Israel Journal of Mathematics, Volume 3, Issue 2 (1965)

MORE ON THE LINEAR SEARCH PROBLEM BY ANATOLE BECK ABSTRACT The linear search problem concerns a search made in the real line for a point selected according to a given probability distribution. The search begins at zero and is made by continuous motion with constant speed along the line, first in one direction and then the other. The problem is to search in such a manner that the expected time required for finding the point according to the chosen plan of search is a minimum. This plan of search is usually conceived of as having a first step, a second, etc., and in that case, this author has previously shown a necessary and sufficient condition on the probability distribution for the existence of a search plan which minimizes the expected searching time. In this paper, we define a notion of search in which there is no first step, but the steps are instead numbered from negative to positive infinity. These new rules change the problem, and under them, there is always a minimizing search procedure. In those cases which satisfy the earlier criterion, the solutions obtained are essentially the same as those obtained previously. Introduction. I n a recent paper by this a u t h o r , [1], the linear search problem is discussed, and a necessary and sufficient c o n d i t i o n is derived for the existence o f a search procedure having minimal expected path. It is shown t h a t when the left and right upper derivatives o f the normalized distribution function are b o t h infinite at 0, then no m a t t e r h o w small the first steps m i g h t be, it is nonetheless advantageous t o add a yet smaller step before them, thus decreasing the expected p a t h length. I n this paper, we consider a modification o f the definition o f search procedure in which there is no first step. The procedures are conceived o f as beginning with an infinitesimal oscillation, as defined below. U n d e r this definition, which is a g e n e r a l i z a t i o n o f the c o n c e p t o f search procedure as defined in [1], a minimizing procedure exists for every distribution with finite first m o m e n t . Furthermore, if minimizing exists in the sense o f [1], then the minimizing procedures derived here are the same ones, in a certain natural sense. Definitions and fundamental notions. We begin with a probability distribution F on the real line which has finite first m o m e n t M l = MI(F) = ae t) F is assumed to be normalized to be continuous f r o m the left in the left half-line, continuous f r o m the right in the right half-line, and continuous at 0, for reasons discussed in I1]. I n that paper, we define a search procedure as a sequence oo with x = {x i}i=l

Y_+:ltl

Received June 4, 1965. * The research work for this paper was supported by the National Science Foundation under grant No. GP 2559 to the University of Wisconsin. 61

62

ANATOLE BECK

[June

•..=<x4<x2x+, while 0 < F(t) < 1 for x - < t < x + . If, for instance, - oo < x - < 0, x + = + oo then we allow the possibility o f a search procedure with x s_ 1 = x - , x j = + oo, 3j, a n d no entries x, for i > j. Similarly if x - = - oo, 0 < x + < + oo. In any case we do n o t allow any search procedures with entries x~ for which x - < x, < x + does n o t hold. F o r each x e ~0, we can define a corresponding element 2 e ~ , by the following: I f . . . < x 4 < x2 < 0 < xl < x 3 " " , then [ xl

if

i> 0

0

if

i_ %, t > ~7_l, so that 0

X,(~,t) >

2 ~

-1

I*,l >

i=-oo

I*,1 > 2kg.

2 Z i=-2k

Thus we have

2ke .P < = XI(; ) = Xl(x ) < 2m I .

It follows that 2k < (2ml/gP), and therefore 2k - 1 is also.

Q.E.D.

6. LEMMA. I f X - < a < _ O < - b < x +, then 3 n = n ( a , b , F ) > O such that for all x e~.l with Xl(x ) < 2m I ,

2j~[a,b],Vj > n. Proof. Let P = min (Pr(t < a), Pr(t > b)). Choose j > 0 and assume that x2j e [a, b]. Then

~o < I~, I+ 1~1 < I ~ l + I ~ l
0 with SRe < iS. Since x[ n)-, Yi,V i, we know that for all n large enough, say all n > n l, we have ]w~" ) - x,(")] < e, Vi. Then Xl(Wt~),t) < Xl(X ("~t) + Ske, V -- oo < t < + oo, V n > n I .

It follows that for n > nx, X l ( w (~)) < X l ( x (")) + ske. Note that w~")-, Yi as n ~ oo, Y i, and define v (~) = {v~)}e ~ t by f

v~" ) = 4 y t

t w~~

ifi - 2k.

Then X t(v (n)) < X l(W t")) + 26, V n, since -2k

X i=--o0

-2k

2iy,[-

X i=--oO

-2k

2[w}")l
k,

) L y j - 2 , Vi Y k + l >= Y k 4 1 - - Y k > - -

K , so that

F(yk_2) -- t ( y k _ l ) < F ( K ) - F( - K ) < 1,

and 1 - F(yk-2) + F(Yk-1) > ½.

