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 Martinez, 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 F-2k such that g(x) = h(x) + xu+ xv, where h(x) = ax(4) + bx(3) + cx(2) + bx + a is a self-reciprocal polynomial with a, b, c is an element of F-2k and (u, v) is an element of {(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 is an element of F-2k so that f(x) is a permutation polynomial for F-22k. Moreover, we show that some known permutation pentanomials are QM equivalent. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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.
Abstract This study investigates the slip-bond behavior and the associated failure mechanisms of carbon textile-reinforced concrete (C-TRC), with a focus on the effects of embedment length, yarn orientation, and yarn grouping configurations of a gridlike carbon textile. A comprehensive experimental approach was adopted that incorporated yarn pullout tests to evaluate the slip-bond mechanical performance, as well as the use of advanced high-resolution 3D scanning, to precisely characterize the yarn geometry. Results show that yarn orientation (warp versus weft) measurably affects bond response and the embedment length required for effective load transfer. Weft yarns, which have a larger perimeter and greater waviness, achieved maximum bond efficiency at a shorter embedment length of 100 mm. In contrast, warp yarns required a longer embedment length of 150 mm to develop sufficient bonding. Yarn geometry and waviness influenced the governing failure mode. Elliptical cross-sections, derived from 3D scanning, yielded more accurate bond strength estimations than conventional circular assumptions. Moreover, scanning electron microscopy revealed orientation-dependent interfacial behavior, with warp yarns showing debonding and pullout, while weft yarns exhibited interfacial delamination and enhanced anchorage. Last, although yarn grouping enhances overall pullout capacity, it induces nonuniform stress redistribution, diminishing the load-carrying efficiency of individual yarns.
