In this paper, we characterize Borel $\unicode[STIX]{x1D70E}$-fields of the set of all fuzzy numbers endowed with different metrics. The main result is that the Borel $\unicode[STIX]{x1D70E}$-fields with respect to all known separable metrics are identical. This Borel field is the Borel $\unicode[STIX]{x1D70E}$-field making all level cut functions of fuzzy mappings from any measurable space to the fuzzy number space measurable with respect to the Hausdorff metric on the cut sets. The relation between the Borel $\unicode[STIX]{x1D70E}$-field with respect to the supremum metric $d_{\infty }$ is also demonstrated. We prove that the Borel field is induced by a separable and complete metric. A global characterization of measurability of fuzzy-valued functions is given via the main result. Applications to fuzzy-valued integrals are given, and an approximation method is presented for integrals of fuzzy-valued functions. Finally, an example is given to illustrate the applications of these results in economics. This example shows that the results in this paper are basic to the theory of fuzzy-valued functions, such as the fuzzy version of Lebesgue-like integrals of fuzzy-valued functions, and are useful in applied fields.

A new generalized class of fuzzy implications, called ($h,f,g$)-implications, is introduced and discussed in this paper. The results show that the new fuzzy implications possess some good properties, such as the left neutrality property and the exchange principle.

We analyse the coverage performance of cognitive radio networks powered by renewable energy. Particularly, with an energy harvesting module and energy storage module, the primary transmitters (PTs) and the secondary transmitters (STs) are assumed to be able to collect ambient renewables, and store them in batteries for future use. Upon harvesting sufficient energy, the corresponding PTs and STs (denoted by eligible PTs and STs) are then allowed to access the spectrum according to their respective medium access control (MAC) protocols. For the primary network, an Aloha-type MAC protocol is considered, under which the eligible PTs make independent decisions to access the spectrum with probability $\unicode[STIX]{x1D70C}_{p}$. By applying tools from stochastic geometry, we characterize the transmission probability of the STs. Then, with the obtained results of transmission probability, we evaluate the coverage (transmission nonoutage) performance of the overlay CR network powered by renewable energy. Simulations are also provided to validate our analysis.

We analyse a parallel (identical) machine scheduling problem with job delivery to a single customer. For this problem, each job needs to be processed on $m$ parallel machines non-pre-emptively and then transported to a customer by one vehicle with a limited physical capacity. The optimization goal is to minimize the makespan, the time at which all the jobs are processed and delivered and the vehicle returns to the machine. We present an approximation algorithm with a tight worst-case performance ratio of $7/3-1/m$ for the general case, $m\geq 3$.

We study a nonpreemptive scheduling on two parallel identical machines with a dedicated loading server and a dedicated unloading server. Each job has to be loaded by the loading server before being processed on one of the machines and unloaded immediately by the unloading server after its processing. The loading and unloading times are both equal to one unit of time. The goal is to minimize the makespan. Since the problem is NP-hard, we apply the classical list scheduling and largest processing time heuristics, and show that they have worst-case ratios, $8/5$ and $6/5$, respectively.

We investigate the state feedback pinning synchronization of fractional-order complex networks. Based on the stability theory of fractional-order differential systems and state feedback control by a single controller, synchronization conditions for fractional-order complex networks are given. We assume that the coupling matrix is irreducible, and provide a numerical example to illustrate the validity of the proposed conclusions.

In recent years, balanced network optimization problems play an important role in practice, especially in information transmission, industry production and logistics management. In this paper, we consider some logistics optimization problems related to the optimal tree structures in a network. We show that the most optimal subtree problem is NP-hard by transforming the connected dominating set problem into this model. By constructing the network models of the most balanced spanning tree problem with edge set restrictions, and by finding the optimal subtrees in special networks, we present efficient computational methods for solving some logistics problems.

We consider a class of network-design problems with minimum sum of modification and network costs for minimum spanning trees under Hamming distance. By constructing three auxiliary networks, we present a strongly polynomial-time algorithm for this problem. The method can be applied to solve many network-design problems. And, we show that a variation model of this problem is NP-hard, even when the underlying network is a tree, by transforming the 0–1 knapsack problem to this model.

Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.

This paper analyses the pseudo almost periodicity of the impulsive neoclassical growth model. We investigate the existence, uniqueness and exponential stability of the pseudo almost periodic solution. Moreover, an example is given to illustrate the significance of the main findings.

This paper investigates interrelated price online inventory problems, in which decisions as to when and how much of a product to replenish must be made in an online fashion to meet some demand even without a concrete knowledge of future prices. The objective of the decision maker is to minimize the total cost while meeting the demands. Two different types of demand are considered carefully, that is, demands which are linearly and exponentially related to price. In this paper, the prices are online, with only the price range variation known in advance, and are interrelated with the preceding price. Two models of price correlation are investigated, namely, an exponential model and a logarithmic model. The corresponding algorithms of the problems are developed, and the competitive ratios of the algorithms are derived as the solutions by use of linear programming.

We discuss discrete stochastic processes with two independent variables: one is the standard symmetric random walk, and the other is the Poisson process. Convergence of discrete stochastic processes is analysed, such that the symmetric random walk tends to the standard Brownian motion. We show that a discrete analogue of Ito’s formula converges to the corresponding continuous formula.

We provide an elementary method for exploring pricing problems of one spread options within a fractional Wick–Itô–Skorohod integral framework. Its underlying assets come from two different interactive markets that are modelled by two mixed fractional Black–Scholes models with Hurst parameters, $H_{1}\neq H_{2}$, where $1/2\leq H_{i}<1$ for $i=1,2$. Pricing formulae of these options with respect to strike price $K=0$ or $K\neq 0$ are given, and their application to the real market is examined.

We investigate delayed state feedback control of a periodic-review inventory management system with perishable goods. The stock under consideration is replenished from multiple supply sources. By using delayed states as well as current states of the inventory system, a delayed feedback $H_{\infty }$ control strategy is developed to mitigate bullwhip effects of the system. Some conditions on the existence of the delayed feedback $H_{\infty }$ controller are derived. It is found through simulation results that the proposed delayed $H_{\infty }$ control scheme is capable of improving the performance of the inventory management significantly. In addition, the delayed controller is better than the traditional delay-free $H_{\infty }$ controller.

The valuation of perpetual timer options under the Hull–White stochastic volatility model is discussed here. By exploring the connection between the Hull–White model and the Bessel process and using time-change techniques, the triple joint distribution for the instantaneous volatility, the cumulative reciprocal volatility and the cumulative realized variance is obtained. An explicit analytical solution for the price of perpetual timer call options is derived as a Black–Scholes–Merton-type formula.

We investigate rare or small probability events in the context of large deviations of the stochastic Camassa–Holm equation. By the weak convergence approach and regularization, we get large deviations of the regularized equation. Then, by stochastic equations exponentially equivalent to the corresponding laws, we get large deviations of the stochastic Camassa–Holm equation.

We study the existence of the invariant region for the equations of nonisentropic gas dynamics. We obtain the mean-integral of the conserved quantity after making an intensive study of the Riemann problem. Using the extremum principle and the Lagrangian multiplier method, we prove that the one-dimensional equations of nonisentropic gas dynamics for an ideal gas possess a unique invariant region. However, the invariant region is not bounded.

Based on the definition of divisibility of Markovian quantum dynamics, we discuss the Markovianity of tensor products, multiplications and some convex combinations of Markovian quantum dynamics. We prove that the tensor product of two Markovian dynamics is also a Markovian dynamics and propose a new witness of non-Markovianity.