(c) F(y~) - F(yk_ 2) ----F(y~) -- F(0) __>/)Yk, with equality holding only if Yk = O. Thus, 1 X ( y ) - X ( x ) > 2 [Yk " - ~ -- K " Dyk] ,

with equality holding only if Yk =- Yk- 1 ~- O. However, X ( y ) - X ( x ) < 0 by assumption on y, while K D < ½ by definition on K, so that 1 ~ Yk -- K D y k > O.

It follows that Yk = Yk-1 = 0.

Q.E.D.

One important feature of an absolute minimum, aside from aesthetic considerations, lies in the fact that a recursion formula for the entries can sometimes be obtained by partial differentiation. Let x C°) e ~1, with X l(x to))= m l. Assume that Fis differentiable at each x~W~- oo < i < + co.Then X l ( x ) i s a differentiable function of each x~ at x = x w), and tOXl(x(°)) = O, V - co < i < + co. t~xi

70

ANATOLE BECK

[June

Since (O(Xl(x))/Oxi) = 2[(xi+ 1 - x~)F'(xi) - ( - 1) * - F(x~) + F(xi_ l)], we have F(xi_

Xi+ l ~-- Xt "4-

1) -

F(x~) - ( -

1) ~

F'(x,)

and F ( x i - l ) = F(xi) + ( - 1) i + (xi - xi+ l)F'(xi). Thus, under proper differentiability conditions, we can derive all the xi of a minimal solution if we have two consecutive ones. Clearly, if F is a strictly increasing function, the two equations above will give us all the xi. To extend the same observation to distributions which are not strictly increasing, we note that o f all the values of x for which F ( x ) takes a given value, only that value o f x having the smallest absolute value can appear as an entry in a minimal search procedure. The formulae hold as well for x ~X1, as for x ~-3~o. Although the problem as originally posed has as yet no solution in a useful sense, even for approximations, the analysis here is too delicate to carry over approximations, and the recurrence relations, which depend strongly on F ' (a very sensitive quantity) do not withstand the approximation process. BIBLIOGRAPHY 1. Anatole Beck, On the linear search problem, Israel Journal of Math. 2 (1964), 221-228.* 2. Wallace Franck, On the optimal search problem, Technical Report No. 44, University of New Mexico, Oct. 1963. THE HEBREWUNIVERSITYOF JERUSALEM, AND THE UNIVERSITYOF WISCONSIN

* Please note the following erratum in [1]: Equation 2 ° on page 224 should read: 20 F ( t ) - F ( O ) _E~b~= ~b~, i = 1,2, so ~ = E(~bI V ~2) - (~bl ~/tk~) >=0; and since E~k = 0, ~k vanishes on EE. So by Stone's proof of the Stone-Weierstrass theorem, A R (and thus A) is an algebra. Consequently A = C(Y) for some factor space Y of Er. Now trivially k = kE e K has support ER c ~ , since if f ~ C(X) vanishes on E E then k f = k E f = kO = O. Since Cp is invariant for each k in K (as for all elemnts of S), k actually yields a well defined operator ( f l E E ~ kflY, E) on A = Cp[ER. So K acts as a group of operators on A, and evidently k ~ k f is strongly continuous as a map into A since it coincides with k ~ for any extension g e C(X) of f , and k ~ kg is strongly continuous. Viewed as operators on C(Y) t h e n , K is a group of nonnegative operators leaving 1 fixed whose identity is the identity operator. So each adjoint k* maps P(Y) onto itself, and thus maps extreme points onto exteme points: with/~y the unit mass at y, k*/~y=/~kty) for some unique k(y) in Y. But

kglr,~

(1)

(k,y) -~ k(y)

is continuous since this amounts to continuity of

( k , y ) ~ f(k(y)) = k * # y ( f ) = kf(y), all f e C(Y), and that follows from the strong continuity of k ~ kf. Hence each element k of K induces a self-homeomorphism k( • ) of Y, which in turn induces the action of k on C(Y): k f = f o k( • ). Thus K, with the action (1), gives rise to a transformation group on Y which yields the action of K on A = Ce I Ee, and in particular that of k o = TE. Having identified C(Y) and EC(X)I EE we can of course compose E f I Ee with an element k ( - ) of our transformation group on Y, and thus write, without ambiguity,(1) k f l E ~ = k(f[Ee) = ( f [ ~ E ) o k(. ) if f = Ef. To obtain this action of T (hence that of TE = ko) on Cp -- EC(X) then we note that for any x in X and f = E f, Tf(x) = TEf(x) = kof(x ) = Ekof(x ) = ex(kof I F,n), so (2)

Tf(x) = ex([fl~E] o go(" )),

f E EC(X).

