An interacting multi-agent system in a network can behave like a nature-inspired smart system (SS) exhibiting the four salient properties of an artificial life system (ALS): (i) Collective, coordinated and efficient (ii) Self-organization and emergence (iii) Power law scaling or scale invariance under emergence (iv) Adaptive, fault tolerant and resilient against damage. We explain how these basic properties can arise among agents through random enabling, inhibiting, preferential attachment and growth of a multiagent system. However,the quantitative understanding of a Smart system with an arbitrary interactive topology is extremely difficult. Hence we cannot design a general purpose programmable Smart system. However, for specific applications and a predefined static interactive topology among the agents, the quantitative parameters can be obtained through simulation to build a specific SS.

This paper uses successive squaring of a composite matrix to approximate the generalised inverse of an m by n matrix A. For m \approx n, the g-inverse of A can be computed in parallel time ranging from O(log n) to O(log^{2}n). The simple structure of the successive matrix squaring algorithm leads to a straightforward parallel implementation. Test results are provided.

Abstract This paper is concerned with the combinator representation of numeral systems with logarithmic space complexity of symbols. The principle used is based on the lexicographic ordering of words over a finite alphabet.

Three different algorithms are described for the conversion of Hensel codes to Farey rationals. The first algorithm is based on the trial and error factorization of the weight of a Hensel code, inversion and range test. The second algorithm is deterministic and uses a pair of different p-adic systems for simultaneous computation; from the resulting weights of the two different Hensel codes of the same rational, two equivalence classes of rationals are generated using the respective primitive roots. The intersection of these two equivalence classes uniquely identifies the rational. Both the above algorithms are exponential (in time and/or space).

A fast iterative scheme based on the Newton method is described for finding the reciprocal of a finite segment p-adic numbers (Hensel code). The rate of generation of the reciprocal digits per step can be made quadratic or higher order by a proper choice of the starting value and the iterating function. The extension of this method to find the inverse transform of the Hensel code of a rational polynomial over a finite field is also indicated.

A unique code (called Hensel's code) is derived for a rational number by truncating its infinite p-adic expansion. The four basic arithmetic algorithms for these codes are described and their application to rational matrix computations is demonstrated by solving a system of linear equations exactly, using the Gaussian elimination procedure.

A unique code (called Hensel's code) is derived for a rational number, by truncating its infinite padic expansion. The four basic arithmetic algorithms for these codes are described and their application to rational matrix computations is demonstrated by solving a system of linear equations exactly, using the Gaussian elimination procedure. A comparative study of the computational complexity involved in this arithmetic and the multiple prime module arithmetic is made with reference to matrix computations. On this basis, a multiple padic scheme is suggested for the design of a highly parallel matrix processor.

A rank-augmnented LU-algorithm is suggested for computing a generalized inverse of a matrix. Initially suitable diagonal corrections are introduced in (the symmetrized form of) the given matrix to facilitate decomposition; a backward-correction scheme then yields a desired generalized inverse.

Algorithms are described for the basic arithmetic operations and square rooting in a negative base. A new operation called polarization that reverses the sign of a number facilitates subtraction, using addition. Some special features of the negative-base arithmetic are also mentioned.

Described here is a deterministic division algorithm in a negative-base number system; here, the divisor is mapped into a suitable range by premultiplication, so that the choice of the quotient digit is deterministic.

The logical design of a polarizer for negative binary numbers is described and compared with the two's complementer used for positive binary numbers.

Properties and construction of a class of test matrices, called " simply invertible matrices," are given.

It is shown that the Wilkes-Harvard and Newton-Raphson iterative division schemes with an order of convergence more than two or three are uneconomical for realization in computers.

This paper describes two sets of algorithms in positive radix arithmetic for conversions between positive and negative integral radix representation of numbers. Each set consists of algorithms for conversions in either direction; these algorithms are mutually complementary in the sense they involve inverse operations depending upon the direction of conversion. The first set of algorithms for conversion of numbers from positive to negative radix (negative to positive radix) proceeds serially from the least significant end of the number and involves complementation and addition (subtraction) of unity on single-digit numbers. The second set of algorithms for conversion of numbers from positive to negative radix (negative to positive radix) proceeds in parallel starting from the full number (the most significant end of the number) and involves complementation and right (left) shift operations. The applications of these algorithms to integers, mixed integer-fractions, floating-point numbers, and for real-time conversions are given.

Modified Meggitt methods (pseudodivision methods) are suggested for evaluating logarithm, arctan, and square root. The modifications described here consist in restricting the magnitude of the pseudopartial remainsler such that the pseudoquotient assumes a form close to the minimal representation in the radix of choice. These methods will become useful for large-scale integrated system design.

This note points out the close relationship between some of the recently described division techniques, in which the divisor is transformed to a range close to unity. A brief theoretical analysis is presented which examines the choice of quotient digit when this type of division technique is used for conventional and signed-digit number systems.

This paper describes division schemes which are derived from the classical functional iterative schemes. These schemes are compared with t he schemes currently used in the high- speed digital computers.