In this paper, a new numerical scheme for optimal control of jump-diffusion model is proposed by using stohastic Runge-Kutta (SRK) method. After discretizing the problem with the SRK method, optimality conditions are obtained by using the discretize-then-optimize approach. It is shown that the constructed numerical scheme is similar to the continuous optimality conditions obtained by using the Hamilton-Jacobi-Bellman equations. Moreover, a numerical scheme for control problems of Ornstein-Uhlenbeck (OU) with jump is presented as a simple version of jump diffusion equations. Some numerical examples are chosen to show the efficiency of the theoretical results.
Let Ps(n) denote the n-th s-gonal number. Consider the Diophantine equation Ps(n)=tm for integers n,s,t and m>2. All solutions to this equation are known for m>2 and s∈{3,5,6,8,10,20}. Here we extend these results to the cases s=2k+4 (where k=4,6 or 5≤k≤97 is a prime number) and s=k+4 (where k=9,15 or 3≤k≤97 is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective abc conjecture) that there will be no additional solutions beyond those explicitly shown in Theorem 1, Theorem 2, Theorem 3.
In this paper, we address two open problems posed by Martínez, Gupta and Quoos in [24]. We solve both problems completely and therefore generalize [28, Theorems 3.4 and 3.5]. We study polynomials of the form f(x)=x3g(xq−1) over the finite field F2k such that g(x)=h(x)+xu+xv, where h(x)=ax4+bx3+cx2+bx+a is a self-reciprocal polynomial with a,b,c∈F2k and (u,v)∈{(3,−1),(1,−1),(2,−2),(2,−1)}. The studied classes of polynomials either generalize some existing pentanomials and hexanomials or they are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on a,b,c∈F2k so that f(x) is a permutation polynomial for F22k. Moreover, we show that some known permutation pentanomials are QM equivalent.
This paper presents a new variable time-step method for the electromagnetic transient (EMT) simulation of power systems. Trapezoidal discretization with a constant time-step is commonly employed in EMT simulations of power systems. During an EMT simulation, power systems typically encounter high-frequency transients that demand small time-steps for accuracy, as well as low-frequency and steady-state conditions that allow larger time-steps for efficiency. With a fixed time-step, either the simulation accuracy or speed is compromised, and achieving a trade-off is not feasible for all cases. Consequently, this paper proposes a novel variable time-step strategy for maintaining both simulation accuracy and speed. The performance of the proposed method is verified through the WECC 240-bus system.
In this study, a biodegradable and sustainable piezoelectric–triboelectric hybrid nanogenerator (HENG) was designed and fabricated using cellulose nanofibrils (CNFs) as a nucleating agent and phycocyanin (PC), an algae-derived protein from Spirulina platensis, as the tribo-positive layer paired with poly(vinylidene fluoride) (PVDF) films. The investigation was conducted in two parts to investigate the effect of CNF incorporation into different layers. In the first part, CNFs were incorporated into the PVDF layer, whereas in second part incorporated into the PC layer. Electromechanical performance was characterized under periodic contact–separation motion. Even in the absence of nanofillers, the PVDF–PC pair exhibited efficient electromechanical behavior, generating an open-circuit voltage (Voc) of 84 V and a short-circuit current (Isc) of 87 µA. Upon CNF incorporation, the output was significantly enhanced. The highest performance was observed when CNFs were added to the PVDF layer (at 20 wt.
