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A recurrent queueing system under multi-optional and state dependent arrivals has been explored wherein the customers/units join in cluster form. The service provided to the units is carried out in duet junctures. The server is unreliable and may collapse. Thereafter, it needs to be mended in d requisite phases. Delay in the mending procedure is also considered. Whenever the system is free of customers, then server either departs for a vacation with a certain probability or pursues to serve the next customer with the complimentary probability. Eminent results are derived by employing Embedded Markov Chain (EMC) approach. Further, convexity of expected system size is also discussed. Also, the results are verified via Adaptive Neuro Fuzzy Interface System (ANFIS). Lastly, cost optimization is carried out and optimal cost is compared via different methods, namely: Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Artificial Bee Colony (ABC) algorithm and Cuckoo Search Algorithm (CSA). [ABSTRACT FROM AUTHOR]

This study focuses on the analytical and numerical solutions of the convexity analysis for fractional differences with exponential and Mittag-Leffler kernels involving negative and nonnegative lower bounds. In the analytical part of the paper, we will give a new formula for ∇ 2 of the discrete fractional differences, which can be useful to obtain the convexity results. The correlation between the nonnegativity and negativity of both of the discrete fractional differences, ( a CFR ∇ α f) (t) and ( a ABR ∇ α f) (t) , with the convexity of the functions will be examined. In light of the main lemmas, we will define the two decreasing subsets of (2 , 3) , namely ℋ k , and ℳ k , . The decrease of these sets enables us to obtain the relationship between the negative lower bound of ( a CFR ∇ α f) (t) and the convexity of the function on a finite time set given by N a + 1 P : = { a + 1 , a + 2 , ... , P } , for some P ∈ N a + 1 : = { a + 1 , a + 2 , ... }. Besides, the numerical part of the paper is dedicated to examine the validity of the sets ℋ k , and ℳ k , in certain regions of the solutions for different values of k and . For this reason, we will illustrate the domain of the solutions by means of several figures in which the validity of the main theorems are explained. [ABSTRACT FROM AUTHOR]

We consider the positivity of the discrete sequential fractional operators $ \left(^{\rm RL}_{a_{0}+1}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ defined on the set $ \mathscr{D}_{1} $ (see (1.1) and Figure 1) and $ \left(^{\rm RL}_{a_{0}+2}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ of mixed order defined on the set $ \mathscr{D}_{2} $ (see (1.2) and Figure 2) for $ \tau\in\mathbb{N}_{a_{0}} $. By analysing the first sequential operator, we reach that $ \bigl(\nabla {f}\bigr)(\tau)\geqq 0, $ for each $ \tau\in{\mathbb{N}}_{a_{0}+1} $. Besides, we obtain $ \bigl(\nabla {f}\bigr)(3)\geqq 0 $ by analysing the second sequential operator. Furthermore, some conditions to obtain the proposed monotonicity results are summarized. Finally, two practical applications are provided to illustrate the efficiency of the main theorems.

Abstract In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the Δ 2 $\Delta ^{2}$ , which will be useful to obtain the convexity results. We examine the correlation between the positivity of ( w 0 RL Δ α f ) ( t ) $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of ( 2 , 3 ) $(2,3)$ , H k , ϵ $\mathscr{H}_{\mathrm{k},\epsilon}$ and M k , ϵ $\mathscr{M}_{\mathrm{k},\epsilon}$ . The decrease of these sets allows us to obtain the relationship between the negative lower bound of ( w 0 RL Δ α f ) ( t ) $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ and convexity of the function on a finite time set N w 0 P : = { w 0 , w 0 + 1 , w 0 + 2 , … , P } $\mathrm{N}_{w_{0}}^{\mathrm{P}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots , \mathrm{P}\}$ for some P ∈ N w 0 : = { w 0 , w 0 + 1 , w 0 + 2 , … } $\mathrm{P}\in \mathrm{N}_{w_{0}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots \}$ . The numerical part of the paper is dedicated to examinin the validity of the sets H k , ϵ $\mathscr{H}_{\mathrm{k},\epsilon}$ and M k , ϵ $\mathscr{M}_{\mathrm{k},\epsilon}$ for different values of k and ϵ. For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem.

In this study, we have presented two new alternative definitions corresponding to the basic definitions of the discrete delta and nabla fractional difference operators. These definitions and concepts help us in establishing a relationship between Riemann-Liouville and Liouville-Caputo fractional differences of higher orders for both delta and nabla operators. We then propose and analyse some convexity results for the delta and nabla fractional differences of the Riemann-Liouville type. We also derive similar results for the delta and nabla fractional differences of Liouville-Caputo type by using the proposed relationships. Finally, we have presented two examples to confirm the main theorems.