Noting that the powers of ko = TE are dense in K = ~E since those of T are dense in S we have proved half of the following THEOREM. A non-negative operator T with T1 = 1 has S = {Tn:n > 1} almost periodic if and only if (i) there is a projection E in the strong operator closure of S, (ii) there is a quotient space Y of~,Efor which EC(X)I ~'r. is precisely C(Y) (as naturally imbedded in C(EE)), (iii) a compact (monothetic) transformation group K acts on Y, with the action o f T on EC(X) that induced by a generator ko of K, as in (2). Q) The second k is our operator on A = Cp] --rE.

1965]

A REMARK ON ALMOST PERIODIC TRANSITION OPERATORS

73

" I f " is easily proved by showing conditional compactness of orbits for f in

( I - E)C(X), and then for f in EC(X), as follows. Suppose the net T n0 ~ E strongly. For f e (I - E)C(X), E f = 0, so given e > 0, [[ T"f[[ _ 1} is conditionally weakly compact for each f i n C(X)), (i) - (iii) still hold if "strong" is replaced by " w e a k " in i). Indeed E is then the identity of the least ideal K of the weak operator closure of S, which is still [2, 8.1] a compact group in the strong operator topology, so that the same proof applies. Needless to say, in this situation Co (the nullity of E) is not so simply described. (More generally the same proof yields (i)-(iii))(with obvious modifications) for any (weakly) almost periodic semigroup S of non-negative T with T 1 = 1 for which the conclusions of [2, 4.11] hold (in particular for S amenable [1]), with Co and Cp the subspaces defined in [2].) 2. Note that if X = ZE, the natural decomposition of Y into orbits lifts to a decompositon ,~ of X for which x e F e ~- implies the support of tx is contained in F. Indeed i f f ~ C(Y), 0 = ( H x , y)=(H1/2x, H1/2y) Oenoting the initial norm

by I1" IIandthenew norm on

x, y e ./g

by Ill Ill, weshallhave

IIIx[II= HH'/2 II l"

Now separating the right-hand side as the difference of two integrals then by virtue of (1), we get (4)

£ F2(a,-n,-n;1,1;x,y) n=O

= (m + 1 ) ( x - y)-I [ F 2 ( a - 1 , - m , - m a

1 ; 1 , 1 ; x , y ) - ~-],

where ~- shows the presence of a similar term with x and y interchanged. This is a summation formula. REFERENCES 1. Earl D. Rainville, Specialfunctions, (1960) New York. UNIVERSITY OF JODHPUR, JODHPUR (INDIA).

SECONDARY FLOW ABOUT A MAGNETIZED SPHERE R O T A T I N G I N VISCOUS C O N D U C T I N G F L U I D BY

SUNIL DATTA ABSTRACT

The problem of secondary motion induced by the steady rotation of a magnetized sphere in an infinite incompressible viscous conducting fluid is considered. It is found that the secondary flow adds nothing to the couple required tomaintain the motion and the effect of the magnetic field is to damp the secondary velocity field. Introduction. The steady rotation of a sphere, magnetized along the axis of rotation, in an infinite incompressible viscous conducting fluid was considered by the author [1] under the assumption that the fluid moves in concentric circles whose centres lie on the axis of rotation. Actually since the centrifugal force is greatest in the neighbourhood of the equator of the sphere, the fluid particles will recede from the sphere at the equator and approach it again at the poles. Thus combined with the motion about the axis of rotation, there will be a circulatory motion in planes containing the axis of rotation. This secondary flow for the case of an incompressible viscous fluid has been investigated by several authors [2-4]. In the present paper the analysis has been extended to the case of a magnetized, sphere. Basic Equations. With the usual notation the basic equations of magnetohydrodynamics in the non-dimensional form are (1 a , b , c , d )

c u r i E = 0 , J = Rmcurl/~, ] = ( E +

17x B),

R( IT"xT)/7 = - V P + ~ 7 2 p + M ( J × B ) ,

(2 a,b)

div/3=0, div 17=0,

where R( = (aZf~/v)) is the Reynolds number, R,,( = 4rctra2~21~) is the magnetic Reynolds number and M ( = (a/pv)a2B 2) is the square of Hartmann number. From (1) we get (3) E = - grad q~ and V 2 q5 = div( 17 x B). To solve the problem we make use of the following perturbation expansions

( ~'= F'o + RPx + MV~ + RM~"2 + ..., i

(4 a,b,c)

i P = Po + RP1 + MP'I + RMP2 + ..., |

L qS= e~o+ Rc~l+ ....

Received Feb. 7, 1965. 89

90

SUNIL DATTA

[June

The magnetic Reynolds number is assumed to be small so that the magnetic field, BI = - V(½(z/ra))], of the sphere, remains unaffected by the velocity field. Inserting this value of/~ and the expansions of 17and ~ in (3) we get [ ~72~0 = div(Vo x B), (5 a,b)

i ~(kt

div(17, x B).

