-Liu's approach into account, some new fuzzy fixed point results for nonlinear fuzzy set-valued $ \theta $-contractions in the framework of metric-like spaces are introduced in this paper without using the usual Pompeiu-Hausorff distance function. Our established concepts complement, unify and generalize a few important fuzzy and classical fixed point theorems in the corresponding literature. A handful of these special cases of our notions are pointed and analyzed. Some of the main results herein are further applied to derive their analogues in metric-like spaces endowed with partial ordering and binary relations. Comparisons and nontrivial examples are given to authenticate the hypotheses and significance of the obtained ideas.

In this paper, we consider the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ where $ a > 0 $ is an integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtained the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution. When a solution exists, assuming the positive solutions of the Pell equation $ x^2(a^2+1)-y^2 = -1 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $.

The purpose of this paper is to establish an existence and uniqueness theorem for the nonlocal fourth-order nonlinear beam differential equations with a parameter $ \begin{equation*} u^{(4)}+A(x)u = \lambda f (x, \ u, \ u''), \ 0 subject to the integral boundary conditions: $ \begin{equation*} u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx, \ u''(0) = u''(1) = \int_{0}^{1}q(x)u''(x)dx, \end{equation*} $ where $ A\in \mathbb{C}[0, 1], $ $ \lambda > 0 $ is a parameter and $ p, q \in \mathbb{L}^{1}[0, 1]. $

It is well-known that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis and fuzzy-interval analysis, the inclusion relation (⊆) and fuzzy order relation $\left(\preccurlyeq \right)$ both are two different concepts, respectively. In this article, with the help of fuzzy order relation, we introduce fractional Hermite-Hadamard inequality (HH-inequality) for h-convex fuzzy-interval-valued functions (h-convex-IVFs). Moreover, we also establish a strong relationship between h-convex fuzzy-IVFs and Hermite-Hadamard Fejér inequality (HH-Fejér inequality) via fuzzy Riemann Liouville fractional integral operator. It is also shown that our results include a wide class of new and known inequalities for h-convex fuzz-IVFs and their variant forms as special cases. Nontrivial examples are presented to illustrate the validity of the concept suggested in this review. This paper's techniques and approaches may serve as a springboard for further research in this field.

A comprehensive study on the (2+1)-dimensional hyperbolic nonlinear Schrödinger (2D-HNLS) equation describing the propagation of electromagnetic fields in self-focusing and normally dispersive planar wave guides in optics is conducted in the current paper. To this end, after reducing the 2D-HNLS equation to a one-dimensional nonlinear ordinary differential (1D-NLOD) equation in the real regime using a traveling wave transformation, its optical solitons are formally obtained through a group of well-established methods such as the exponential and Kudryashov methods. Some graphical representations regarding optical solitons that are categorized as bright and dark solitons are considered to clarify the dynamics of the obtained solutions. It is noted that some of optical solitons retrieved in the current study are new and have been not retrieved previously.

In this paper, we consider a Cauchy problem of a coupled linearly-damped wave equations with nonlinear sources. In the whole space, we establish the local existence and show that there are solutions with negative initial energy that blow up in a finite time. Moreover, under some conditions on the initial data, we estimate a lower bound of that time.

In this paper, a class of absolute value equations (AVE) $ Ax-B|Cx| = d $ with $ A, B, C\in \mathbb{R}^{n\times n} $ is considered, which is a generalized form of the published works by Wu [1], Wu and Shen [2] and Mezzadri [3]. Some conditions to guarantee the unique solution of the above AVE are gained. The corresponding results of the above published works are generalized.

In the paper, the authors define a notion of geometric-arithmetic-$ F $-convex functions and, via an integral identity and other analytic techniques, establish several integral inequalities of the Hermite-Hadamard type for geometric-arithmetic-$ F $-convex functions.

In this paper, we study a three step iterative scheme to estimate fixed points of non-expansive mappings in the framework of Banach spaces. Further, some convergence results are proved for such mappings. A nontrivial numerical example is presented to verify our assertions and main results. Finally, we approximate the solution of a boundary value problem of second order differential equation.

In this paper, we introduce the reversal of generalized Banach contraction principle and mean Lipschitzian mapping respectively. Secondly, we prove the existence and uniqueness of fixed points for these expanding type mappings. Further, we extend Wardowski's idea of $ F $-contraction by introducing the reversed generalized $ F $-contraction mapping and use our obtained result to prove the existence and uniqueness of its fixed point. Finally, we apply our results to prove the existence of a unique solution of a non-linear integral equation.

