Complexity of Some Generalized Operations on Networks

In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. In this paper we define a class of pyramid graphs and derive simple formulas of the complexity, number of spanning trees, of these graphs, using linear algebra, Chebyshev polynomials, and matrix analysis techniques.

The connected and acyclic components contained in a network are identified by the computation of its complexity, where complexity of a network refers to the total number of spanning trees present within. The article in hand deals with the enumeration of the complexity of various networks’ operations such as sum ( K 2, n + W 3 , K 2, n + n K 1 , K n + S n ), product ( K 2, n ⊠ K 2 , K 2, n ⋉ K 2 , K n × K 2 , K n ⊠ K 2 ), difference ( K 2, n ⊖ K 2 ), and the conjunction of S n with K 2 . All our computations have been concluded by implementation of the methods of linear algebra and matrix theory. Our derivations will also be highlighted with the assistance of 3D plots at the end of this article.

Qualitative Problems in Matrix Theory

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

Functional models in linear algebra

No Abstract.

Linear algebra is an important basic course in university mathematics, finite dimensional vector space and linear transformation theory mainly on matrix theory, and the combination of matrix. Concept in linear algebra is defined directly by the mathematical symbol, rarely by example to import. How to make students master the abstract and difficult course, first of all is to let the students find the course in our daily life, to stimulate interest in learning.

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".

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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

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky.

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.

Linear Algebra and Polynomial Matrices