Again using the perturbation expansions (4) in (2a) we get the following equation

f

0 = - VPo + V2Vo,

(~o.V) 17o= - vPl (6 a,b,c,d)

I

+

v~17~,

o= - v v i + v217; + (•o+ 17o× n) × ~, (17o.V)~, +(17i.v)~7~ = -

v p 2 + v2172+ + (E~ + 17~ x B) x/~.

The problem is to be solved subject to the following boundary conditions (i) No-slip condition at the surface of the sphere (ii) Continuity of normal component of current density vector and continuity of tangential component of electric intensity vector at the surface of the sphere. (iii) Vanishing of the quantities at infinity. Solution. Cylindrical polar coordinates (&,O,z) with velocity components (u, v, w) will be used in writing down the soutions. The equation (6a)with Vo = c5 at r( = x/~--2-~z 2) = 1 yields 1-5]

(7)

~o = ( o , ~ , o).

Again the solution of (6b) can be written down 13] in terms of a stream function (8)

~b~= r52z ( r - l ) 2 171 = ( 1 ~¢1 8 rs ' t5 Oz '0' When P1 is inserted in (5b) we get

I ~h~) t5 &5 "

~7~~bl = 0, which together with the boundary condition gives/~ = 0. The solution of equation (6c) has been obtained elsewhere I-1] and is

~7i = (0,,~g,, 0), where

l(z 2 1) 3,Z(z 2 l) ~ gl=~ ~+7-7)- +ig 7~+3--~ +g+v

(Sz 2 1) r7 r5 '

9I

FLOW ABOUT A MAGNETIZED SPHERE

19651 with 2= -~

1(l+a,) 3+2a,

E-

l+3a, 140(3+2ar)'F=

'

3-2a, 480(3+2a,)'

a, being the ratio of the conductivities of the sphere and the fluid medium. Using the above results, equation (6d) in cylindrical polar-coordinates wit axial symmetry can be written as 2> r--Y-- -

Op2 ~

"~ V 2 U 2

×Lr, g - -r4+

g

u2

th {3z 2

(..~2

i-6 \ r s

r3

r5

r6 + P-

2

~

1 )

+2 ~ rs

x

(9 a,b,c)

2 ,_z2(3r-T - r - g8- + ~ - 5)}]

r-5 . - r---X + ~

0---~ V 2 / ) 2

=

/)2

th 2

-- ~-~ dr- V 2 W 2 -~- 1~6

3 Z2(~ 2 2 rs

x

r3 ]

I~3

r---f - L~ rS .

r--~ -

i.--~ -t- - ~

,) ')1]

5) + 21(3z2 rs

8 + ~r6

r3

{1r--~ - r---g 2 + ~1_z2(3r---g- r--g8 + ~-

The boundary conditions are that, on r = 1, tl 2 ~--- 1.72 = W2 ~ O,

and each of the functions tends to zero as r tends to infinity. The equation for v2 is satisfied by taking v z = 0 throughout the fluid. To solve the equations for u 2 and w2, let

u2

1 c~2 60 Oz ' w2--

1 ~¢2 c3 c ~

Substituting these values the equation for ~b2 is obtained from (9 a, c) to be (10)

A4~,2 = &2[z/l(r ) + zaf2(r)],

92

SUNIL DATTA

[June

where 02

A2 ~

06~2

1 0 02 & O& + Oz2'

1 / " 1600 F A

=

6 6 2 - 15

28

7 )

+

fz =

12E rs

'

36F 62 + 3 rio + 16rt~

17 9 21r12 + 16r~----T,

and ~2 = (O~b2/Or) = 0, on r = 1 and ¢2 -~ 0 as r ~ oe. Writing ~2 = (°2[zFa(r)+z3F2(r)] we get ordinary linear differential equations for the functions Fl(r) and F2(r). These are 1(1600F 662-15 28 7 ) D(0-2)(0+7)(0+9)Fz = ]-~ 7 + ~ r +~ +~ ' and

D(D - 2)(D + 3)(D + 5)F 1 -

12E r4

36F 62 + 3 r6 + l l r ~

17 9 21r8 + 16r---g

- (12r4F~ + 96raFt), where D = (1/r)(d/dr). The equations have been solved to give the functions Fx and F2 which satisfy the boundary conditions F 1 = F 2 = F] = F~ = 0, on r = 1, and tend to zero as r tends to infinity. The functions F 1 and Fz for tr, = 1 are given below Fx(r ) = [ 976 2857 7435 243 3778 In r3 r ~ + r5 r6 r7 r (11)

1378 r7

2133 152] ~r- + r 9 J × 10 -6,

and (12)

F2(r)=

-

6062 521 298 8900, 4861 r---~-+-~+-~-+--~-mr+-~-+-~

382 ]

× 1 0 -6

The stream function for the secondary flow is given by 8 - - gr~

+ M zFl(r) + z3F2(r)}

It is easy to see that the secondary velocity field, as obtained above, contributes nothing to the couple required to maintain the motion.

