On the Classification of Weierstrass Elliptic Curves over Zn\mathbb{Z}_n

The development of secure cryptographic protocols and the subsequent attack mechanisms have been placed in the literature with the utmost curiosity. While sophisticated quantum attacks bring a concern to the classical cryptographic protocols present in the applications used in everyday life, the necessity of developing post-quantum protocols is felt primarily. In post-quantum cryptography, elliptic curve-base protocols are exciting to the researchers. While the comprehensive study of elliptic curves over finite fields is well known, the extended study over finite rings is still missing. In this work, we generalize the study of Weierstrass elliptic curves over finite ring $\mathbb{Z}_n$ through classification. Several expressions to compute critical factors in studying elliptic curves are conferred. An all-around computational classification on the Weierstrass elliptic curves over $\mathbb{Z}_n$ for rigorous understanding is also attached to this work.Comment: 12 pages, 2 figures, draf

Results on Vanishing Polynomials and Polynomial Root Counting

We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close connections to both ring theory and the technical applications of polynomials, along with numerous applications to other mathematical and engineering fields. We first determine the minimum degree of monic vanishing polynomials over a specific infinite family of rings of a specific form and consider a generalization of the notion of a monic vanishing polynomial over a subring. We then present a partial classification of the ideal of vanishing polynomials over general commutative rings with identity of prime and prime square orders. Finally, we prove some results on rings that have a finite number of roots and propose a technique that can be utilized to restrict the number of roots polynomials can have over certain finite commutative rings.Comment: 14 page

Design and evaluation of neural classifiers

In this paper we propose a method for design of feed-forward neural classifiers based on regularization and adaptive architectures. Using a penalized maximum likelihood scheme we derive a modified form of the entropic error measure and an algebraic estimate of the test error. In conjunction with Optimal Brain Damage pruning the test error estimate is used to optimize the network architecture. The scheme is evaluated on an artificial and a real world problem. INTRODUCTION Pattern recognition is an important aspect of most scientific fields and indeed the objective of most neural network applications. Some of the by now classic applications of neural networks like Sejnowski and Rosenbergs "NetTalk" concern classification of patterns into a finite number of categories. In modern approaches to pattern recognition the objective is to produce class probabilities for a given pattern. Using Bayes decision theory, the "hard" classifier selects the class with the highest class probability, henc..

Real Lie Algebras of Differential Operators and Quasi-Exactly Solvable Potentials

We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on $R^2$.Comment: 33 pages, plain TeX. To apper in Phil. Trans. London Math. Soc. Please typeset only the file rf.te