A STUDY OF SOLVING SYSTEM OF LINEAR EQUATION USING DIFFERENT METHODS AND ITS REAL LIFE APPLICATIONS:

Solving a system of linear equations (or linear systems or, also simultaneous equations) is a common situation in many scientific and technological problems. Many methods either analytical or numerical, have been developed to solve them so, in this paper, I will explain how to solve any arbitrary field using the different – different methods of the system of linear equation for this we need to define some concepts. Like a general method most used in linear algebra is the Gauss Elimination or variation of this sometimes they are referred as “direct methods “Basically it is an algorithm that transforms the system into an equivalent one but with a triangular matrix, thus allowing a simpler resolution, Other methods can be more effective in solving system of the linear equation like Gauss Elimination or Row Reduction, Gauss Jordan and Crammer’s rule, etc. So, in this paper I will explain this method by taking an example also, in this paper I will explain the Researcher’ works that how they explain different –different methods by taking different examples. And I worked on using these different methods in solving a single example, i.e. I will use these methods in an example. In this paper, I will explain the real-life application that how a System of Linear Equation is used in our daily life.

DETERMINAN SUATU MATRIKS TOEPLITZ K-TRIDIAGONAL MENGGUNAKAN METODE REDUKSI BARIS DAN EKSPANSI KOFAKTOR

This paper discusses the determinants of a k-tridiagonal Toeplitz matrix using row reduction and cofactor expansion methods. The analysis was carried out recursively from the general form of the determinant of the tridiagonal Toeplitz matrix, the determinant of the 2-tridiagonal Toeplitz matrix, and the determinant of the 3-tridiagonal Toeplitz matrix. In the end, the general form of the determinant of the k-tridiagonal Toeplitz matrix is obtained.

Ontogenetic and static allometry in the skull and cranial units of nine-banded armadillos (Cingulata: Dasypodidae: Dasypus novemcinctus)

A large part of extant and past mammalian morphological diversity is related to variation in size through allometric effects. Previous studies suggested that craniofacial allometry is the dominant pattern underlying mammalian skull shape variation, but cranial allometries were rarely characterized within cranial units such as individual bones. Here, we used 3D geometric morphometric methods to study allometric patterns of the whole skull (global) and of cranial units (local) in a postnatal developmental series of nine-banded armadillos (Dasypus novemcinctus ssp.). Analyses were conducted at the ontogenetic and static levels, and for successive developmental stages. Our results support craniofacial allometry as the global pattern along with more local allometric trends, such as the relative posterior elongation of the infraorbital canal, the tooth row reduction on the maxillary, and the marked development of nuchal crests on the supraoccipital with increasing skull size. Our study also reports allometric proportions of shape variation varying substantially among cranial units and across ontogenetic stages. The multi-scale approach advocated here allowed unveiling previously unnoticed allometric variations, indicating an untapped complexity of cranial allometric patterns to further explain mammalian morphological evolution.

Two new species of Rhinophis Hemprich (Serpentes: Uropeltidae) from Sri Lanka

Two new species of uropeltid (shieldtail) snake are described from Sri Lanka; Rhinophis lineatus sp. nov. from Harasbedda, near Ragala, and Rhinophis zigzag sp. nov. from Bibilegama, near Passara. The new species are distinguished from congeners in morphometric and meristic external characters, and in having very distinctive colour patterns. Scale-row reduction data are presented for the two new species; this is a new development for uropeltid systematics, and its potential utility is highlighted. The nature of the overlap between the two anal scales is also highlighted as a potentially useful character. The two new species were included in previous phylogenetic analyses of allozyme and albumin immunological data, but their phylogenetic relationships are not yet well resolved.

Factor rings of the Gaussian integers

Whereas the homomorphic images of Z (the ring of integers) are well known, namely Z, {0} and Zn (the ring of integers modulo n), the same is not true for the homomorphic im-ages of Z[i] (the ring of Gaussian integers). More generally, let m be any nonzero square free integer (positive or negative), and consider the integral domain Z[ √m]={a + b √m | a, b ∈ Z}. Which rings can be homomorphic images of Z[ √m]? This ques-tion offers students an infinite number (one for each m) of investigations that require only undergraduate mathematics. It is the goal of this article to offer a guide to the in-vestigation of the possible homomorphic images of Z[ √m] using the Gaussian integers Z[i] as an example. We use the fact that Z[i] is a principal ideal domain to prove that if I =(a+bi) is a nonzero ideal of Z[i], then Z[i]/I ∼ = Zn for a positive integer n if and only if gcd{a, b} =1, in which case n = a2 + b2 . Our approach is novel in that it uses matrix techniques based on the row reduction of matrices with integer entries. By characterizing the integers n of the form n = a2 + b2 , with gcd{a, b} =1, we obtain the main result of the paper, which asserts that if n ≥ 2, then Zn is a homomorphic image of Z[i] if and only if the prime decomposition of n is 2α0 pα1 1 ··· pαk k , with α0 ∈{0, 1},pi ≡ 1(mod 4) and αi ≥ 0 for every i ≥ 1. All the fields which are homomorpic images of Z[i] are also determined.