1965]

FLOW ABOUT A MAGNETIZED SPHERE

i

/~

V,

\~

'~\ \

\,

\',

",~, \'~\

4,,

"~. ".¢.

-,,, . \\

93

--..~-._

"~...

~5o

~ - - - - ~

\\

"-~

"%. "-%.

....

I

I

r:1

I

-

i r--~

Fig. Stream line pattern for the secondary flow. -- M = 0, - - - M ---- 1 Stream line pattern for the secondary flow, in the plane containing the axis, is given in the figure for M = 0 and M = 1. The graph suggests that the secondary velocity field decreases on account o f the magnetic field. Acknowledgement. I am grateful to Prof. R a m Ballabh for his help and guidance in the preparation of this paper. REFERENCES 1. SUNIL DATTA, J'. Phys. Soc. Japan, 19 (1964), 392.

2. 3. 4. 5.

W.G. BICKLEY,Phil. Mag. 25 (1938), 746. W.D. COLLINS, Mathematika 2 (1955), 42. W.L. HABERMAN,Phys. of Fluids, 5 (1962), 625. H. LAMB, Hydrodynamics, p. 588-9.

DEPARTMENT OF MATHEMATICSAND ASTRONOMY, LUCKNOW UNIVERSITY, LUCKNOW, INDIA

A MODIFIED NEWTON-RAPHSON METHOD FOR THE SOLUTION OF SYSTEMS OF EQUATIONS BY

ADI BEN-ISRAEL* ABSTRACT

An implicit function theorem and a resulting modified Newton-Raphson method for roots of functions between finite dimensional spaces, without assuming non-singularity of the Jacobian at the initial approximation. Introduction. The Newton-Raphson method for solving an equation

f(y) = 0

(1)

is based upon the convergence, under suitable conditions, of the sequence (2)

y~+l = y~

y f(Yp) ,(yp)

p = 0,1,...

to the solution of (1), where Yo is an initial approximation to that solution. The modified Newton-Raphson method uses, instead of (2), the sequence (3)

Yp+ 1 = Yp

f (Yp) f'(Yo)

p = 0, 1,...

These methods are described in detail in [7-1, [6] and [4]. Extensions to systems of equations

(4)

fI(Yl,Y2,'",Yn) :

= 0 :

fm(Yl,Y2,'",Yn) = 0 are immediate in case: m = n, e.g. [3] and [4]. The analogs of (2), (3) are respectively:

(5)

y~ + ~ = yp - ( J ( y ~ ) ) - ~f(y~)

p = 0,1,...

(6)

yp+l = y ~ - ( J ( y o ) ) - ~ f ( y , )

p = 0,1,...

where y is the vector with components y j, j = 1,..., n Received February 28, 1965. * Research supported by the Swope Foundation. 94

1965]

METHOD FOR THE SOLUTION OF SYSTEMS OF EQUATIONS

95

f(y) is the vector with components fi(Y), i = 1, ..., n J(y) is the Jacobian matrix, whose (i,j)th element is t~fi(Y) tgyj and Yo is an initial approximation to a solution of (4). The composite Newton-Raphson gradient method of Hart and Motzkin [2], is applicable also if m # n, provided rank J(y) = n at the solution. In this note the Algorithm (6) is extended to general systems of equations, by using the generalized inverse [8] of the Jacobian matrix. Conditions of convergence, as well as bounds on the convergence rate, are stated in Theorem 2. These are based on theorem 1, which is an implicit function theorem following from a classical result of Hildebrandt and Graves [5, Theorem 3], and is of independent interest. NOTATIONS.: Let

Ek

be the k-dimensional vector space with the Euclidean norm

Ilxll = ( x , x ) 1'2. Let Em×" be the space of m × n complex matrices, with the norm IIh II--max {x/2:2 an eigenvalue of A ' A } , A* being the conjugate transpose of A. These norms satisfy [I A x II --< IIa II IIx II for every A E E m×~,x ~ E". By R(A),! N ( A ) we denote the range space respectively null space of A, and by A + the generalized inverse of A, [8]. For x o e E k and a real positive r, S(xo, r) = { x e E k ; II x - Xo[I < r}, the open ball of radius r around x o. The components of a function F : E " - - * E m are denoted by fi(Y), i = 1 , . . . , m . The Jacobian of F at y e E" is the m × n matrix

