A GENERALIZATION OF A THEOREM BY CHEO AND YIEN CONCERNING DIGITAL SUMS

For a non-negative integer n, let s(n) denote the digital sum of n. Cheo and Yien proved that for a positive integer x, the sum of the terms of the sequence{s(n):n=0,1,2,…,(x−1)}is (4.5)xlogx


INTRODUCTION.
In Cheo and Yien [1] it was proven that for a positive integer x xs(n) (4.5)xlogx + 0(x) (1.1) where s(n) denotes the digital sum of n.Here, we will show that, in fact, for any positive integer k, xs(kn) (4.5)xlogx + 0(x) (1.2) where the constant implicit in the big-oh notation is dependent on k.
The following notation will be used to facilitate the proof of (1.2).For integers x and y, x rood y (I .3)will be the remainder when x is divided by y and, as usual, square brackets will denote the integral part operator.In addition, for non-negative integers m, i, and j we let Thus, the j right-most digits of m are given by (1.4) and the number determined by dropping the i right-most digits of m is given by (1.5).Therefore, the number determined from the jth right-most digit of m to the (i + l)st right-most digit of m is given by (1.6).(2.4) Also, since each s([kn] L) is bounded by a constant (dependent on k), we have that the second term of (2.1) is 0(x) Continuing in this manner and combining terms, we have (2.7) x-L x.
s([knl L) . i s ([knlL+l-i)  To determine the value of the first term of (2.8), we need the following lemma.
Its proof is straight forward and will not be given.(2.12) By this lemma and the fact that x. L+2-i.

CONCLUSION.
For any positive integer k, there exists non-negative integers a, b, and r such that k 2a5br with (r,10) i.Note that if k r, then we have (1.2).However, by use of the following generalization to Lemma 2, and some technical modifications, it can be shown that the restriction that k and I0 be relatively prime can be removed in the derivation of (2.1).That is,

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:

LEMMA 2 .
Let d and i be non-negative integers.Then for (k,10) any positive integer k.LEMMA 3. Let k 2a5br with (r,lO) and i max {a,b}.Then for any non-