ON CODING THEOREM CONNECTED WITH ’ USEFUL ’ ENTROPY OF ORDER -

Guiasu and Picard (I) introduced the mean length for 'useful' codes. They called this length as the 'useful' mean length. Longo (2) has proved a noiseless coding theorem for this 'useful' mean length. In this paper we will give two generalizations of 'useful' mean length. After then the noiseless coding theorems are proved using these two generalizations.


INTRODUCTION.
Bells and Guiasu [3] consider the following model for a finite random experiment (or information source A: where X is the alphabet, P the probability distribution and U (u l,u2,...,un) u. > 0 i-s the utility distribution.They introduced the measure n H(P,U) r. uiPi log Pi i=l (1.2) about the scheme (I.I).
They called it 'useful' information provided by a source letter.
Guiasu and Picard [I] have considered the problem of encoding the letters output by the source (I.I) by means of a single letter prefix code, whose codewords CI,C2,...,Cn have lengths I"'" 'n satisfying the Kraft's [4]  In this communication two generalizations of (1.4) have been studied and then the bounds for these generalizations are obtained in terms of 'useful' entropy of type , which is given by (m,u) which is the generalization of Kraft's inequality (1.4).
2. TWO GENERALIZATIONS OF 'USEFUL' MEAN LENGTH AND THE CODING THEOREMS.
Let us introduce the measure of length: It is easy to see that (1.6) lim L (u) L(U).
In the followgng theorem we obtain lower bound for (2.1) in terms of H(P,U).
THEOREM I.If I'2'''" 'n denote the lengths of a code satisfying (1. Let 0 < B < I. Raising both sides of (2.6) to the power (B-I), we get n .
The proof for < B < follows on the same lines.It is clear tht equality in for each i and D 2, that is, the codes are binary, then (2.2) reduces to the result proved by Van der Lubbe [5].
In the following theorem, we will give an upper bound for L131(U) in terms of H13(P,U).
PARTICULAR CASE.Let u.
for each i and D 2, that is, the codes are 1 binary codes, then (2.11) reduces to the result proved by Van der Lubbe [5].REMARK.When 13 I, (2.2) and (2.1 I) give H(P,.U) < L(U) < H(PU) + I, where L(U) is the 'useful' mean length function (1.4), Longo [2] gave the lower and upper bounds on L(U) as follows: H(PU-) u Io u + u .logu < L(U) < H(P'U) -'u'l"'[ u + u log u + I, where the bar means the value with respect to probability distribution P (PI"'''Pn)" Since x log x is a convex U function, the inequality u log u > u log u holds and therefore H(P,U) does not seem to be as basic in (2.16) as in (2.15).Now we will define another measure of length related to (P,U).We define the 13(U) by measure of length L 2 (2.17) It is easy to see that lira L2(U) L(U).
It is clear that the equality in (2.18) is true if and only if (2.22)   which implies that i lgD (I/pi)" Thus it is always possible to have a code word satisfying the requirement :og D!__< <:OD:  Pi Pi PARTICULAR CASE.Let u. for each i and D 2, then (2.18) reduces to the result proved by Math and Nittal [6].
Next we obtain a result giving the upper bound to the 'useful' mean length L82(U).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: the size of the code alphabet.They defined the following quantity n it 'useful' mean length of the code They also derived a lower bound for it.
E uzP/iE B uiP i) < D z < Dp B( uiPi/ I i=l i=l i=l uiPi (2.10) PARTICULAR CASE.Let u.
which is equivalent to

First
Round of Reviews March 1, 2009