• E V Krishnamurthy

Articles written in Proceedings – Section A

• “Līlāvatī”—A new analogue computer for solving linear simultaneous equations and related problems - Part I. General principles and design of model I

• Līlāvatī—A new analogue computer for solving linear simultaneous equations and related problems - Part II. Design of model II and its application to the solution of secular equations

• Algorithmic line-notation for the representation of knots

An algorithmic line-notation is suggested for the unambiguous description of the planar projection of a knot made with a single string. This may have applications in automatic knot-craft.

• Nega-base periodic decimals

Some properties of the periods of prime-reciprocals in a general negative base representation are described.

• Formal description, compression and transformation of digital pictures - I. Picture encoding and decoding

This paper describes the application of vector spaces over Galois fields, for obtaining a formal description of a picture in the form of a very compact, non-redundant, unique syntactic code. Two different methods of encoding are described. Both these methods consist in identifying the given picture as a matrix (called picture matrix) over a finite field. In the first method, the eigenvalues and eigenvectors of this matrix are obtained. The eigenvector expansion theorem is then used to reconstruct the original matrix. If several of the eigenvalues happen to be zero this scheme results in a considerable compression.

In the second method, the picture matrix is reduced to a primitive diagonal form (Hermite canonical form) by elementary row and column transformations. These sequences of elementary transformations constitute a unique and unambiguous syntactic code-called Hermite code—for reconstructing the picture from the primitive diagonal matrix. A good compression of the picture results, if the rank of the matrix is considerably lower than its order. An important aspect of this code is that it preserves the neighbourhood relations in the picture and the primitive remains invariant under translation, rotation, reflection, enlargement and replication. It is also possible to derive the codes for these transformed pictures from the Hermite code of the original picture by simple algebraic manipulation.

This code will find extensive applications in picture compression, storage, retrieval, transmission and in designing pattern recognition and artificial intelligence systems.

• Formal description, compression and transformation of digital pictures - II. Picture algebra and geometry

In an earlier paper (Part I) we described the construction of Hermite code for multiple grey-level pictures using the concepts of vector spaces over Galois Fields. In this paper a new algebra is worked out for Hermite codes to devise algorithms for various transformations such as translation, reflection, rotation, expansion and replication of the original picture.

Also other operations such as concatenation, complementation, superposition, Jordan-sum and selective segmentation are considered.

It is shown that the Hermite code of a picture is very powerful and serves as a mathematical signature of the picture. The Hermite code will have extensive applications in picture processing, pattern recognition and artificial intelligence.

• Finite segmentp-adic number systems with applications to exact computation

A fractional weighted number system, based on Hensel’sp-adic number system, is proposed for constructing a unique code (called Hensel’s code) for rational numbers in a certain range. In this system, every rational number has an exact representation. The four basic arithmetic algorithms that use the code for the rational operands, proceed in one direction, giving rise to an exact result having the same code-wordlength as the two operands. In particular, the division algorithm is deterministic (free from trial and error). As a result, arithmetic can be carried out exactly and much faster, using the same hardware meant forp-ary systems.

This new number system combines the best features and advantages of both thep-ary and residue number systems. In view of its exactness in representation and arithmetic, this number system will be a very valuable tool for solving numerical problems involving rational numbers, exactly.

• p-Adic arithmetic procedures for exact matrix computations

Computer procedures are described for error-free matrix computations, using thep-adic arithmetic. As an example, the exact solution of a highly ill-conditioned linear system of equations is obtained, by using the Gaussian elimination method.

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