SIAM REVIEWVol. 22, No. 1, January 1980

1980 Society for Industrial and Applied Mathematics

0036-1445/80/2201-0006501.00/0

PROBLEMS AND SOLUTIONS

EDITED BY MURRAY S. KLAMKIN

COLLABORATING EDITORS: HENRY E. FETTIS (1885 California, Mountain View, CA 94041),YUDELL L. LUKE (University of Missouri, Kansas City, MO),CECIL C. ROUSSEAU (Memphis State University, TN),OTTO G. RUEHR (Michigan Technological University, Houghton, MI).

All problems and solutions should be sent, typewritten in duplicate, to Murray S. Klamkin, Department ofMathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2GI. An asterisk placed beside aproblem number indicates that the problem was submitted without solution. Proposers and solvers whose solutionsare published will receive 10 reprints of the corresponding problem section. Other solvers will receive just onereprint provided a self-addressed stamped envelope is enclosed. Proposers and solvers desiring acknowledgmentof their contributions should include a self-addressed stamped postcard (no stamp necessary outside Canada).Solutions should be received by April 15, I980.

PROBLEMS

A Determinant and an Identity

Problem 80-1 by A. V. BOYD (University of the Witwatersrand, Johannesburg, SouthAfrica).(a) Prove that

where r, s 1, 2,. , n,

det IAI (-1)"+’(22" -2)Bz./(2n)!

{ lf(2r-2s +3)!,A 0

and Bn is the Bernoulli number defined by

e -1 ,,=o n!

s-<r+l,s>r+l,

(b) Prove that if n is odd,,,-)/2 1_22’n-1 ( )t"= Z

n hZm_lBzm{(t+h)n+_zm__(t__h)n+l_Zm}..,=o n-2m+l 2m

A Closed Form Integration

Problem 80-2" by A. VARMA (University of Notre Dame).The separable differential equation

du { k + uol /2U uo + k log k+"uJ u(0)=u0>0dx

arises in the solution of a diffusion-reaction problem. Can one obtain a closed formsolution?

A Matrix Stability Problem

Problem 80-3* by K. SOURISSEAU (University of Minnesota)and M. F. DOHERTY(University of Massachusetts).

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98 PROBLEMS AND SOLUTIONS

Let"A1C2

B1B2

be a real square matrix of order 2N partitioned into N2 2 2 blocks. The 2 2 blockshave the following structure"

,-(Pll + L,,) (R, + Vii)all

( -,,:/ -(R. + Vll

-(Rll + V,, )llH.

-(Pll + L,, (Rll + Vll)5.H. H

Lll+l Lll+"1..I Wn-lOln-1 Wn-lfn-o -h--i--

0Cn-"

Vn_l,]/n_l Wn_atn_H.A, H H.

with a,Sll yn/3,. All the entries are nonzero.positive constants with the exception of Plland Rll, n 1, 2,..-, N which are nonnegative. The 2 2 blocks not shown containonly zeros and the eigenvalues of the nonzero blocks are"

(Pll +L,)All:

H.(P,,+Lll)

A2 (1 +all +Sll) <0;HLn+lBll" A1 A2 >0;H.

C,,’ Ai

Wn-l (oln_l .+. ,_l) > O=HWhat additional conditions must the elements of the matrix J satisfy in order that

the eigenvalues of J have negative real parts?The problem arose in considering the stability of a system of first order nonlinear

ordinary differential equations describing the dynamics of a fractionation process.

Eigenvalues o| a Tri-diagonal Matrix

Problem 80-4 by D. K. Ross (La Trobe University, Victoria, Australia).Prove that the real tri-diagonal matrix A JlAiill of order n has only real simple

eigenvalues if aiiaji > 0 for j + 1.

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