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.

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.

We analyse the asymptotic behaviour of a biological system described by a stochastic competition model with $n$ species and $k$ resources (chemostat model), in which the species mortality rates are influenced by the fractional Brownian motion of the extrinsic noise environment. By constructing a Lyapunov functional, the persistence and extinction criteria are derived in the mean square sense. Some examples are given to illustrate the effectiveness of the theoretical result.

We address the construction and approximation for feed-forward neural networks (FNNs) with zonal functions on the unit sphere. The filtered de la Vallée-Poussin operator and the spherical quadrature formula are used to construct the spherical FNNs. In particular, the upper and lower bounds of approximation errors by the FNNs are estimated, where the best polynomial approximation of a spherical function is used as a measure of approximation error.

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.

Understanding how seasonal patterns change from year to year is important for the management of infectious disease epidemics. Here, we present a mathematical formalization of the application of complex demodulation, which has previously only been applied in an exploratory manner in the context of infectious diseases. This method extracts the changing amplitude and phase from seasonal data, allowing comparisons between the size and timing of yearly epidemics. We first validate the method using synthetic data that displays the key features of epidemic data. In particular, we analyse both annual and biennial synthetic data, and explore the effect of delayed epidemics on the extracted amplitude and phase. We then demonstrate the usefulness of complex demodulation using national notification data for influenza in Australia. This method clearly highlights the higher number of notifications and the early peak of the influenza pandemic in 2009. We also identify that epidemics that peaked later than usual generally followed larger epidemics and involved fewer overall notifications. Our analysis establishes a role for complex demodulation in the study of seasonal epidemiological events.

The $L_{r}$ convergence and a class of weak laws of large numbers are obtained for sequences of $\widetilde{\unicode[STIX]{x1D70C}}$ -mixing random variables under the uniform Cesàro-type condition. This is weaker than the $p$ th-order Cesàro uniform integrability.

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.

An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain break down, because the associated Poisson equation is no longer $H^{2}$ -regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation.

We present a new iterative model order reduction method for large-scale linear time-invariant dynamical systems, based on a combined singular value decomposition–adaptive-order rational Arnoldi (SVD-AORA) approach. This method is an extension of the SVD-rational Krylov method. It is based on two-sided projections: the SVD side depends on the observability Gramian by the resolution of the Lyapunov equation, and the Krylov side is generated by the adaptive-order rational Arnoldi based on moment matching. The use of the SVD provides stability for the reduced system, and the use of the AORA method providesnumerical efficiency and a relative lower computation complexity. The reduced model obtained is asymptotically stable and minimizes the error ( $H_{2}$ and $H_{\infty }$ ) between the original and the reduced system. Two examples are given to study the performance of the proposed approach.

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 evolutionary theory, a key issue in selection theory is the expected time for a given amount of allele frequency change to occur. Crow and Kimura, by assuming weak selection, presented explicit results for several important cases of the directional selection and of the stochastic process. Those results played an important role in the theory of population genetics. In this paper, first we show that the weak selection assumption can be removed from most of the results of Crow and Kimura, and then we generalize those results to the most general selection model. Next, we estimate the errors of the deterministic formulae produced by proving that the deterministic formulae are limits of the corresponding stochastic formulae when the size of the population tends to infinity. Finally, we present a result which removes the restriction of Kimura’s corresponding results for a favourite recessive selection model, and we also observe that the conclusion made by Kimura about the favourite dominant selection might not be correct.

Selecting important variables and estimating coordinate covariation have received considerable attention in the current big data deluge. Previous work shows that the gradient of the regression function, the objective function in regression and classification problems, can provide both types of information. In this paper, an algorithm to learn this gradient function is proposed for nonidentical data. Under some mild assumptions on data distribution and the model parameters, a result on its learning rate is established which provides a theoretical guarantee for using this method in dynamical gene selection and in network security for recognition of malicious online attacks.

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.

The error of a distributed algorithm for big data classification with a support vector machine (SVM) is analysed in this paper. First, the given big data sets are divided into small subsets, on which the classical SVM with Gaussian kernels is used. Then, the classification error of the SVM for each subset is analysed based on the Tsybakov exponent, geometric noise, and width of the Gaussian kernels. Finally, the whole error of the distributed algorithm is estimated in terms of the error of each subset.

The central projection transform can be employed to extract invariant features by combining contour-based and region-based methods. However, the central projection transform only considers the accumulation of the pixels along the radial direction. Consequently, information along the radial direction is inevitably lost. In this paper, we propose the Mellin central projection transform to extract affine invariant features. The radial factor introduced by the Mellin transform, makes up for the loss of information along the radial direction by the central projection transform. The Mellin central projection transform can convert any object into a closed curve as a central projection transform, so the central projection transform is only a special case of the Mellin central projection transform. We prove that closed curves extracted from the original image and the affine transformed image by the Mellin central projection transform satisfy the same affine transform relationship. A method is provided for the extraction of affine invariants by employing the area of closed curves derived by the Mellin central projection transform. Experiments have been conducted on some printed Chinese characters and the results establish the invariance and robustness of the extracted features.

We solve the problem of concept learning using a semi-tensor product method. All possible hypotheses are expressed under the framework of a semi-tensor product. An algorithm is raised to derive the version space. In some cases, the new approach improves the efficiency compared to the previous approach.

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.

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.