Revising the link between L-Chu correspondences and completely lattice L-ordered sets

Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets.     

Distributed Zero-Gradient-Sum (ZGS) consensus optimisation over networks with time-varying topologies

This paper addresses a distributed consensus optimisation problem over networks with time-varying topologies based on Zero-Gradient-Sum (ZGS) algorithm. First, the exponential convergence of the algorithm is guaranteed under a new condition on network topologies, called cooperatively connected. This condition does not require the topology constantly connected or jointly connected but only requires the integral of the Laplacian matrix of the network topology over a period of time is connected. Hence, it is suitable for more general time-varying topologies. Second, by establishing a key mathematical lemma, we develop a convergence analysis technique which is based on the difference of the Lyapunov function rather than its differentiation. Finally, a simulation example is also provided to verify the results obtained in this paper.     

On boolean functions and connected sets

Theory of Computing Systems -     

Estimating motion and structure from correspondences of linesegments between two perspective images

Presents an algorithm for determining 3D motion and structure from correspondences of line segments between two perspective images. To the author's knowledge, this paper is the first investigation of use of line segments in motion and structure from motion. Classical methods use their geometric abstraction, namely straight lines, but then three images are necessary for the motion and structure determination process. In this paper the author shows that it is possible to recover motion from two views when using line segments. The assumption used is that two matched line segments contain the projection of a common part of the corresponding line segment in space, i.e., they overlap. Indeed, this is what the author uses to match line segments between different views. This assumption constrains the possible motion between two views to an open set in motion parameter space. A heuristic, consisting of maximizing the overlap, leads to a unique solution. Both synthetic and real data have been used to test the proposed algorithm, and excellent results have been obtained with real data containing a relatively large set of line segments     

On partitions of functions on finite sets

Let D and R be finite sets with cardinality n and m respectivelyR ^D be the set of all functions from D into R, and G and H be permutation groups acting on D and R respectively. Two functions f and g in R ^D are said to be related if there exists a σ in G and a τ in H with f(σd) = τg(d) for every d in D. Since the relation is an equivalence relation, R ^D is partitioned into disjoint classes. Roughly, by using the cycle indices of G and H, de Bruijn's theorem determines the number of equivalence classes, and Pólya's theorem, with H being the identity group, gives the function counting series, Pólya-de Bruijn's theorem has many applications (for instance, see Pólya and Read [6]). The theorem and its applications, basically, centered around the partitions of functions. Here, we present an algorithm to determine which functions in R ^D belong to the same equivalent class. Our algorithm does not use the cycle indices of G and H (to compute the cycle index of a given group, in general, is difficult), but it uses the generators of G and H, and the m-nary numbers to code the functions in R ^D . Our algorithm also gives the function counting series and the number of equivalence classes. An important application is that for each positive integer n, we use our algorithm and the symmetric group S _n to determine all isomorphic and nonisomorphic graphs and directed graphs with n vertices.     

Some sets of orthogonal polynomial kernel functions

Kernel methods provide high performance in a variety of machine learning tasks. However, the success of kernel methods is heavily dependent on the selection of the right kernel function and proper setting of its parameters. Several sets of kernel functions based on orthogonal polynomials have been proposed recently. Besides their good performance in the error rate, these kernel functions have only one parameter chosen from a small set of integers, and it facilitates kernel selection greatly. Two sets of orthogonal polynomial kernel functions, namely the triangularly modified Chebyshev kernels and the triangularly modified Legendre kernels, are proposed in this study. Furthermore, we compare the construction methods of some orthogonal polynomial kernels and highlight the similarities and differences among them. Experiments on 32 data sets are performed for better illustration and comparison of these kernel functions in classification and regression scenarios. In general, there is difference among these orthogonal polynomial kernels in terms of accuracy, and most orthogonal polynomial kernels can match the commonly used kernels, such as the polynomial kernel, the Gaussian kernel and the wavelet kernel. Compared with these universal kernels, the orthogonal polynomial kernels each have a unique easily optimized parameter, and they store statistically significantly less support vectors in support vector classification. New presented kernels can obtain better generalization performance both for classification tasks and regression tasks.     

Effective optimisation of continuum topologies through a multi-GA system

The success of GA-based topology optimisation methods for continuum structures has been limited. Though a number of methods exist, the results are far from comparable to solutions from more established methods in terms of quality and fidelity. This had led many structural optimisation researchers to dismiss the GA as a viable method for topology optimisation of continuum structures. The authors believe however that the deficiencies of previous applications lie not in the GA but in the manner in which it was applied. Based on insights gained from these applications, a new GA-based method for the continuum topology optimisation problem is herein presented which yields high-fidelity solutions comparable to those produced through more well-known and established methods. The method is centred on a number of novel concepts, most notably the simultaneous use of multiple genetic algorithms, and the use of a high-level string coding based on structural response information. In this work, the basis and methodology of the multi-GA approach are explained, and key algorithms detailed. The method was also applied to a number of structural problems to show its efficacy, and the results presented and discussed.     

