An alternative approach to orthogonal graph theoretical invariants

Given a symmetric bilinear form F on a finite dimensional real vector space V we attempt to find ‘nice’ bases where the matrix representation of the form is as simple as possible, in the same sort of way as we did in the Gram-Schmidt orthogonalization process for inner products. The theorem which says that this is possible and which describes all the possible matrix representations one can get is called Sylvester’s law of inertia. (Sylvester’s law of inertia has nothing to do with inertia: the story goes that Sylvester said, ‘If Isaac Newton can have a law of inertia, then so can I’, and consequently gave this theorem the name it has been known by ever since.) This ‘law’ is exactly analogous to Theorem 2.33 which classified all finite dimensional vector spaces using dimension; it turns out that symmetric bilinear forms are classified by the dimension of the underlying vector space and two further numbers, called the rank and the signature of the form.

Chapter 5 - Vector Spaces

The loss of orthogonality in the Gram-Schmidt orthogonalization process

Two-dimensional orthogonal polynomials for vibration of rectangular composite plates

In the previous chapter, the theory and methods of elementary linear algebra were extended to linear spaces, in that finite-dimensional vector spaces were extended to include infinite-dimensional sequence and function spaces, that the inner product was applied to derive such mathematical tools as orthogonal projection and the Gram-Schmidt orthogonalization process, and that the concept of completeness was studied to introduced Hilbert and Banach spaces.

Structure–taste relationships for 25 acyclic and 20 cyclic carbosulfamates were investigated by means of pattern recognition using different graph theoretical invariants as molecular substituent descriptors. The SIMCA method was used to classify the compounds into sweet and nonsweet classes. All selected graph theoretical invariants that are related to the “rooted” vertex were found to give promising results. Using the weighted path numbers and self‐returning walks for the rooted atom as descriptors of substituents, we found 87% of acyclic compounds were correctly classified. Using the atomic path numbers for the rooted atom as descriptors of substituents, we found 81% of cyclic compounds were correctly classified. These results are better than previously used shape and size substituent descriptors. It may be concluded that the graph theoretical descriptors have great potential in encoding structure components in structure–activity studies (SAR) studies.

An efficient model correction method is proposed by using the modal measurement from a structural system. The method corrects/updates the mass and stiffness matrix without imposing any parameterization. It considers the information from both the nominal finite-element model and the measurement of modal frequencies and mode shapes. The method is computationally very efficient and it does not require computation of the complete set of eigenvalues and eigenvectors of the nominal model. Instead, only the nominal eigenvalues and eigenvectors of the modes to be corrected are needed. The Gram-Schmidt orthogonalization process is used to construct a basis that satisfies the mass orthogonality condition. This basis is used to transform the eigenvectors of the nominal model so that the corrected model is compatible with the measurement. A thousand-degree-of-freedom chainlike system and a 1,440-degree-of-freedom structural frame are used to illustrate the proposed method.

Study on flexural vibration of triangular composite plates influenced by fibre orientation

In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision.

The Modified Gram-Schmidt (MGS) orthogonalization process -- used for example in the Arnoldi algorithm -- constitutes often the bottleneck that limits parallel efficiencies. Indeed, a number of communications, proportional to the square of the problem size, is required to compute the dot-products. A block formulation is attractive but it suffers from potential numerical instability.In this paper, we address this issue and propose a simple procedure that allows the use of a Block Gram-Schmidt algorithm while guaranteeing a numerical accuracy similar to MGS. The main idea is to dynamically determine the size of the blocks. The main advantage of this dynamic procedure are two-folds: first, high performance matrix-vector multiplications can be used to decrease the execution time. Next, in a parallel environment, the number of communications is reduced. Performance comparisons with the alternative Iterated CGS also show an improvement for moderate number of processors.

BackgroundMuscle morphology, age and gender may be determinants of muscle strength in older adults. However, very few research studies have directly examined such correlation in the ageing spine. The aim of the study was to examine the correlation between lumbar extensor muscle strength, its muscle fibre angles, thoracolumbar curvature, age and gender in the older and younger adults.MethodsMuscle fibre angles of lumbar extensor muscles, thoracolumbar curvature and lumbar extensor muscle strength were examined in 26 young (mean age 27.9, SD 5.2) and 26 older (mean age 72.1, SD 5.9) participants. Pearson’s correlation was employed to determine the association among lumbar extensor muscle fibre angle, thoracolumbar curvature, age, gender and lumbar extensor muscle strength. Multiple stepwise linear regression analysis was used to identify significant determinants of lumbar extensor muscle strength.ResultsThe results demonstrated a significant correlation between lumbar extensor muscle strength, muscle fibre angle, age and gender. In the step wise regression analysis, both gender and age were identified as the most robust determinant for lumbar extensor muscle strength in older adults. However, gender was the only determinant of muscle strength in the young.ConclusionThese results suggest that the decline in the lumbar extensor muscle strength in older adults was more dependent on age when compared to younger adults.

In this work we prove a theorem which makes it possible to obtain a basis of graph theoretical invariants. The differences with Randic's method of orthogonalization are pointed out.

The main philosophy of MD calculation is a measurement scale used to discriminate and interpret the relationships among data samples (abnormal vs normal).Mahalanobis distance is the multivariate generalization of finding how many standard deviations away a point is from the mean of the multivariate distribution. Taguchi introduced the Mahalanobis Distance into the process of defining a reference group and measuring individual subsets.Mahalanobis-Taguchi System MTS or Mahalanobois-Taguchi-Gram-Schmidt process (MTGS) is an alternative way of evaluating MD with the help of the Gram Schmidt orthogonalization process (GSP) is to develop and optimize a diagnostic system with a measurement scale of abnormality and for casting using multivariate data.This paper evaluates the value of Mahalanobis Distance MD, on some different random sample sizes by using the Mahalanobois-Taguchi-Gram-Schmidt process (MTGS) and another new modification way to evaluating MD called kawa-MTGS (k-MTGS) methods then comparing the result with other

Puberty and Microvascular Function in Healthy Children and Adolescents

Cross-mode bioelectrical impedance analysis in a standing position for estimating fat-free mass validated against dual-energy x-ray absorptiometry