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Here, the authors correct the proof in the reference when explaining that the produced plateaued functions have no non-zero linear structures. Moreover, a new class of plateaued functions with the best algebraic degree is given.

For improved discrimination of a ballistic warhead from a decoy, a simple 3D feature vector (3DFV) is proposed that is composed of the fundamental frequency, the bandwidth, and the sinusoidal moment of the time–frequency image. Compared to two existing methods, the 3DFV was more efficient and gave results that were less sensitive to noise.

A new characterization of balanced rotation symmetric (*n*, *m*)-functions is presented. Based on the characterization, the nonexistence of balanced rotation symmetric (*p* ^{ r }, *m*)-functions is determined, where *p* is an odd prime and *m* ≥ 2. And there exist balanced rotation symmetric (2^{ r }, *m*)-functions for 2 ≤ *m* ≤ 2^{ r } − *r*. With the help of these results, we also prove that there exist rotation symmetric resilient (2^{ r }, *m*)-functions for 2 ≤ *m* ≤ 2^{ r } − *r* − 1.

In recent years, two new types of irreducible pentanomials, i.e. Type C.1 and Type C.2 pentanomials, and their associated generalised polynomials bases (GPBs) have been proposed to yield efficient bit-parallel multiplier architectures. The GPB squarer for Type C.1 pentanomial is also investigated previously. But no GPB squarer for Type C.2 pentanomial is given as these pentanomials are far more complicated. In this Letter, the authors give explicit GPB squarer formulae for all Type C.2 pentanomials by re-classifying these pentanomials into certain sub-groups, which is based on the parities of pentanomial parameters. As the main contribution of this Letter, the authors show that the GPB squarers for most Type C.2 pentanomials match the fastest results.

Using the linear representation of symmetric group in the structure vector of finite games as its representation space, the inside structures of several kinds of symmetric games are investigated. First of all, the symmetry, described as the action of symmetric group on payoff functions, is converted to the product of permutation matrices with structure vectors of payoff functions. Second, in the light of the linear representation of the symmetric group in structure vectors, the algebraic conditions for the ordinary, weighted, renaming and name-irrelevant symmetries are obtained as the invariance under the corresponding linear representations. The semi-tensor product of matrices is a fundamental tool in this approach.

Toeplitz matrix–vector product (TMVP) decomposition is an important approach for designing and implementing subquadratic multiplier. In this Letter, a symmetric matrix (SM), which is the sum of a symmetric TM and Hankel matrix, is proposed. Applying the symmetry property, 2-way, 3-way and *n*-way splitting methods of SMVP is presented. On the basis of 2-way splitting method, the recursive formula of SMVP is presented. Using the two cases *n* = 4 and 8, the SMVP decomposition approach has less space complexity than 2-way TMVP, TMVP block recombination and symmetric TMVP for even-type Gaussian normal basis multiplication.

The right and left con-Sylvester matrices for two polynomials are defined, and then the criteria for right and left coprimeness of two polynomials in the framework of conjugate product are given in terms of the determinant of the right and left con-Sylvester matrices, respectively. The results in this study can be viewed as a generalisation of the well-known Sylvester resultant criterion for coprimeness of two polynomials in the framework of ordinary product.

This study is concerned with the finite-horizon bounded synchronisation and state estimation for the discrete-time complex networks with missing measurements based on the local performance analysis. First, a new local description of the bounded synchronisation performance index is proposed, which considers only the synchronisation errors among neighbours. In addition, a more general sector-bounded condition is presented, where the parameter matrices are different for different node. Next, by establishing the vector dissipativity-like for the complex network dynamics, the synchronisation criterion is derived in term of the locally coupled conditions for each node. These conditions implemented in a cooperative manner can judge whether the complex network reaches synchronisation. Similarly, the existence conditions for the estimator on each node are obtained, and then the estimator parameters are designed via the recursive linear matrix inequalities. Notably, these conditions on each node by cooperation among neighbours can achieve the desirable performance index. The distinctive features of the authors' algorithms are low complexity, scalability, and distributed execution. At last, two numerical examples are utilised to verify the effectiveness and applicability of the proposed algorithms.

The positive fraction vector fitting (PFVF) is a special method to guarantee the passivity of rational models such as frequency-dependent network equivalents. It involves constraints that enforce each fraction of the rational model to be passive, which are much stricter than the original passive requirements. PFVF lacks theoretical foundation but works well in practise. This study explains the rationality of PFVF by revealing important features of rational models that the complex-pole fractions corresponding to dominant resonance peaks can be adjusted passive through a minor change. The numerical case corroborates the theoretical analysis.

Sequential order one negative exponential (SOONE) function is used to measure the sparsity of a two-dimensional (2D) signal. A 2D gradient projection (GP) method is developed to solve the SOONE function and thus the 2D-GP-SOONE algorithm is proposed. The algorithm can solve the sparse recovery of 2D signals directly. Theoretical analysis and simulation results show that the 2D-GP-SOONE algorithm has a better performance compared with the 2D smoothed L0 algorithm. Simulation results also show that the proposed algorithm has a better performance and requires less computation time than 2D iterative adaptive approach.

This appendix contains a short overview of the concept of vectors and linear transformations (dyadics) of vectors, and how these are represented in terms of their components. Transformations between different rotated coordinate systems are also reviewed as well as the corresponding transformation of the components of a dyadic. The appendix ends with a short overview of quaternions and their use to represent rotations.