We consider the positivity of the discrete sequential fractional operators ... defined on the set ... (see (1.1) and Figure 1) and ... of mixed order defined on the set ... (see (1.2) and Figure 2) for τ ∈ ℕa0. By analysing the first sequential operator, we reach that (∇ f) (τ) ≧ 0, for each τ ∈ ℕa0. Besides, we obtain (∇ f) (3) ≧ 0 by analysing the second sequential operator. Furthermore, some conditions to obtain the proposed monotonicity results are summarized. Finally, two practical applications are provided to illustrate the efficiency of the main theorems. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the Δ 2 , which will be useful to obtain the convexity results. We examine the correlation between the positivity of ( w 0 RL Δ α f) (t) and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of (2 , 3) , H k , ϵ and M k , ϵ . The decrease of these sets allows us to obtain the relationship between the negative lower bound of ( w 0 RL Δ α f) (t) and convexity of the function on a finite time set N w 0 P : = { w 0 , w 0 + 1 , w 0 + 2 , ... , P } for some P ∈ N w 0 : = { w 0 , w 0 + 1 , w 0 + 2 , ... } . The numerical part of the paper is dedicated to examinin the validity of the sets H k , ϵ and M k , ϵ for different values of k and ϵ. For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem. [ABSTRACT FROM AUTHOR]

In this study, we have presented two new alternative definitions corresponding to the basic definitions of the discrete delta and nabla fractional difference operators. These definitions and concepts help us in establishing a relationship between Riemann-Liouville and Liouville-Caputo fractional differences of higher orders for both delta and nabla operators. We then propose and analyse some convexity results for the delta and nabla fractional differences of the Riemann-Liouville type. We also derive similar results for the delta and nabla fractional differences of Liouville-Caputo type by using the proposed relationships. Finally, we have presented two examples to confirm the main theorems. [ABSTRACT FROM AUTHOR]

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. To overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, and this, in turn, yields a convex-concave decomposition of the physical energy in the ternary system. Consequently, both the unique solvability and the unconditional energy stability of the proposed numerical scheme are established at a theoretical level. In addition, an optimal rate convergence analysis in the ℓ ∞ (0 , T ; H N - 1) ∩ ℓ 2 (0 , T ; H N 1) norm is provided, with Fourier pseudo-spectral discretization in space, which is the first such result in this field. To deal with the nonlinear implicit equations at each time step, we apply an efficient preconditioned steepest descent (PSD) algorithm. A second order accurate, modified BDF scheme is also discussed. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

In cluster analysis, regions of high local density of data points, which might correspond to significant clusters, can be found from the modes of the underlying probability density function (pdf). However, due to irregularities in the data distribution, the modes and the valleys of the pdf are often ill defined so that mode detection can lead to poor results. In this paper, an anisotropic diffusion process is proposed in order to reinforce the smoothing of the pdf in the modal regions where it is concave and in the valleys where it is convex, while preserving the boundaries between them. This adaptive smoothing procedure is combined with a strategy which consists in applying a forward diffusion when the pdf is concave and a backward diffusion when it is convex. Iterations of this convexity dependent anisotropic diffusion tend to enhance the modes and to deepen the valleys of the underlying pdf, so that mode detection becomes trivial. Experiment and comparative results with some well-known clustering algorithms over simulated datasets show the effectiveness of the proposed clustering method. [ABSTRACT FROM AUTHOR]

Abstract: Chemical and phase equilibria were considered for closed multicomponent reactive systems at: (a) constant pressure and temperature; (b) constant pressure and enthalpy. Equilibrium at constant P and T was found by minimization of G, while equilibrium at constant P and H was found by maximization of S or minimization of −S, all with respect to the number of moles of each component in each phase. Both cases could be handled as optimization problems, satisfying the restrictions imposed by mole or atom balances, and non-negativity of number of moles. Convexity analyses were carried out, and the conditions were found in order to guarantee global minimum, for one liquid phase, one gas phase, and a number of solid phases. The minimum point was then found either by analytical methods or by direct minimization methods. These strategies were tested for a number of cases, with good results. [Copyright &y& Elsevier]

Abstract: An approach is presented to calculate solid–liquid phase equilibrium for binary mixtures, using expressions for the temperature as a function of the molar fraction. For Margules model the expression gives explicitly the temperature, while for other liquid phase activity models an iterative procedure is required to calculate the temperature. The method is very easy to apply and it can be used for mixtures that have peritectic and eutectic points, or just a eutectic point. The approach was applied to five case studies with binary mixtures of fatty acids and triglycerides. The results were in good agreement with experimental data. [Copyright &y& Elsevier]

