Group Invariant Theory Research Articles

Maximum eigenvalue‐based target detection for the K‐distributed clutter environment

Information geometry-based matrix constant false alarm rate (CFAR) detector is an efficient solution to target detection, especially for the K-distributed clutter environment with respect to a small bunch of pulses. The main reason for the matrix CFAR detector to achieve better detection performance is covariance matrix captures the correlations between the data. However, most existing matrix CFAR detectors suffer from heavy computational complexity, which leads to a limitation in practical detection scenarios. Motivated by this, the authors utilise the eigenvalues of the covariance matrix to capture the correlation and propose an eigenvalue-based detection method with lower computational complexity. Based on the Neyman-Pearson criterion, they first analyse the likelihood ratio test (LRT) in the eigenvalue domain and derive the relationship between the LRT test statistic and the maximum eigenvalue. To meet the practical requirement, they further design a totally blind scheme: the maximum eigenvalue-based matrix CFAR detector. By employing the group invariant theory, they show that the proposed detector presents the CFAR property. In addition, the theoretical performance analysis is also provided. Simulation results based on the numerical experiment and real sea clutter data verify that the proposed method with a low computational complexity can achieve a better detection performance.

Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods

This paper is concerned with two examples on the application of the free boundary formulation to BVPs on a semi-infinite interval. In both cases we are able to provide the exact solution of both the BVP and its free boundary formulation. Therefore, these problems can be used as benchmarks for the numerical methods applied to BVPs on a semi-infinite interval and to free BVPs. Moreover, we emphasize how for two classes of free BVPs, we can define non-iterative initial value methods, whereas BVPs are usually solved iteratively. These non-iterative methods can be deduced within Lie's group invariance theory. Then, we show how to apply the non-iterative methods to the two introduced free boundary formulations in order to obtain meaningful numerical results. Finally, we indicate several problems from the literature where our non-iterative transformation methods can be applied.

Magnetic field-induced martensitic phase transformation in magnetic shape memory alloys: Modeling and experiments

In this work, a continuum based model of the magnetic field induced phase transformation (FIPT) for magnetic shape memory alloys (MSMA) is developed. Hysteretic material behaviors are considered through the introduction of internal state variables. A Gibbs free energy is proposed using group invariant theory and the coupled constitutive equations are derived in a thermodynamically consistent way. An experimental procedure of FIPT in NiMnCoIn MSMA single crystals, which can operate under high blocking stress, is described. The model is then reduced to a 1-D form and the material parameter identification from the experimental results is discussed. Model predictions of magneto-thermo-mechanical loading conditions are presented and compared to experiments.

On the equivalence of non-iterative transformation methods based on scaling and spiral groups

The non-iterative numerical solution of nonlinear boundary value problems is a subject of great interest. The present paper is concerned with the theory of non-iterative transformation methods (TMs). These methods are defined within group invariance theory. Here we prove the equivalence between two non-iterative TMs defined by the scaling group and the spiral group, respectively. Then, we report on numerical results concerning the steady state temperature space distribution in a non-linear heat generation model. These results improve the ones, available in the literature, obtained by using the invariance with respect to a spiral group. Copyright © 2009 John Wiley & Sons, Ltd.

Numerical transformation methods: a constructive approach

Within group invariance theory we consider a constructive approach in order to define numerical transformation methods. These are initial-value methods for the solution of boundary value problems governed by ordinary differential equations. Here we consider the class of free boundary value problems governed by the most general second-order equation in normal form. For this class of problems the main theorem is concerned with the definition of an iterative transformation method. The definition of a noniterative method, applicable to a subclass of the original class of problems, follows as a corollary. Therefore, the proposed constructive approach allows us to establish a unifying framework for noniterative and iterative transformation methods.

The falkneer-skan equation: Numerical solutions within group invariance theory

The iterative transformation method, defined within the framework of the group invariance theory, is applied to the numerical solution of the Falkner-Skan equation with relevant boundary conditions. In this problem a boundary condition at infinity is imposed which is not suitable for a numerical use. In order to overcome this difficulty we introduce a free boundary formulation of the problem, and we define the iterative transformation method that reducess the free boundary formulation to a sequence of initial value problems. Moreover, as far as the value of the wall shear stress is concerned we propose a numerical test of convergence. The usefulness of our approach is illustrated by considering the wall shear stress for the classical Homann and Hiemenz flows. In the Homann's case we apply the proposed numerical test of convergence, and meaningful numerical results are listed. Moreover, for both cases we compared our results with those reported in literature.

Latin square determinants II

The theory of latin square determinants may be regarded as a direct continuation of the line of research which led Frobenius to introduce group characters. A previous paper introduced the basic ideas and indicated how the theory relates to quasigroup character theory. The work here sets out further developments. The linear factors of a latin square determinant are characterised. Results on a lower bound for the number of irreducible factors are obtained, and methods to factorise determinants with various kinds of symmetries are given, as well as determinants arising as extensions. A ‘Molien series’ for a latin square is defined, generalising that arising in group invariant theory. A determinant arising out of a pair of squares is discussed, and when the pair of squares is an orthogonal pair arising from a finite field it is shown that this determinant has a special property. Further examples have been calculated using symbolic manipulation packages.