A note on projective modules over real affine algebras

Let A be an affine algebra over the field of real numbers of dimension d. Let f \in A be an element not belonging to any real maximal ideal of A. Let P be a projective A-module of rank \geq d-1. Let (a,p) \in A_f \oplus P_f be a unimodular element. Then the projective A_f module Q=A_f \oplus P_f/(a,p)A_f is extended from A.Comment: 11 page

Stability result for projective modules over blowup rings

Let R be an affine algebra of dimension n \geq 3 over an algebraically closed field k. Suppose char k =0 or char k =p \geq n. Let g,f_1,...,f_r be a R-regular sequence and A=R[f_1/g,...,f_r/g]. Let P be a projective A-module of rank n-1 which is extended from R. Let (a,p) \in (A \op P) be a unimodular element and Q=A\op P/(a,p)A. Then, Q is extended from R. A similar result for affine algebras over reals are also proved.Comment: 15 page

Password Based a Generalize Robust Security System Design Using Neural Network

Among the various means of available resource protection including biometrics, password based system is most simple, user friendly, cost effective and commonly used. But this method having high sensitivity with attacks. Most of the advanced methods for authentication based on password encrypt the contents of password before storing or transmitting in physical domain. But all conventional cryptographic based encryption methods are having its own limitations, generally either in terms of complexity or in terms of efficiency. Multi-application usability of password today forcing users to have a proper memory aids. Which itself degrades the level of security. In this paper a method to exploit the artificial neural network to develop the more secure means of authentication, which is more efficient in providing the authentication, at the same time simple in design, has given. Apart from protection, a step toward perfect security has taken by adding the feature of intruder detection along with the protection system. This is possible by analysis of several logical parameters associated with the user activities. A new method of designing the security system centrally based on neural network with intrusion detection capability to handles the challenges available with present solutions, for any kind of resource has presented

Budgetary constraints and growth scenario in Uttar Pradesh.

Economic growth ; Uttar Pradesh