In this paper, coefficient bounds of bi-univalent functions in certain two subclasses, which are defined by subordination are estimated. Some special outcomes of the main results are also presented. Moreover, it is remarked that the given bounds improve and generalize some of the pervious results.

Int his paper, we study and analysis the complex Ginzburg-Landau model or CGL model to obtain some new solitary wave structures through the modified $ (G'/G) $-expansion method. Those solutions can explain through hyperbolic, trigonometric, and rational functions. The graphical design makes the dynamics of the equations noticeable. Herein, we state that the examined method is important, powerful, and significant in performing numerous solitary wave structures of various nonlinear wave models following in physics and engineering as well.

In this paper, a new derivative-free approach for solving nonlinear monotone system of equations with convex constraints is proposed. The search direction of the proposed algorithm is derived based on the modified scaled Davidon-Fletcher-Powell (DFP) updating formula in such a way that it is sufficiently descent. Under some mild assumptions, the search direction is shown to be bounded. Subsequently, the convergence result of the proposed method is established. The performance of the proposed algorithm on a collection of some test problems as well as signal recovery problems is demonstrated in comparison with some existing algorithms with similar characteristics. The results of the numerical experiments confirm the efficiency as well as the robustness of the proposed algorithm by comparing it with some existing methods in the literature.

The Piatetski-Shapiro sequences are sequences of the form $ (\left\lfloor {{n^c}} \right\rfloor)_{n = 1}^\infty $ for $ c > 1 $ and $ c \not\in \mathbb{N} $. It is conjectured that there are infinitely many primes in Piatetski-Shapiro sequences for $ c \in (1, 2) $. For every $ R \ge 1 $, we say that a natural number is an $ R $-almost prime if it has at most $ R $ prime factors, counted with multiplicity. In this paper, we prove that there are infinitely many $ R $-almost primes in Piatetski-Shapiro sequences if $ c \in (1, c_R) $ and $ c_R $ is an explicit constant depending on $ R $.

In data science, it is frequent that new and sophisticated computational methods and tools are used to build predictive models to perform time to event data analysis. Such predictive models based on previously collected data from patients can support decision-making and prediction for the clinical data. Hence, this paper introduced a novel superior distribution, namely a new lifetime Weibull (NLT-W) distribution, using an efficient method to generate new distributions called the T-X method for generating new distributions. Parameter estimation has been done through maximum likelihood estimation (MLE) to show the significance of this proposed model over other competitive models. Comparison to two-parameter Weibull, Exponentiated Weibull (EW), and the and the Kumaraswamy Weibull (Ku-W) indicates that the proposed model could preform better to model various types of survival.

Supertree methods are tree reconstruction techniques that combine several smaller gene trees (possibly on different sets of species) to build a larger species tree. The question of interest is whether the reconstructed supertree converges to the true species tree as the number of gene trees increases (that is, the consistency of supertree methods). In this paper, we are particularly interested in the convergence rate of the maximum likelihood supertree. Previous studies on the maximum likelihood supertree approach often formulate the question of interest as a discrete problem and focus on reconstructing the correct topology of the species tree. Aiming to reconstruct both the topology and the branch lengths of the species tree, we propose an analytic approach for analyzing the convergence of the maximum likelihood supertree method. Specifically, we consider each tree as one point of a metric space and prove that the distance between the maximum likelihood supertree and the species tree converges to zero at a polynomial rate under some mild conditions. We further verify these conditions for the popular exponential error model of gene trees.

Interactive dynamics between effector-tumor-normal cells in a mathematical model related to the growth of cancer in presence of immunotherapy has been discussed in the present paper. Adoptive immunotherapy has been added to the original model proposed by De Pillis et al. [1]. This has been done to get rid of the tumor cells. Different dynamical behaviours of the modified systems have been studied. The existence of the solution and global stability conditions of the healthy equilibrium point is addressed. Corresponding optimal control problem has been formulated to find the best possible way of administration of adoptive immunotherapy by which cancer cells can be eradicated without putting the patient at any health-related risk. To achieve this purpose, the quadratic control principle has been adopted. The dynamical behaviour of the effector-tumor-normal cells model with control is also numerically verified and demonstrated. Through numerical simulations, it is formally shown that the optimal regimens eradicate the tumor load in less time without putting the patientso health at any risk.

