Permutation polynomials have important applications in cryptography, coding theory and combinatorial designs. In this letter, we construct four classes of permutation polynomials over 𝔽2n × 𝔽2n, where 𝔽2n is the finite field with 2n elements.

This paper focuses mainly on issues related to the pricing of American options under a fuzzy environment by taking into account the clustering of the underlying asset price volatility, leverage effect and stochastic jumps. By treating the volatility as a parabolic fuzzy number, we constructed a Levy-GJR-GARCH model based on an infinite pure jump process and combined the model with fuzzy simulation technology to perform numerical simulations based on the least squares Monte Carlo approach and the fuzzy binomial tree method. An empirical study was performed using American put option data from the Standard & Poor's 100 index. The findings are as follows: under a fuzzy environment, the result of the option valuation is more precise than the result under a clear environment, pricing simulations of short-term options have higher precision than those of medium- and long-term options, the least squares Monte Carlo approach yields more accurate valuation than the fuzzy binomial tree method, and the simulation effects of different Levy processes indicate that the NIG and CGMY models are superior to the VG model. Moreover, the option price increases as the time to expiration of options is extended and the exercise price increases, the membership function curve is asymmetric with an inclined left tendency, and the fuzzy interval narrows as the level set α and the exponent of membership function n increase. In addition, the results demonstrate that the quasi-random number and Brownian Bridge approaches can improve the convergence speed of the least squares Monte Carlo approach.

Nowadays portable devices with multiple wireless interfaces and using multimedia services are becoming more popular on the Internet. This paper describes a family of multipath binomial congestion control algorithms for audio/video streaming, where a low variant of transmission rate is important. We extend the fluid model of binomial algorithms for single-path transmission to support the concurrent transmission of packets across multiple paths. We focus on the extension of two particular algorithms, SQRT and IIAD, for multiple paths, called MPSQRT and MPIIAD, respectively. Additionally, we apply the design technique (using the multipath fluid model) for multipath TCP (MPTCP) into the extension of SQRT and IIAD, called fbMPSQRT and fbMPIIAD, respectively. Both two approaches ensure that multipath binomial congestion control algorithms achieve load-balancing, throughput improvement, and fairness to single-path binomial algorithms at shared bottlenecks. Through the simulations and comparison with the uncoordinated protocols MPSQRT/MPIIAD, fbMPSQRT/fbMPIIAD and MPTCP, we find that our extended multipath transport protocols can preserve lower latency and transmission rate variance than MPTCP, fairly share with single-path SQRT/IIAD, MPTCP and TCP, and also can achieve throughput improvements and load-balancing equivalent to those of MPTCP under various scenarios and network conditions.

Bayesian methods are often applied for estimating the event rate from a series of event occurrences. However, the Bayesian posterior distribution requires the computation of the marginal likelihood which generally involves an analytically intractable integration. As an event rate is defined in a very high dimensional space, it is computationally demanding to obtain the Bayesian posterior distribution for the rate. We estimate the rate underlying a sequence of event counts by deriving an approximate Bayesian inference algorithm for the time-varying binomial process. This enables us to calculate the posterior distribution analytically. We also provide a method for estimating the prior hyperparameter, which determines the smoothness of the estimated event rate. Moreover, we provide an efficient method to compute the upper and lower bounds of the marginal likelihood, which evaluate the approximation accuracy. Numerical experiments demonstrate the effectiveness of the proposed method in terms of the estimation accuracy.

For a prime p with p≡3 (mod 4) and an odd number m, the Bentness of the p-ary binomial function fa,b(x)=Tr1n(axpm-1)+Tr12 is characterized, where n=2m, a ∈ F*pn, and b ∈ F*p2. The necessary and sufficient conditions of fa,b(x) being Bent are established respectively by an exponential sum and two sequences related to a and b. For the special case of p=3, we further characterize the Bentness of the ternary function fa,b(x) by the Hamming weight of a sequence.

This paper presents a method to monitor information of a large-sized multicast group that can follow the group's dynamics in real-time while avoiding feedback implosion by using probabilistic polling. In particular, this paper improves the probabilistic-polling-based approach by deriving a reference mean value as the reference control value for the number of expected feedback from the properties of a binomial estimation model. As a result, our method adaptively changes its estimation parameters depending on the feedback from receivers in order to achieve a fast estimate time with high accuracy, while preventing the possible occurrence of feedback implosion. Our experimental implementation and evaluation on PlanetLab showed that the proposed method effectively controls the number of feedback and accurately estimates the size of a dynamic multicast group.

The expected lengths of the parsed segments obtained by applying Lempel-Ziv incremental parsing algorithm for i.i.d. source satisfy simple recurrence relations. By extracting a combinatorial essence from the previous proof, we obtain a simpler derivation.