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Quality of Minimal Sets of Prime Implicants of Boolean Functions

Prasad V. C.

International Journal of Electronics and Telecommunications

2017

Vol. 63, No. 2

165--169

Two new problems are posed and solved concerning minimal sets of prime implicants of Boolean functions. It is well known that the prime implicant set of a Boolean function should be minimal and have as few literals as possible. But it is not well known that min term repetitions should also be as few as possible to reduce power consumption. Determination of minimal sets of prime implicants is a well known problem. But nothing is known on the least number of (i) prime implicants (ii) literals and (iii) min term repetitions , any minimal set of prime implicants will have. These measures are useful to assess the quality of a minimal set. They are then extended to determine least number of prime implicants / implicates required to design a static hazard free circuit. The new technique tends to give smallest set of prime implicants for various objectives.

The generalized residue classes and integral monoids with minimal sets

Rycerz A.

Opuscula Mathematica

2000

Vol. 20

65-69

In this note we consider the integral monoids mon(A) = {Ax : x ∈ Zn+}, where A ∈ Zmxn, with the swelling points. Such monoids appear in the natural way in the generalized Frobenius problem in Zm. An integral element g ∈ mon(A) is called the swelling point if (g + cone(A)) ∩ Zm ⊆ mon(A), where cone(A) = {Ax : x ∈ Rn+}. The aim of this note is to characterize the structure of the set of swelling points in terms of the generalized residue classes.

W pracy rozważane są całkowite monoidy mon(A) = {Ax : x ∈ Zn+}, gdzie A ∈ Zmxn, z punktami źródłowymi, które w naturalny spo sób pojawiają się przy próbach uogólniania jednowymiarowego problemu Frobeniusa na przypadek wielowymiarowy. Punkt g ∈ mon(A) jest punktem źródłowym, jeśli (g + cone(A)) ∩ Zm ⊆ mon(A), gdzie cone(A) = {Ax : x ∈ Rn+}. Głównym rezultatem pracy jest charakteryzacja struktury zbioru punktów źródłowych za pomocą uogólnionych klas reszt.

On the location of critical points of some complex polynomials

Dronka J., Sokół J.

Demonstratio Mathematica

1999

Vol. 32, nr 2

297-302

Let P(a,n) be the set of all complex polynomials of degree n which have all their roots in the closed unit disk and one fixed root at a, 0 < a < 1. In this paper we show thiat for n > 3 all critical points of the polynomial f(z)= (zn-l+ l)(z - a) lie outside the set K(a,n) consisting of all b such that for some c the polynomial p(z)= (z - b) n - c belongs to P(a,n). Hence we infer that minimal sets satisfying the Sendov property (i.e. containing at least one critical point of each pzawiera P(a, n) ) exist but they are not unique.