In this paper, we consider the following magnetic Laplace nonlinear Choquard equation $ \begin{equation*} -\Delta_A u+V(x)u = (I_{\alpha}*F(|u|))\frac{f(|u|)}{|u|}u, \, \, \text{in}\, \, \mathbb{R}^N, \ \end{equation*} $ where $ u: \mathbb{R}^N\rightarrow C $, $ A: \mathbb{R}^N\rightarrow \mathbb{R}^N $ is a vector potential, $ N\geq 3 $, $ \alpha\, \in\, (N-2, N) $, $ V:\, \mathbb{R}^N \rightarrow \mathbb{R} $ is a scalar potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\, \in\, (N-2, N) $. Under certain assumptions on $ A(x) $, $ V(x) $ and $ f(t) $, we prove that the equation has at least a ground state solution by variational methods.

The vertex arboricity $ a(G) $ of a graph $ G $ is the minimum number of colors required to color the vertices of $ G $ such that no cycle is monochromatic. The list vertex arboricity $ a_l(G) $ is the list version of this concept. In this paper, we prove that if $ G $ is a planar graph without 5-cycles intersecting with 6-cycles, then $ a_l(G)\le 2 $.

This paper is devoted to consider some new characteristics and properties of the weak group inverse and the weak group matrix. First, we characterize the weak group inverse of a square matrix based on its range space and null space. Also several different characterizations of the weak group inverse are presented by projection and the Bott-Duffin inverse. Then by using the core-EP decomposition, we investigate the relationships between weak group inverse and other generalized inverses. And some new characterizations of weak group matrix are obtained.

In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bezier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bezier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bezier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) L , bending energy E and curvature variation energy V are presented. The superiority of cubic GHT-Bezier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated.

This paper deals with some existence of random solutions and the Ulam stability for a class of Katugampola random fractional differential equations in Banach spaces. A random fixed point theorem is used for the existence of random solutions, and we prove that our problem is generalized Ulam-Hyers-Rassias stable. An illustrative example is presented in the last section.

In this paper, we introduce a modified implicit relation and obtain some new fixed point results for $ \sigma $-implicit type contractive conditions in relational metric-like spaces. We present some nontrivial examples to illustrative facts and compare our results with the related work. We also discuss sufficient conditions for the existence of a unique positive definite solution of the non-linear matrix equation $ \mathcal{U} = \mathcal{D} + \sum_{i = 1}^{m}\mathcal{A}_{i}^{*}\mathcal{G} \mathcal{(U)}\mathcal{A}_{i} $, where $ \mathcal{D} $ is an $ n\times n $ Hermitian positive definite matrix, $ \mathcal{A}_{1} $, $ \mathcal{A}_{2} $, $\dots$, $ \mathcal{A}_{m} $ are $ n \times n $ matrices, and $ \mathcal{G} $ is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Finally, we discuss a couple of examples, convergence and error analysis, average CPU time analysis and visualization of solution in surface plot.

In this paper, the Hermite-Hadamard inequality for $ s $-convex functions in the third sense is provided. In addition, some integral inequalities for them are presented. Also, the new functions based on the integral and double integral of $ s $-convex functions in the third sense are defined and under certain conditions, the third sense $ s $-convexity of these functions are shown and some inequality relations for these are expressed.

In this paper, we define type-1 quasi-hyperfilters (resp., type-1 strong quasi-hyperfilters, type-2 quasi-hyperfilters, type-2 strong quasi-hyperfilters, interior hyperfilters, strong interior hyperfilters, interior semihyperfilters, strong interior semihyperfilters) of an ordered LA-semihypergroup. We applied rough set theory to quasi and interior hyperfilters in ordered LA-semihypergroups.

In this paper, a promising modeling strategy for data fitting is derived from a new general class of univariate continuous distributions. This class is governed by an original logarithmically-exponential one-parameter transformation which has the ability to enhance some modeling capabilities of any parental distribution. In relation to the current literature, it appears to be a "limit case" of the well-established truncated generalized Fréchet generated class. In addition, it offers a natural alternative to the famous odd inverse exponential generated class. Some special distributions are presented, with particular interest in a novel heavy-tailed three-parameter distribution based on the Lomax distribution. Functional equivalences, modes analysis, stochastic ordering, functional expansions, moment measures, information measures and reliability measures are derived. From generic or real data, our modeling strategy is based on the new class combined with the maximum likelihood approach. We apply this strategy to the introduced modified Lomax model. The efficiency of the three parameter estimates is validated by a simulation study. Subsequently, two referenced real data sets are adjusted according to the rules of the art; the first one containing environmental data and the second one, financial data. In particular, we show that the proposed model is preferable to four concurrents also derived from the Lomax model, including the odd inverse exponential Lomax model.

