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In this paper, we investigate the position/orientation formation problem of multiple nonholonomic agents. Coordinate transformations are first presented to obtain the matrix form of formation error model, then a distributed smooth time-varying control law is designed based on a Lyapunov-like function. We prove that the closed-formation-system is globally asymptotically stable by Barbalat’s lemma if the communication graph is undirected, time-invariant and connected. Simulation results verify the effectiveness of the proposed control scheme.

Although the laws governing the trace-element behaviour are not linear, it is still possible, in some cases, to “invert” them and constrain the proportion of minerals present in a cumulate or a restite based on melt’s composition and partition coefficients. This chapter presents the relevant equations.

In this paper, we proposed an initial quantization parameter (QP) decision method based on frame complexity with multiple objectives of GOP (Group of Pictures) for H.264 rate control algorithm. Primarily, we choose four video sequences with different characteristics to constitute a sample space and find their optimal initial QPs which can guarantee to generate video sequences with consistent quality by minimizing the variation of QPs in a GOP, while ensuring the minimizing actual encoding bit rate closer to the target bit rate in various bit rates. And then we calculate the spatial characteristic of tested video sequences using the proportion of the number of complex MBs in the first I-frame. The optimal initial QPs are represented in a two-dimensional matrix form arranged in spatial characteristic and target bit rate according to proposed selection method of optimal initial QP. When any video sequence is given under any target bit rate, its spatial characteristic is calculated and mapped to one of four samples through the proposed mapping method. Finally, its optimal initial QP is determined by picking an element of matrix according to the mapped spatial characteristic and given target bit rate. Simulation results show that the proposed method achieves more obvious consistency in objective PSNRs and has secured encoding bit rate than noted algorithms.

According to linear algebra [1,2], a linear multi-variable simultaneous equation can always be written in a matrix form below

How we organize elements when reading a table was examined in a psychological experiment using a modified spatial-cuing paradigm. Table-like stimuli consisting of 16 square elements arranged in a four-by-four matrix form were used. Participants were instructed to discriminate whether the presented stimuli could be read as containing either one element or two elements in accordance with the induced reading direction. The results showed that when two elements were presented along with the induced direction, it was easier to read as such than when two elements were presented orthogonal to the induced direction. Although there was no contour line in the stimuli, participants were able to mentally segment and organize them into global units lying in the particular direction, which was instrumental to reading the tables efficiently.

Business intelligence focuses on the discovery of useful retail patterns by combining both historical and prognostic data. Ultimate goal is the orchestration of more targeted sales and marketing efforts. A frequent analytic task includes the discovery of associations between customers and products. Matrix co-clustering techniques represent a common abstraction for solving this problem. We identify shortcomings of previous approaches, such as the explicit input for the number of co-clusters and the common assumption for existence of a block-diagonal matrix form. We address both of these issues and present techniques for automated matrix co-clustering. We formulate the problem as a recursive bisection on Fiedler vectors in conjunction with an eigengap-driven termination criterion. Our technique does not assume perfect block-diagonal matrix structure after reordering. We explore and identify off-diagonal cluster structures by devising a Gaussian-based density estimator. Finally, we show how to explicitly couple co-clustering with product recommendations, using real-world business intelligence data. The final outcome is a robust co-clustering algorithm that can discover in an automatic manner both disjoint and overlapping cluster structures, even in the preserve of noisy observations.

The inverse problem of the thermal sounding of the atmosphere is formulated. The matrix form of the problem is considered. Features of ill-posed inverse problem are analyzed. Two approaches for solution are proposed.

Cultural relics are often damaged and incomplete due to various reasons. For the purpose of helping archaeological studies, we present a novel method for simultaneously restoring the original shapes of a group of similar objects. Based on the assumption that similar shapes are approximately linearly correlated, we use a matrix recovery technique to achieve the restoration. In order to represent input shapes in a matrix form, vectorization of each aligned sample is carried out by stacking coordinates of dense corresponding points that are generated by a surface matching scheme using non-rigid deformation. An experiment using 3D scans of facial sculptures from Bayon is conducted, and the result verifies the feasibility and effectiveness of our method.

The most significant turbulence parameters of a turbulent flow are specified by corresponding Reynolds stresses that are given in the matrix form at Eq. (2.11). The knowledge about the turbulent shear stresses for instance \(\tau _{\textrm{uv}}\) basically demands, according to Eq. (5.7), coincident measurements of two velocity components u and v.

In this chapter we will investigate the solution of sets of linear algebraic equations. First, we show a method which will be used in most practical cases. This is the Gaussian method of elimination and its refinements. The basic idea is quite clear and elementary. Notation in matrix form will prove to be helpful.

The object-oriented approach applied to the Finite Element Method leads to better structured numerical codes. The purpose of such an approach is to facilitate the maintenance and the extension of the code. Among the early work in this domain one may note [1] [2], the authors’ contribution in [3] [4] [5]. After this work, which was restricted to the direct coding of the matrix form, an object-oriented approach applied to the symbolic derivations of Finite Element models and automatic programming was proposed [6] [7]. Many attempts in this domain already exist ([8] [9] [10] [11] for example), but they were restricted to the derivation of specific elemental matrices. The advantage of using the high level of abstraction of Object-Oriented Programming is that it leads, on one hand to an environment which is easy to extend, and on the other hand to quite natural derivations. Simple applications in linear mechanics were shown in references [7] and [12]. The present paper introduces a first extension of the environment to non-linear mechanics; a simple model of Navier-Stokes flow illustrates the new approach.

A new method of homology design based on finite element sensitivity analysis is proposed in static problems. The stiffness equation is separated into two equations by introducing homologous constraint. The governing equation of the design variables is derived from the condition for the two equations to hold. The Moore-Penrose generalized inverse is employed to determine the design variables, as the governing equation is expressed in a rectangular matrix form. The validity of the proposed method is demonstrated by the numerical example concerning with out-of-plane bending of a planar frame structure, for which the homologous constraint is set to keep a member straight.

As an extension of the input-output analysis, we have introduced in the previous chapter a new concept which might be called the “interrnational income multiplier” in matrix form. It was designed to analyse the interrelationships among various income-groups in the process of income formation, and in this respect it tells us how much of one group’s income is generated by another group’s expenditure from one unit of additional income via the medium of industrial production activity. Although this multi-sector multiplier follows from Leontief’s “interindustry matrix multiplier”, it is formulated by the inclusion of the income generation process, which is omitted in the standard input-output open model, and by projecting the multiplier process into the income-determination side rather than the output-determination side.

This is the first in a projected series of papers on solutions to games in matrix and extensive form. The predominant solution concept in the literature is that of the noncooperative equilibrium put forward by Nash (1951).