In this chapter, we consider the representation of vector functions (often referred to as “vector fields”) with low-order (constant and linear) polynomial basis functions on simple cells, such as triangles or quadrilateral cells in two dimensions or tetrahedrons and bricks in three dimensions. As will soon be apparent, there are multiple ways of defining vector basis functions, and therefore the approach requires some consideration. The proper representation of a function depends on what will be done with it-do we need to compute the curl of the function, for instance? If so, the representation might be different than if we need to compute the divergence of the function. We use the term curl conforming to denote the space of vector functions that maintain first-order tangential-vector continuity throughout the domain and can be differentiated via the curl operation, without producing unbounded or generalized functions (Dirac delta functions) in the process. The term divergence conforming is used to denote the complementary space of vector functions that maintain first-order normal-vector continuity throughout the domain and can therefore be differentiated via the divergence operation. (First order or C_{0} continuity is continuity of the function itself, but not necessarily continuity of its first derivatives.) The simple low-order polynomial vector basis functions in widespread use are either curl conforming or divergence conforming; seldom we will use functions that maintain complete continuity and belong to both the curl-conforming and divergence-conforming spaces, although it is possible to define such functions.

This paper describes an efficient implementation of a form of linear semi-infinite programming (LSIP). We look at maximizing (minimizing) a linear function over a set of constraints formed by positive trigonometric polynomials. Previous studies about LSIP are formulated using semi-definite programming (SDP), this is typically done by using the Kalman Yakubovich Popov (KYP) lemma or using a trace operation involving a Grammian matrix, which can be computationally expensive. The proposed algorithm is based on simplex method that directly solves the LSIP without any parameterization. Numerical results show that the proposed LISP algorithm is significantly more efficient than existing SDP solvers using KYP lemma and Grammian matrix, in both execution time and memory.

Given a graph G = (V, E) with a set W ⊆ V of vertices, we enumerate colorings to W such that for every two enumerated colorings c and c' the corresponding colored graphs (G, c) and (G, c') are not isomorphic. This problem has an important application in the study of isomers of chemical graphs such as generation of benzen isomers from a tree-like chemical graph structure. The number of such colorings can be computed efficiently based on Polya's theorem. However, enumerating each from the set of these colorings without using a large space is a challenging problem in general. In this paper, we propose a method for enumerating these colorings when the automorphisms of G are determined by two axial symmetries, and show that our algorithm can be implemented to run in polynomial delay and polynomial space.

Word search is a classical puzzle to search for all given words on a given assignment of letters to a rectangular grid (matrix). This problem is clearly in P. The inverse of this problem is more difficult, which asks to assign letters in a given alphabet to a matrix of given size so that every word in a given wordset can be found horizontally, vertically, or diagonally. This problem is in NP; it admits a trivial polynomial-size certificate. We prove its NP-hardness. It turns out to be so even under the following restrictions: 1) the alphabet size is 2 (binary) and 2) all the words to be found are of length at most 2. These results are optimal in the sense that decreasing these bounds 2 to 1 makes the problem be trivially in P.

Polynomial parahermitian matrices can accurately and elegantly capture the space-time covariance in broadband array problems. To factorise such matrices, a number of polynomial EVD (PEVD) algorithms have been suggested. At every step, these algorithms move various amounts of off-diagonal energy onto the diagonal, to eventually reach an approximate diagonalisation. In practical experiments, we have found that the relative performance of these algorithms depends quite significantly on the type of parahermitian matrix that is to be factorised. This paper aims to explore this performance space, and to provide some insight into the characteristics of PEVD algorithms.

This study deals with the design of a robust fault estimation and fault-tolerant control for vehicle lateral dynamics subject to external disturbance and unknown sensor faults. Firstly, a descriptor state and fault observer is designed to achieve the system state and sensor fault estimates simultaneously. Secondly, based on the information of on-line fault estimates, a robust fault-tolerant controller based on static output-feedback controller (SOFC) design approach is developed. To provide linear matrix inequalities of less conservatism, the results are conducted in the non-quadratic framework dealing with unmeasurable premise variables case. Simulation results show the effectiveness of the proposed control approach when the vehicle road adhesion conditions change and the sideslip angle is unavailable for measurement.

This study considers the pinning synchronisation in a network of coupled Lur’e dynamical systems under directed topology. By using tools from M-matrix theory, S-procedure and Lyapunov functional method, some simple pinning criteria in terms of linear matrix inequalities, whose dimensions are just determined by the size of a single Lur’e node, are derived for Lur’e networks with fixed and designed inner coupling matrices, respectively. A selective pinning scheme is proposed for a directed Lur’e network such that the network can be globally asymptotically pinned to a homogeneous state. Simulation results are provided to illustrate the effectiveness of the theoretical analysis.

We consider the computation of r-th roots in finite field-s. For the computation of square roots, there are two typical probabilistic methods: the Tonelli-Shanks method and the Cipolla-Lehmer method. The former method can be extended to the case of r-th roots, which is called the Adleman-Manders-Miller(AMM) method. The latter method had been generalized to the case of r-th roots with r prime. In this paper, we extend the Cipolla-Lehmer to the case of r-th root with r prime power and give the expected running time of our algorithm.

An approach for eigenvalue assignment in linear descriptor systems via output feedback is proposed. Sufficient conditions in order that a given set of eigenvalues is assignable are established. Parametric form of the desired output feedback gain matrix is also given. The approach assigns the full number of generalised eigenvalues, guarantees the closed-loop regularity and overcomes the defects of some previous works.