In this paper, we study the transcendental entire solutions for the nonlinear differential-difference equations of the forms: \begin{document}$ \begin{eqnarray*} f^{2}(z)+\widetilde{\omega} f(z)f'(z)+q(z)e^{Q(z)}f(z+c) = u(z)e^{v(z)}, \end{eqnarray*} $\end{document} and \begin{document}$ \begin{eqnarray*} f^{n}(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c) = p_{1}e^{\lambda_{1} z}+p_{2}e^{\lambda_{2} z}, \quad n\geq 3, \end{eqnarray*} $\end{document} where $ \omega $ is a constant, $ \widetilde{\omega}, c, \lambda_{1}, \lambda_{2}, p_{1}, p_{2} $ are non-zero constants, $ q, Q, u, v $ are polynomials such that $ Q, v $ are not constants and $ q, u\not\equiv0 $. Our results are improvements and complements of some previous results.

In this paper, we discussed some new properties on the newly defined family of Genocchi polynomials, called poly-Genocchi polynomials. These polynomials are extensions from the Genocchi polynomials via generating function involving polylogarithm function. We succeeded in deriving the analytical expression and obtained higher order and higher index of poly-Genocchi polynomials for the first time. We also showed that the orthogonal version of poly-Genocchi polynomials could be presented as multiple shifted Legendre polynomials and Catalan numbers. Furthermore, we extended the determinant form and recurrence relation of shifted Genocchi polynomials sequence to shifted poly-Genocchi polynomials sequence. Then, we apply the poly-Genocchi polynomials to solve the fractional differential equation, including the delay fractional differential equation via the operational matrix method with a collocation scheme. The error bound is presented, while the numerical examples show that this proposed method is efficient in solving various problems.

In this paper, we discuss the existence and uniqueness of boundary value problems for sequential $ \psi $-Hilfer fractional pantograph differential equations with mixed nonlocal boundary conditions. The existence results are obtained via the well known Krasnoselskii's fixed point theorem while the uniqueness is demonstrated by using the Banach's contraction mapping principle. Some examples are also given to demonstrate the application of the main results.

A class of Dirichlet-Morrey spaces $ D_{\beta, \lambda} $ is introduced in this paper. For any positive Borel measure $ \mu $, the boundedness and compactness of the identity operator from $ D_{\beta, \lambda} $ into the tent space $ \mathcal{T}_s^1(\mu) $ are characterized. As an application, the boundedness of the Volterra integral operator $ T_g: D_{\beta, \lambda} \to F(1, \beta-s, s) $ is studied. Moreover, the essential norm and the compactness of the operator $ T_g $ are also investigated.

The existence of periodic bouncing solutions for sublinear impact oscillator is proved by using Poincare-Birkhoff twist theorem. The approach of this paper is based on a well defined successor map and the phase-plane analysis of the spiral properties.

The properties of solutions of the following differential equation $ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) $ are studied, where $ A_{j}(z) $ and $ F(z) $ are analytic in the unit disc $ \mathbb{D} = \{z:|z| < 1\} $, $ j = 0, 1, \ldots, k-1 $. First, the growth of solutions of the equation is estimated. Second, some coefficient's conditions such that the solution of the equation belong to Hardy type spaces are showed. Finally, some related question are studied in this paper.

This paper deals with the attractor problem of Hopfield neural networks with multiple time-varying delays. The mathematical expression of the networks cannot be expressed in the vector-matrix form due to the existence of the multiple delays, which leads to the existence condition of the attractor cannot be easily established by linear matrix inequality approach. We try to derive the existence conditions of the linear matrix inequality form of pullback attractor by employing Lyapunov-Krasovskii functional and inequality techniques. Two examples are given to demonstrate the effectiveness of our theoretical results and illustrate the conditions of the linear matrix inequality form are better than those of the algebraic form.

In this paper, a mathematical model of generalized fractional-order is constructed to study the problem of human immunodeficiency virus (HIV) infection of CD$ 4^+ $ T-cells with treatment. The model consists of a system of four nonlinear differential equations under the generalized Caputo fractional derivative sense. The existence results for the fractional-order HIV model are investigated via Banach's and Leray-Schauder nonlinear alternative fixed point theorems. Further, we also established different types of Ulam's stability results for the proposed model. The effective numerical scheme so-called predictor-corrector algorithm has been employed to analyze the approximated solution and dynamical behavior of the model under consideration. It is worth noting that, unlike many discusses recently conducted, dimensional consistency has been taken into account during the fractionalization process of the classical model.