Improving feasibility of robotic milling through robot placement optimisation

Milling performed with robots is quite demanding, even for low-strength materials, due to the high accuracy requirements, the generally high and periodically varying milling forces and the low stiffness of robots compared to CNC machine tools. In view of the generally improved recently robot stiffness, it is desirable to perform the milling operation in regions of the robot’s workspace where manipulability, both kinematic and dynamic, is highest, thereby exhausting the robot’s potential to cope with the process. In addition, by selecting the most suitable initial pose of the robot with respect to the workpiece, a reduction in the range of necessary joint torques may be reached, to the extent of alleviating the heavy requirements on the robot. Two genetic algorithms (GAs) are employed to tackle these problems. The values of several robot variables, such as joint positions and torques, which are needed by the genetic algorithms, are calculated using inverse kinematics and inverse dynamics models. In addition, initial positions and poses leading to singularities along the milling path are penalized and, thus, avoided. The first GA deals solely with robot kinematics to maximize manipulability. The second GA takes into account milling forces, which are computed numerically according to the particular milling parameters, to minimise joint torque loads.     

Distributed algorithms for reaching consensus on general functions

This paper presents analysis and design results for distributed consensus algorithms in multi-agent networks. We consider continuous consensus functions of the initial state of the network agents. Under mild smoothness assumptions, we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus. This characterization is the building block to obtain various design results for networks with weighted, directed interconnection topologies. We first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus. We apply this result to the family of weighted power mean functions, and characterize the exponential convergence properties of the resulting algorithms. We establish the validity of these results for scenarios with switching interconnection topologies. Finally, we conclude with two discontinuous distributed algorithms that achieve, respectively, max and min consensus in finite time.     

Hierarchical optimisation for non-linear dynamical systems with non-separable cost functions

In this paper the previous hierarchical optimisation algorithm of Hassan and Singh for non-linear interconnected dynamical systems with separable cost functions is extended to the case of non-linear and non-separable cost functions. This ensures that any decomposition could be used and makes the new algorithm suitable for the optimisation of general non-linear problems.     

Realizing Boolean functions on disjoint sets of variables

For switching functions f let C(f) be the combinational complexity of f. We prove that for every ε>0 there are arbitrarily complex functions f:{0,1}ⁿ→{0,1}ⁿ such that C(f×f)? (1+ε)C(f) and arbitrarily complex functions f:{0,1}ⁿ→{0,1} such that C(v°(fxf)? (1+ε)C(f). These results and the techniques developed to obtain them are used to show that Ashenhurst decomposition of switching functions does not always yield optimal circuits, and to prove a new result concerning the gap between circuit size and monotone circuit size.     

Basis functions and parameter optimisation in high-order iterative learning control

In this paper a new parameter-optimal high-order Iterative Learning Control (ILC) algorithms is proposed to extend the work of Owens and Feng [Parameter optimisation in iterative learning control. International Journal of Control 14(11), 1059-1069]. If the original plant is positive, this new algorithm will result in convergent learning where the convergence is monotonic to zero tracking error. If the original plant is not positive, it can be shown that by adding a suitable set of basis functions into the algorithm, the tracking error will again converge monotonically to zero. This provides a considerable improvement to earlier work on parameter-optimal ILC as it opens up the possibility of globally convergent algorithms for any linear plant G. The number of parameters needed to ensure convergence could, however, become large. The paper shows that the use of low-order parameterisations is capable of achieving much of the benefit achieved in the ‘ideal’ case.     

Constructing sets of functions which have a givenF-cardinality

The termF-cardinality of (=F-card()) is introduced whereF: ^{n n} is a partial function and is a set of partial functionsf: ^{n n} . TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs (n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ^{n n} and a natural numberk, it is almost always possible to construct a set such that itsF-cardinality has the valuek; in addition, can be required to be closed under composition of functionsf,g . Moreover, ifF is continuous, then consists of continuous functions.     

Satisfying constraint sets through convex envelopes

In this article, we present a general representation for constraint satisfaction problems with disjunctive relations called cluster constraint systems (CCS). For this representation, we develop a novel and simple approach for solving CCSs using convex envelopes. These envelopes can be used to decompose the feasible space of the CCS through convex approximations. We explore interval reasoning as a case study of CCS. Our experimental results demonstrate that such CCS can be effectively and efficiently solved through convex enveloping with very modest branching requirements in comparison to other generic as well as specialized algorithms for interval reasoning. In fact, convex enveloping solves significantly more cases and more efficiently than other methods used in our test bed.     

Finite-time scaled consensus through parametric linear iterations

This paper deals with finite-time scaled consensus problems over undirected and directed topologies, wherein agents’ states reach prescribed ratios in a finite time. We develop distributed linear iterations as a function of a linear operator on the underlying network and present necessary and sufficient conditions guaranteeing scaled consensus in a fixed number of steps equal to the number of distinct eigenvalues of a related linear operator. We identify the dependence of the final consensus states on the initial state condition, which can be conveniently and freely tuned by designing suitable parameters. Our results extend the recently developed approach on successive nulling of eigenvalues from complete consensus to scaled consensus, and from undirected topologies to directed topologies. Numerical examples and comparison studies are provided to illustrate the effectiveness of our theoretical results.