On the mapping xy→(xy)n in an associative ring by Scott J. Beslin, Awad Iskander

“…There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R2, and f(xy)=(xy)n(x,y) for some positive integer n(x,y)>1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for non-Artinian rings. …”
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A New Roper-Suffridge Extension Operator on a Reinhardt Domain by Jianfei Wang, Cailing Gao

“…We introduce a new Roper-Suffridge extension operator on the following Reinhardt domain given by where f is a normalized locally biholomorphic function on the unit disc D, pj are positive integer, aj are complex constants, and j=2,…,n. Some conditions for aj are found under which the operator preserves almost starlike mappings of order α and starlike mappings of order α, respectively. …”
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Riesz potential operators and inverses via fractional centred derivatives by Manuel Duarte Ortigueira

“…Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. …”
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A closer look at some new identities by Geoffrey B. Campbell

“…Among these is ∏k=3∞(1−Z)φ,(k)/k=11−Zexp(Z32(1−Z)2−12Z−12Z(1−Z)),  |Z|<1, in which φ3(k) denotes for fixed k, the number of positive integer solutions of (a,b,k)=1 where a<b<k, assuming (a,b,k) is the gcd(a,b,k).…”
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Development of Optimization System of Two-stage Cylindrical Gear Standard Reducer by Wang Yong, Jiang Jing, Guo Gang, Pan Junyuan, Huang Taohong, Shi Wenjie

“…Based on the platform of Visual Basic 6. 0,Visual Fox Pro 6. 0 and Visual LISP,the optimal design system is developed by establishing mathematical model for optimization design of two- stage cylindrical gear standard reducer and using integer optimal method and visual combination software development technology. …”
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-Regular Modules by Areej M. Abduldaim, Sheng Chen

“…An -module is called -regular if for each and , there exist and a positive integer such that . The notion of -pure submodules was introduced to generalize pure submodules and proved that an -module is -regular if and only if every submodule of is -pure iff   is a -regular -module for each maximal ideal of . …”
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Algorithm for the cloud service workflow scheduling with setup time and deadline constraints by Hong SHEN, Xiao-ping LI

“…Cost minimization with deadline constraints and setup time is a new resource allocation optimization problem in cloud computing.An integer programming model is constructed for this NP-hard problem.A novel probability model and updating mechanism is suggested for sampling the solution space.A heuristic method is investigated for generating good initial population.A hybrid estimation of distribution algorithm (HEDA) is proposed for the considered problem.The global best solution is adopted to balance the diversification and intensification of HEDA.Experimental results show that the proposed method is effective to optimize the workflow running cost within reasonable CPU time.…”
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Differentiation Theory over Infinite-Dimensional Banach Spaces by Claudio Asci

“…We study, for any positive integer k and for any subset I of N⁎, the Banach space EI of the bounded real sequences xnn∈I and a measure over RI,B(I) that generalizes the k-dimensional Lebesgue one. …”
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Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation by Benedict Barnes, Anthony Y. Aidoo, Joseph Ackora-Prah

“…The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. …”
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Adaptation of Differential Transform Method for the Numeric-Analytic Solution of Fractional-Order Rössler Chaotic and Hyperchaotic Systems by Asad Freihat, Shaher Momani

“…A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.…”
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A Branch and Bound Algorithm for the Critical Grid Coverage Problem in Wireless Sensor Networks by Maher Rebai, Matthieu Le Berre, Faicel Hnaien, Hichem Snoussi, Lyes Khoukhi

“…We verify by computational experiments that the proposed B&B algorithm is more efficient, in terms of computation time, than the integer linear programming model developed by Rebai et al. (2013), for the same problem.…”
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Collaborative task offloading and resource allocation optimization for intelligent edge devices by Xian LI, Suzhi BI, Hongru ZENG, Bin LIN, Xiaohui LIN

“…In order to deal with the increasingly scarce computing resources, a cooperative edge computing scheme was proposed, which makes full use of the idle resources among users to improve the overall data processing performance.To maximize the user utility, the target problem was formulated as an MINLP (mixed integer non-linear programming), and a learning-optimization-integrated method was proposed to jointly optimize the resource allocation and user offloading decisions.Simulation results show that the proposed scheme can produce a near-optimal solution in sub-second and effectively improve the system utility at least 85.4% compared to the considered benchmark methods.…”
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New pseudorandom functions based on SIS by He-feng CHEN, Wen-ping MA, Sheng GAO, Cheng-li ZHANG

“…Pseudorandom functions are vital tools in the construction of cryptographic primitives.Under the hard assumption of SIS (short integer solution),two lattice-based pseudorandom functions are proposed.The first one has parallel structure by the ideal of tree-like pseudorandom synthesizer,and the second one is serial structure whose public key size is reduced.Both constructions have small modulus and provable security.Compared with A Banerjer,C Peikert and A Rosen’s construction (EUROCRYPT 2012),their key sizes are asymptotically smaller,and efficiency are improved by avoiding the “rounding” technology.…”
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Multiplicative polynomials and Fermat's little theorem for non-primes by Paul Milnes, C. Stanley-Albarda

“…Fermat's Little Theorem states that xp=x(modp) for x∈N and prime p, and so identifies an integer-valued polynomial (IVP) gp(x)=(xp−x)/p. Presented here are IVP's gn for non-prime n that complete the sequence {gn|n∈N} in a natural way. …”
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Non-Single-Valley Solutions for p-Order Feigenbaum’s Type Functional Equation f(φ(x))=φp(f(x)) by Min Zhang

“…This work deals with Feigenbaum’s functional equation f(φ(x))=φp(f(x)), φ(0)=1, 0≤φ(x)≤1, x∈ [0, 1], where p≥2 is an integer, φp is the p-fold iteration of φ, and f(x) is a strictly increasing continuous function on [0, 1] that satisfies f(0)=0, f(x)<x, (x∈(0, 1]). …”
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Positive Definite Solutions of the Matrix Equation Xr-∑i=1mAi∗X-δiAi=I by Asmaa M. Al-Dubiban

“…We investigate the nonlinear matrix equation Xr-∑i=1mAi∗X-δiAi=I, where r is a positive integer and δi∈(0,1], for i=1,2,…,m. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. …”
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Stability of a Quartic Functional Equation by Abasalt Bodaghi

“…We obtain the general solution of the generalized quartic functional equation f(x+my)+f(x-my)=2(7m-9)(m-1)f(x)+2m2(m2-1)f(y)-(m-1)2f(2x)+m2{f(x+y) + f(x-y)} for a fixed positive integer m. We prove the Hyers-Ulam stability for this quartic functional equation by the directed method and the fixed point method on real Banach spaces. …”
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Orthant spanning simplexes with minimal volume by Michele Elia

“…Furthermore, limited to dimension 3, parametric representations are given to points A to which correspond triangles of minimal area with integer vertex coordinates and area.…”
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Controlling the dynamics of Burgers equation with a high-order nonlinearity by Nejib Smaoui

“…We show existence of an absorbing ball in L2[0,1] and uniqueness of steady state solutions for all integer n≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. …”
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Approximate Solution of Tuberculosis Disease Population Dynamics Model by Abdon Atangana, Necdet Bildik

“…We examine possible approximate solutions of both integer and noninteger systems of nonlinear differential equations describing tuberculosis disease population dynamics. …”
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