On norm preserving conditions for local automorphisms of commutative banach algebras

The history of commutative algebra first appeared in 1890 by David Hilbert which was then followed by Banach spaces in 1924 since localization reduces many problems of geometric special case into commutative algebra problems of local ring. So far, many studies on preserver problems have been focusing on linear preserver problems (LPPs) especially LPPs in matrix theory. Also in consideration has been the characterization of all linear transformation on given linear space of matrices that leave certain functions, subsets and relations invariant. Clearly, we also have spectrum preserver problem or transmission. Kadison and Sourour have also shown that the derivation of local derivation of Von Neumann algebra R are continous linear maps if it coincides with some derivation at each point in the algebra over C. We employ the concept of 2-local automorphisms introduced by Serml that if we let A be an algebra, then the transformation  is called a 2-local automorphism if for all x, y  A there is an automorphism (xy) of A for which x,y(x) and x,y(y). In this paper, we characterize commutativity of local automorphism of commutative Banach algebras, establish the norm preserver condition and determine the norms of locally inner automorphisms of commutative Banach algebras. We use  Hahn-Banach extension theorems and the great ideas developed by Richard, and Sorour to develop the algebra of local automorphisms, then integrate it with norm preserver conditions of commutative Banach algebras. The results of this work have a great impact in explaining the theoritical aspects of quantum mechanics especially when determining the distance of physical quantities

Norms and numerical radii inequalities for ( ) - normal transaloid operators

The studies on Hilbert spaces for the last decade has been of great interest to many mathematicians and researchers, especially on operator inequalities related to operator norms and numerical radii for a family of bounded linear operators acting on a Hilbert spaces. Results on some inequalities for normal operators in Hilbert spaces for instance numerical ranges W(T), numerical radii w(T) and norm ||.|| obtained  by Dragomir and Moslehian among others due to some classical inequalities for vectors in Hilbert spaces. The techniques employed to prove the results are elementary with some special vector inequalities in inner product spaces due to Buzano, Goldstein, Ryff and Clarke as well as some reverse Schwarz inequalities. Recently, the new field of operator theory done by Dragomir and Moslehian on norms and numerical radii for ( ) - normal operators developed basic concepts for our Statement of the problem on normal transaloid operators. M. Fujii and R. Nakamoto characterize transaloid operators in terms of spectral sets and dilations and other non-normal operators such as normaloid, convexoid and spectroid. Furuta did also characterization of normaloid operators. Since none has done on norms and numerical radii inequalities for ( ) – normal transaloid operators, then our aim is to characterize ( )- normal  transaloid  operators, characterize norm inequalities for ( )- normal transaloid operators and to characterize numerical radii for ( )- normal transaloid operators.  We use the approach of the Cauchy-Schwarz inequalities, parallelogram law, triangle inequality and tensor products. The results obtained are useful in applications in quantum mechanics

Symmetry Group Approach to the Solution of Generalized Burgers Equation: U t + UU x = λU xx

Abstract Symmetry of a system of differential equations is a transformation that maps any solution to another solution of the system. In Lie's framework such transformations are groups that depend on continuous parameters and consist of point transformations (point symmetries), acting on the system's space of independent and dependent variables, or, more generally, contact transformations (contact symmetries), acting on independent and dependent variables as well as on all first derivatives of the dependent variables. Lie groups, and hence their infinitesimal generators, can be naturally prolonged to act on the space of independent variables, dependent variables, and derivatives of the dependent variables. We present a Lie symmetry approach in solving Burgers Equation:U t + UU x = λU xx which is a nonlinear partial differential equation, which arises in model studies of turbulence and shock wave theory. In physical application of shock waves in fluids, coefficient λ, has the meaning of viscosity. So far in both analytic and numerical approaches the solutions have only been established for 0 ≤ λ ≤ 1. In this paper, we give a global solution to the Burgers equation with no restriction on λ i.e. for λ ∈ (−∞, ∞)