(0f, y) ] J(Y) =

,~yj

:

i = l,...,m j

= 1,..., n

Let Y be an open set in E n. Following Hildebrandt and Graves [5] we say that a function F : E n ~ E m is in the class C ' ( Y ) if the mapping: E n-~ E m×, given by: y ~ J(y) is continuous for every y ~ Y. The modulus of continuity of J(y) at Yo, 6(yo,e) is defined by I]Y - Yo I] < 6(Yo,e) =~ IIJ(y) - J(yo)II - ~. THEOREM 1. Let X o be an open set in E p, Yo a vector in E n, F a function, F : X o × S(yo, r ) ~ E m, T a linear transformation, T : E"-~ Em, M a real positive n u m b e r , such that:

(7)

M II T+ II < 1

(8)

I[zfyl-r2)-F(x,r )+F(x, y2)[l 0 where

(1)

[

and we m a y add the condition

+ ] b,[ + ]//,- i,/] < e.

Is,l
A~Ai+ t of the rhomb. If the rank of the matrix

A =

I

-

0

- cos#2

cos~3 + sin#a

sin/~

0

- costa

-

- sin#4

°

cos#4 + sin#4

1 _

is three, then there exist neighborhoods V(ai) c L with the following property: On all quadrupels of analytic arcs ci such that the tangent elements to c~< a r e in V(a~) it is possible to find quadrupels Bi(t) which for constant to form the vertices of a rhomb. Bi(t) is a continuous function of t where t is identified with

110

H. GUGGENHEIMER

[June

one of the arclengths s~ of the arcs % For the proof one simply chooses the fixed s~ so that the resulting system has a non-vanishing Jacobian determinant. Step two then follows as before (rank A -- 2 gives four conditions) and also step three (the theorem is trivial for ellipses) by the boundedness of the declensions. The reader may easily fill in the detail s. Schnirelmann also gives a theorem about possible degeneracies if the curve is only supposed to be C'. 6. It is worthwhile to investigate n-dimensional generalizations of the square theorem. The square is the two-dimensional member both of the series of n-dimensional cubes B, and of their duals, the n-dimensional 2n-cells Cn. (For n > 4, the regular simplices An, B,,, and Cn are the only regular polyhedra). In general, a smooth closed surface in R 3 does not contain an inscribed cube. But by Schnirelmann's method we may prove. THEOREM: Every C a hypersurface in R", C3-diffeomorphic to S "-1 , contains 2n points which are the vertices of a regular Cn. The regular Cn has 2n vertices and 2n(n-1) edges. All its two-dimensional faces are triangles. (n __>3). A piece of hypersurface may be described by n-1 variables; for pieces laid out about all the vertices we need 2n(n-1) variables. The equality of the edges is expressed by 2 n ( n - 1 ) - 1 equations which can be expanded so that the Jacobian matrix contains only first powers of the trigonometric functions of the Euler angles of the elements of hypersurface. These conditions alone suffice to make the polytope a regular C~. (This situation is parallel to that in § 5, and a deeper discussion probably would yield a proof that each closed C 2+~ hypersurface admits a continuum of inscribed Cn, n __>3.) By hypothesis, the function which describes the surface, as well as its Gauss curvature, can be considered as univalent differentiable functions on the sphere S n- 1. Therefore they can be developed into series of spherical harmonics and, by compactness, we again have the possibility of C 2 approximation by finite polynomials of ( n - 1 ) spherical harmonics. These polynomials again may be characterized by points of an open set Gl(n) of a finite dimensional cartesian space, in which there is a dense, connected subset which contains ellipsoids and for which the main (prolongation) lemma holds. The theorem holds for ellipsoids. A vertex of C~ in an ellipsoid of half-axes al, "', as has distance p from the center, where (See [2]) 1

pz

1 ~ 1 k i a]

The Gauss curvature of our surfaces is continuous, hence bounded, and uniformly bounded on the approximating surfaces to one C 3 surface. The approximation of a C a surface by surfaces given by spherical polynomials yields a bounded set of

1965]

FINITE SETS ON CURVES AND SURFACES

111

C's which, by the Blaschke Auswahlsatz, is relatively compact in the space of ovals in R". A converging sequence of C',s must converge to a non-degenerate C, since otherwise the Gauss curvature of the appro ximating surfaces could not be bounded. This completes the outline of the proof. A count of constants shows that also in dimensions 3 and 4 the A, and C, are the only universally inscribable regular polytopes. 7. According to Schnirelmann's theorem, every smooth curve contains a square. It is well known that every oval contains a symmetric hexagon, but not every oval contains a centrally symmetric octagon. In this connection, we can prove: THEOREM: Every simple C' curve can be C'-approximated by a curve which has an inscribed centrally symmetric 2N-gon. Let At (i = 1, ..., 2N) be the vertices of a 2N-gon in cyclic order. The condition of central symmetry is a)

