Computation of the Complexity of Networks under Generalized Operations

An Introduction to the Theory of Equations, by F. Cajori; Linear Algebra and Matrix Theory, by Robert R. Stoll. Pp. ix, 239 and ix, 272 respectively. £1.32 each. 1970. (Constable.) - Volume 55 Issue 391

The number of spanning trees in a network determines the totality of acyclic and connected components present within. This number is termed as complexity of the network. In this article, we address the closed formulae of the complexity of networks’ operations such as duplication (split, shadow, and vortex networks of Sn), sum (Sn + W3, Sn + K2, and (Cn∘K2) + K1), product (Sn⊠K2 and Wn∘K2), semitotal networks (Q(Sn) and R(Sn)), and edge subdivision of the wheel. All our findings in this article have been obtained by applying the methods from linear algebra, matrix theory, and Chebyshev polynomials. Our results shall also be summarized with the help of individual plots and relative comparison at the end of this article.

Functional models in linear algebra

Excerpt The vector-space representation opens the doors for utilizing the well-developed mathematical fields of linear algebra and matrix theory. Cohen, Horn, and Trussell, among others, have elegantly put forth the fundamental properties of the vector-space color representation. They showed the existence of the color match, the color additivity, the identity, the transformation of primaries, the equivalent primaries, the metameric relationship, the effect of illumination, and many imaging constraints. The abilities of representing spectral sensitivity of human vision and related visual phenomena in matrix-vector form provide the foundation for a major branch of computational color technology, namely, the studies of the phenomenon of surface reflection. This approach deals with signals reflected from an object surface under a given illumination into a human visual pathway; it has no interest in the physical and chemical interactions within the object and substrate. On the other hand, the physical interactions of the light with objects and substrates form the basis for another major branch of the computational color technology, namely, the physical color-mixing models that are used primarily in the printing industry. With the vector-space representation and matrix theory, in this chapter, we lay the groundwork for these computational approaches by revisiting the Grassmann's law of color mixing and reexamining color matching as well as other properties. 2.1 Visual Sensitivity and Color-Matching Functions Visual spectral sensitivity of the eye measured as the spectral absorption characteristics of human cones is given in Fig. 2.1. The sampled visual spectral sensitivity is represented by a matrix of three vectors V = [V1, V2, V3], one for each cone type, where Vi is a vector of n elements. Compared to the CMF curves of Fig. 1.2, they differ in several ways: first, the visual spectral sensitivity has no negative elements; second, the overlapping of green (middle-wavelength) and red (long-wavelength) components is much stronger. The sensor responses to the object spectrum η(λ) can be represented by γ=V T Î⋅.

Short Retraction Notice The paper does not meet the standards of "Advances in Linear Algebra & Matrix Theory". This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused. Editor guiding this retraction: Prof. Mubariz Garayev and Prof. Qingwen Wang (EiC of ALAMT) Please see the article page for more details. The full retraction notice in PDF is preceding the original paper which is marked "RETRACTED".

An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Assume that a pair of general Linear Random-effects Models (LRMs) are given with a correlated covariance matrix for their error terms. This paper presents an algebraic approach to the statistical analysis and inference of the two correlated LRMs using some state-of-the-art formulas in linear algebra and matrix theory. It is shown first that the best linear unbiased predictors (BLUPs) of all unknown parameters under LRMs can be determined by certain linear matrix equations, and thus the BLUPs under the two LRMs can be obtained in exact algebraic expressions. We also discuss algebraical and statistical properties of the BLUPs, as well as some additive decompositions of the BLUPs. In particular, we present necessary and sufficient conditions for the separated and simultaneous BLUPs to be equivalent. The whole work provides direct access to a very simple algebraic treatment of predictors/estimators under two LRMs with correlated covariance matrices.

Matrices and the structure of random number sequences

No Abstract.

It has been shown that the illuminant and object spectra could be approximated to a very high degree of accuracy by linearly combining a few principal components. As early as 1964, Judd, MacAdam, and Wyszecki reported that various daylight illuminations could be approximated by three components (see Section 1.5) and Cohen showed that the spectra of Munsell color chips could be accurately represented by a few principal components. Using principal component analysis (PCA), Cohen reported that the first basis vector alone accounted for 92.72% of the cumulative variance, and merely three or four vectors accounted for 99% variance or better. As a result, principal-component analysis became an extremely powerful tool for the computational color technology community. These findings led to the establishment of a finite-dimensional linear model. Combining with the rich contents of linear algebra and matrix theory, the model provides powerful applications in color science and technology. Many color scientists and researchers have contributed to the building and applying of this model to numerous color-image processing areas such as color transformation, white-point conversion, metameric pairing, indexing of metamerism, object spectrum reconstruction, illuminant spectrum reconstruction, color constancy, chromatic adaptation, and targetless scanner characterization. In this chapter, we present several methods for spectrum decomposition and reconstruction, including orthogonal projection, smoothing inverse, Wiener inverse, and principal component analysis. Principal components from several publications are compiled and evaluated by using a set of fourteen spectra employed for the color rendering index. Their similarities and differences are discussed. New methods of spectrum reconstruction directly from tristimulus values are developed and tested.

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.

Quantum mechanics has a history more than 100 years, and there are still many challenging open-ended questions. The problems encountered in modern physics such as matrix mechanics and quantum information often involve both quantum theory and open-ended problems in modern mathematics such as linear algebra. Distillability problem is an important open-ended problem in the field of quantum information and linear algebra. This article recalls the development history and application of linear algebra, summarizes the background and progress of distillability conjecture, and attempts to explore this problem by summarizing and using the knowledge of linear algebra and matrix theory learned in the course.

To this point, we have studied how Fourier transform methods are used in image reconstruction. This is the approach taken in the seminal work of Cormack [13] and used in the algorithms of today’s CT scan machines. However, the first CT scanner, designed in the late 1960 s by Godfrey Hounsfield, used an approach grounded in linear algebra and matrix theory to generate an image from the machine readings. Algorithms that adopt this point of view are known as algebraic reconstruction techniques, or ART, for short. In this chapter, we look at a few basic mathematical elements of ART.

Linear algebra is the core of modern applied mathematics. Its humble origins are to be found in the need to solve “elementary” systems of linear algebraic equations. But its ultimate scope is vast, impinging on all of mathematics, both pure and applied, as well as numerical analysis, statistics, data science, physics, engineering, mathematical biology, financial mathematics, and every other discipline in which mathematical methods are required. A thorough grounding in the methods and theory of linear algebra is an essential prerequisite for understanding and harnessing the power of mathematics throughout its multifaceted applications

Deconvolutions based on singular value decomposition and the pseudoinverse: a guide for beginners.