In this paper, we establish some new Hermite-Hadamard type inequalities for differential exponential type convex functions and discuss several special cases. Moreover, in order to give the efficient of our main results, some applications for special means and error estimations are obtain.

A labeling of a connected, simple and undirected graph G(V, E) is a map that assigns the elements of a graph G with positive numbers. Many types of labeling for graph are found and one of them is a total edge irregularity strength (TEIS) of G, which denoted by tes(G). In the current paper, we defined a new type of family of graph called uniform theta snake graph, $\theta_{n}(t, m)$. Also, the exact values of total edge irregularity strengths for some special types of the new family have been determined.

Let $ F $ be a finite field of characteristic $ p $ having $ q = p^n $ elements and $ G $ be an abelian group. In this paper, we determine the structure of the group of units of the group algebra $ FG $, where $ G $ is an abelian group of order $ 17\leq |G|\leq 20 $.

In this paper, we first define the generalized (F, $ \varphi $, $ \alpha - \psi $)-contraction mappings. In the following, we consider the conditions in which these mappings have a $ \varphi $-fixed point and also we present examples and applications of these mappings in partial metric space and integral equations.

In this paper, we use the generalized progressive hybrid censoring sample from the Burr Type-Ⅻ distribution to estimate the unknown parameters, reliability and hazard functions. We apply the maximum likelihood (ML) and the Bayesian estimation under different prior distributions and different loss functions; namely; are the squared error, Linex and general entropy. Also, we construct the classical and credible intervals of the unknown parameters as well as for the survival and hazard functions. In addition, we investigate the performance of the point estimation by using the mean square error (MSE) and expected bias (EB) and performance of the interval estimation using the average length and coverage probability. Further, we develop the Bayesian one- and two- samples Bayesan prediction for the non-observed failures in the progressive censoring. In order to show the performance and usefulness of the inferential procedures, we carry out some simulation experiments using MCMC Algorithm for the Bayesian approach based on different prior distributions. Finally, we apply the theatrical finding to some real life data set.

In this paper, we study the following kind of Schrodinger-Poisson system in $ { \mathbb{R}}^{2} $ $ \begin{equation*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi u = K(x)f(u), \ \ \ x\in{ \mathbb{R}}^{2}, \\ \Delta \phi = u^{2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in{ \mathbb{R}}^{2}, \end{array}\right. \end{equation*} $ where $ f\in C({ \mathbb{R}}, { \mathbb{R}}) $, $ V(x) $ and $ K(x) $ are both axially symmetric functions. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution for the above planar system. Our result improves and extends the existing works.

Let $ \mathscr{G} $ be a molecular graph, the eccentricity $ e(w) $ of the vertex $ w $ in $ \mathscr{G} $ is the maximum distance of $ w $ from any other vertex of $ \mathscr{G} $. The non-self-centrality number (NSC) of a graph $ \mathscr{G} $ is defined by $ N(\mathscr{G}) = \sum_{w\not = z}|e(w)-e(z)|, $ where summation goes over all the unordered pairs of vertices of $ \mathscr{G} $. We determine non-self-centrality number of $ TUC_{4}C_{8} $ and $ V $-phenylenic nanotubes in this paper.

Consider two digital spaces $ (X_i, k_i), i \in \{1, 2\} $, (in the sense of Rosenfeld model) satisfying the almost fixed point property(AFPP for brevity). Then, the problem of whether the AFPP for the digital spaces is, or is not necessarily invariant under Cartesian products plays an important role in digital topology, which remains open. Given a Cartesian product $ (X_1 \times X_2, k) $ with a certain $ k $-adjacency, after proving that the AFPP for digital spaces is not necessarily invariant under Cartesian products, the present paper proposes a certain condition of which the AFPP for digital spaces holds under Cartesian products. Indeed, we find that the product property of the AFPP is strongly related to both the sets $ X_i $ and the $ k_i $-adjacency, $ i \in \{1, 2\} $. Eventually, assume two $ k_i $-connected digital spaces $ (X_i, k_i), i \in \{1, 2\} $, and a digital product $ X_1 \times X_2 $ with a normal $ k $-adjacency such that $ N_k^\star(p, 1) = N_k(p, 1) $ for each point $ p \in X_1 \times X_2 $ (see Remark 4.2(1)). Then we obtain that each of $ (X_i, k_i), i \in \{1, 2\} $, has the AFPP if and only if $ (X_1 \times X_2, k) $ has the AFPP