Existence of a Nash equilibrium in a noncooperative game governed by the one-dimensional Burgers equation, proposed in the case of pointwise controls in Ref. 1, is proved under data qualifications that guarantee the diffusion term in the Burgers' equation to be dominant enough with respect to the uniform convexity of the payoffs. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the theory and algorithm on a class of optimal curve fitting problems which can be extensively applied to the engineering are established and completed. With regard to the curve fitting problems, the available theories and methods are largely concerned with the spline interpolation and linear or nonlinear least squares methods, and there are lots of ripe results. To my best knowledge, however, in view of the complexity, the third method, that is, to resort to solve functional extremal problems remains completed both in theories and algorithms. Regarding the class of this problems, that is here called as the optimal curve fitting problems under relaxation constraints, the analytic expression of solution and solving algorithms were discussed in [P. Cheng, K.C. Zhang, On a class of constrained functional minimization problems and their numerical solution, Journal of optimization theory and applications, 78(2) (1993) P267–P287; P. Cheng, KC. Zhang, Unified method for a kind of constrained functional minimization, Computing technology and automation, 10(2) (1991) 13–25; K.C. Zhang, A numerical method for solving a class of variation problems with constraint and its application to engineering, Numerical Mathematics, A Journal of Chinese Universities,11(2) (1989) 1–9; K.C. Zhang, The Algorithm and Analysis of Numerical Computation, Beijing, Science Press, 2003, pp. 172–184]. But the existence of optimal solution has not been solved so far. In the paper, with help of variational principle and convexity analysis, we give the proof of existence in details and a different proof about the analytic expression of optimal solution. Moreover, we prove that, for every smoothing parameter m, this class of optimal curve fitting problems is equivalent to solve a quadratic programming with a spherical constraint which usually appears in the trust-region and SQP subproblems, and its algorithms have been quite ripe. Therefore, a novel algorithm different from Zhang (2003) for this class problems is established. Finally, we give an example of m =2 to validate our theory and method. Numerical test shows that the method is effective and practical. [Copyright &y& Elsevier]

This paper shows how to compute the nonholonomic distance between a pointwise car-like robot and polygonal obstacles. Geometric constructions to compute the shortest paths from a configuration (given orientation and position in the plane of the robot) to a position (i.e., a configuration with unspecified final orientation) are first presented. The geometric structure of the reachable set (set of points in the plane reachable by paths of given length /spl Lscr/) is then used to compute the shortest paths to straight-line segments. Obstacle distance is defined as the length of such shortest paths. The algorithms are developed for robots that can move both forward and backward (Reeds and Shepp's car) or only forward (Dubins' car). They are based on the convexity analysis of the reachable set. [ABSTRACT FROM PUBLISHER]

• Design of an efficient numerical scheme for the strongly anisotropic Cahn-Hilliard equation. • Implementation of the numerical method in 2D and 3D. • Ability of the method to deal with missing orientations through regularization. • Optimal order convergence analyses in strong and weak anisotropy regimes. • Implementation of fast solvers using preconditioned steepest descent and pseudo-spectral methods. In this paper we propose and analyze a weakly nonlinear, energy stable numerical scheme for the strongly anisotropic Cahn-Hilliard model. In particular, a highly nonlinear and singular anisotropic surface energy makes the PDE system very challenging at both the analytical and numerical levels. To overcome this well-known difficulty, we perform a convexity analysis on the anisotropic interfacial energy, and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant. This subtle fact enables one to derive an energy stable numerical scheme. Moreover, a linear approximation becomes available for the surface energy part, and a detailed estimate demonstrates the corresponding energy stability. Its combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear, energy stable scheme for the whole system. In particular, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, and no auxiliary variable needs to be introduced. This has important implications, for example, in the case that the method needs to satisfy a maximum principle. More importantly, with a careful application of the global bound for the second order functional derivatives, an optimal rate convergence analysis becomes available for the proposed numerical scheme, which is the first such result in this area. Meanwhile, for a Cahn-Hilliard system with a sufficiently large degree of anisotropy, a Willmore or biharmonic regularization has to be introduced to make the equation well-posed. For such a physical model, all the presented analyses are still available; the unique solvability, energy stability and convergence estimate can be derived in an appropriate manner. In addition, the Fourier pseudo-spectral spatial approximation is applied, and all the theoretical results could be extended for the fully discrete scheme. Finally, a few numerical results are presented, which confirm the robustness and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]