AtA~+ a

=

AN+iAN+i+I (indices mod 2N)

b) .~A t = ~AN+ t Because of (b), condition (b) is equivalent (b) AtAt+ 2 = AN+t AN+i+ 2 Therefore, all conditions are given by relations between distances which yield a Jacobian matrix linear in the trigonometric functions of the angles of the line elements, and a Main Lemma will hold if the number of variables st is not less then the number of conditions. There are 2N variables st, N conditions (a) and N conditions (b). If we start from a 2N-gon with distinct vertices, we get 2N-gons with distinct vertices by the limit process needed to establish the Main Lemma. However, we are not able to control the sizes of the angles, and, therefore, the theorem has to be understood in such a way that 2N-gons with distinct vertices but angles n are admissible. The approximation by trigonometric polynomials and the definition of a G2(n) for whose curves an inscribed 2N-gon exists does not present any difficulties. On the other hand, since we cannot control the angles we cannot be sure that no points will merge in the final approximation process, even for analytic curves. Therefore, the theorem cannot be improved by Schnirelmann's methods, and I would even conjecture that with every reasonable measure in the space of C' curves the curves admitting a 2N-gon would fill a subset of measure zero. The proof of the previous theorem shows that for the complete success of Schirelmann's method a control of at least some angles of the inscribed polygon is necessary. In this direction, we have for instance the following theorem: THEOREM: On every simple, closed curve of declension of bounded variation

112

H. GUGGENHEIMER

there are five points which are the vertices of an axially symmetric pentagon with three equal edges and fixed base angles ~ > rc / 2. I f the vertex A1 is on the axis o f symmetry, the pentagon is characterized by A I A 2 = AIA5 A2A 3 = A3A 4 = A4A 5 A2A 4 = 2 A2A a sin

A3A 5 = 2 A2A3"sin

Gt

These are five equations for the five variables st. The existence o f the pentagon for an ellipse follows f r o m a simple continuity argument. The reader will easily fill in the details of the proof. NOTE. (Added in Proof). The full Sehnirelmann Theorem, for analytic curves only, was proved in a different way by R. P. Jerrard, Inscribed squares in plane curves, Trans. Amer. Math. Soc. 98 1961, 234-241 (Reference supplied by the Referee). BIBLIOGRAPHY 1. C. Carath6odory, Die Kurven mit baschriinkten Biegungen. Sitz. Ber. Preuss. Akad. Wiss. Berlin, Math. Phys. Kl., (1933), 102-125. (Gesammelte Werke, vol. II) 2. G. Salmon and W. Fiedler, Analitische Geometric des Raumes, 2 Aufl. B. G. Teubner, Leipzig, (1874); see. 95. 3. H. Whitney, Differentiable manifolds. Ann. of Math. (2)37 645-680, 1936; and remark by S. S. Chern, La g6ometrie des sous-vari6t6s d'un espace euclidien ~ plusieurs dimensions, Ens. Math. 40 26--46, 1954. 4. L. G. ~nirel'man, O nekotoryh geometri~eskih svoistvah zamknutyh krivyh. Sbornik rabot matemati~eskogo razdela sekcii estestvennyh i to~nyh nauk Komakademii, Moskva 1929. Reproduced in Uspehi Matemati6eskih Nauk, 10, (1944), 34-44. UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MINNESOTA

ON SOME E X T R E M A L

PROBLEMS

IN GRAPH

THEORY

BY P. E R D O S

ABSTRACT

The author proves that if C is a sufficientlylarge constant then every graph of n verticesand [Cn3/2] edgescontains a hexagon X1, )(2, X3, )(4, )(5, X6 and a seventh vertex Y joined to X1, X3 and )is. The problem is left open whether our graph contains the edges of a cube, (i.e. an eight vertex Z joined to X2, )(4 and )(6). Throughout this paper G, G' will denote graphs, V(G) denotes the number of edges, n(G) the number of vertices of G. G(n; m) is a graph of n vertices and m edges. Vertices will be denoted by xt'"Yl"'" edges by (x, y). {Xl,'", xn} denotes a path whose edges are (x l, x2),'-',(xn-x,X~), the vertices Xx,'", x, are assumed distinct, n - 1 is the length of the path, similarly ( x D ' " , x,) is a circuit of length n whose edges are (Xl, x2), ..., (x~_ t, xn), (x n, Xl). v(x), the valency of x is the number of edges incident to x. G(xl, ...,x~) is the subgraph of G spanned by(xt, ..-,x~). In an even graph all circuits have even length. It is well known and easy to see that the vertices of an even graph can be divided into two classes A and B so that every edge joins a vertex of A to a vertex of B. C, c, e l ' " denote suitable positive absolute constants. Recently several papers appeared which discussed various extremal problems in graph theory [1]. Denote by f ( n ; k , l) the smallest integer for which every G(n;f(n;k;l)) contains a G(k,l). Two years ago Turfin asked me to determine or estimate the smallest integer m for which every G(n;m) contains the various graphs determined by the vertices and edges of the regular polyhedra. For the tetrahedron the problem was solved many years ago by Turfin himself [6], for the octahedron I proved several years ago that (n 2/4) + en 3/2 < m < (n2/4) + Cn 3/2, details of the proof have not been published [1] and in this note we do not discuss the octahedron. The question for the dodecahedron and icosahedron seems difficult. It is well known that f ( n ; 4 , 4 ) > cn 3/2, but for a sufficiently large C every f ( n ; [On 3/2 ]) contains a rectangle [2]. One might conjecture that for a sufficiently large C every G(n; [Cn3/2]) contains a cube. In fact I proved t h a t f ( n ; 8,12) < Cn 3/2, and I even showed that every G(n; [Cn 3/2]) contains a G(8; 12) having the vertices, Received February 24, 1965. 113

