308 PROBLEMS AND SOLUTIONS

(3) + 9,)n--0 - {K(v/’)- E(v/-) }2 Vx e [0, 1[,

(4) E q,(x)/(n + 1) K2(x/), Vx [0, 1[,n--0

K(k) and E(k) are the complete elliptic integrals of the first and second kind, respec-tively.

These formulas have resulted as a by-product from the theoretical treatment of aproblem in microwave electronics, i.e., the calculation of the frequency shift resultingfrom the perturbation of an electromagnetic TMwave-mode by a small metallic cylinderin a resonant cavity.

A Set of Maxima Problems

Problem 92-8, by K.S. MURRAY (Brooklyn, N.Y.).Determine the maximum values of (a) x2, (b) y2, (c) x2 + y2, (d) x2 + z2, (e) x2 +

y2 + x2, and (f) x + y2 + z2 + w2 for all real x, y, z, w, satisfying

(1) X2 ._]_ y2 d’- z2 ’ w2 xy yz k2.

An Extremal Problem

Problem 92-9, by PAUL ERDOS (Hungarian Academy of Sciences).Let

xi xj Xi+l Xi

where the maximum is taken over all real sequences x < x2 < < Xn. It is easy tosee that

f(n) In n + O (1)

(a) Find

lim sup{f(n) -In n}.no

(b) Describe the extremal sequences.This problem admits an electrostatic interpretation. Suppose n unit charges are

placed on a line. Find the positions of these charges which maximizes the ratio of thetotal potential energy to that part which comes from nearest neighbor interaction alone.

Asymptotic Solution of the Telegraph Equation

Problem 92-10, by MARK A. PINSKY (Northwestern University).Let u(x; t) be a solution of the telegraph equation utt if" 2Ztt ltxx defined for

t > 0,-oc < x < oc with smooth and sufficiently rapidly decreasing initial data:u(x; 0+) fl(x), ut(x; 0+) re(x). Show that there is a solution of the heat equation

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PROBLEMS AND SOLUTIONS 309

2vt vxx defined for t > 0,- < x < such that u(x; t) v(x; t) O(t-1), t T o.Specify v(x; t) in terms of its initial data v(x; 0+) and find minimal smoothness and decayconditions on f and f2.

SOLUTIONS

Optimum Multiple Access Coding

Problem 91-2", by D.B. JEVTIt (University of Alaska, Fairbanks).Let Pi be a set of distinct nonnegative integers which are no greater than 2’ 1, n >_

2. The collection {Pl, P2,’-’, PT} such that Pi fq Pj {0}, i # j, is called a codebook.Let P 791 792 x PT and let F" PA/" {0, 1, 2,...} be defined as

F(xx, x2,’", XT) Xl At- X2 nt- At- XT, Xi )i"

Furthermore, let 171 denote the number of elements in 79i, and let

T

R(T, n) n- Elg2 IPil.i--1

Given n > 1 and T > 2, find the collection (P, 792, PT) such that F is aninjection and R(T, n) is maximum.

Comment. The problem described above is, in fact, a design ofoptimum uniquely de-codable codebooks for synchronized noiseless multiple access adder channels [1]. Goodcodebooks all have R(T, n) > 1. The problem is intimately related to the subset-sumproblem in additive number theory [2]. For T 2 the solution isP (0, 1, 2, 3,. , 2’-2},P2 {0,2’- 1}, andR(2,n) n-1og2(2n+-2). Clearly, 1 < R(2, n) <1 + n-. The highest value of R(T, n) obtained so far is R(4, 3) for the codebook{{o, {o, {o, {o, 7)).

REFERENCES

[1] D.B. JEVTI(,A T-player multiple access codinggame, IEEE Trans. Inform. Theory, May 1992.[2] P. ERDOS, Problemsandresults in additive numbertheory, Colloque sur la Theorie des Nombres, Bruxelles,

1955, Liege and Paris, 1956, pp. 127-137, esp. p. 137.

Partial solution by PAUL ERDOS (Hungarian Academy of Sciences) and the proposer.If F is an injection, then the 2T sums E/T=1 ixi, where e E {0, 1} and xi E Pi for1, 2,..., T, are all distinct. Due to x < 2n, none of the 2T distinct sums exceeds

T2’, and thus

(1) 2T < T. 2n.Furthermore, T < 2’ and (1) implyT < 2n, which, in turn, implies 2T < n2’+l, and thusT < 1 +n+log2 n. Once more we replace T on the left-hand side of (1) by 1 +n+log2 nto obtain

(2) T < n + log2(1 + n + log2 n).If F is an injection, then IT’[ [F(P)I and, since IF(P)[ < T2’, it follows from (2) andthe definition of R(T, n) that R(T, n) < G(n), where

(3) G(n) 1 + n- log2(n + log2(1 + n + log2 n)).

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