i 14

P. ERDOS

[June

xl,x2,x3,x4;yl,Y2,ya,y , and the edges (xi, yj) where min(i,j)__< 2 [3]. But at present I can not prove that it must contain a cube. I can prove the much weaker result that it contains a G(7,9) consisting of a hexagon (Xl, "", x6) and a vertex y joined to xl,x a and x s. To prove the existence of a cube we would need an eighth vertex z joined to x2, x , and x6, and I have not succeeded in showing this. More precisely I am going to prove the following THEOREM. Let n > no(k). Then every G(n; lO[kl/En3/2]) contains a

G(2k + 1 ; 4k - 2) which has a path of length 2k{xDyl,..',yg, Xk+l} and the further edges (xl,Yi),(yl, x~), 2 3. To prove our Theorem we need two lemmas. LEMMA 1. Every G(n;m) has an even subgraph having at least m/2 edges. We prove the L e m m a by induction for n. It is clearly true for n _< 2. Assume that it is true for n - 1, we shall show it for n. Denote the vertices of G(n;m) by X l , ' " , xn. Since the lemma is true for n - 1, we can split the vertices xl ... x,_ 1 into two classer A and B so that the number of edges joining a vertex of A to a vertex of B is at least ½V(G(x 1,'", x,_ 1))"Without loss of generality we can assume that the number of edges joining x, to the vertices of B is at least ½v(x.). But then the even graph spanned by the vertices A u X . and B has at least ½(V(G(x i..., x,_ ~) + v (x.)) > (m/2) edges, which proves the Lemma. By a slightly more careful induction process we can prove that if the graph G(n;m) has no vertices of valency 0 then it contains an even graph having at

leastI2+4]edges.

ThecompletegraphofnverticesG(n;(2))showsthat

this result is in general best possible. It seems probable that if we know that our G(n; m) contains no triangle, the lemma can be considerably strengthened i.e. m/2 can perhaps be improved to cm for some c > 1/2, but I did not succeed in doing this. LEMMA 2. Every G(n;m) contains a subgraph G' every vertex of which has

valency (in G') greater that [mini.

1965]

ON SOME EXTREMAL PROBLEMS IN GRAPH THEORY

115

The Lemma is known [-4]. The proof is very simple. Now we can prove our Theorem. By Lemmas 1 and 2 our G(n;lO[kl/2n3/2]) contains an even subgraph every vertex of which has valency greater than 5kl/2n 1/2. Let x l , . . . , x , ; Yl "",Yv u + v < n be the vertices of G'. Let Yx, "",Yt, t > 5kl/2nl/2 be the vertices joined to x~ and let x 2 , ' " , x u , , u' < u be the other x's joined to a Yi, 1 < i -< t. G" is the subgraph of G' spanned by Yl, "'" Yt, x2 "" x,. Clearly each y in G" has valency > 5kl/2n 1/2 - 1 > 4kl/2n 1/2, i.e. each Yi has valency (in G') greater then 5kl/2nl/2. Thus (1)

V(G") > 4tkU2 n 1/2.

Denote by X2, "'" X u. the x~ with (2)

v(xi) > 2tk 1/2/n 1/2.

Let G~' be the subgraph of G" spanned by x2,"',x.,, ; Yl,'",Yv By (1), (2) and u" < n we have (3)

V(G") > V(G") - 2tkl/2n 1/2 > 2tk~/2n t/2,

By (3) one of the y's has valency > 2kl/2n 1/2 (in G~'). Let this vertex be Yl and let x2, " ' x t + l 1 > 2kl/2nl/2 be the vertices joined to Yl- Consider finally the graph G ' ( x 2 , . . . x l + l , Y2"",Yt), each xi has by (2) valency greater than 2tk'2/n w - 1 > tkl/2/n 1/2 (t > 4kl/2nl/2). Thus by a simple computation (4)

V(G'(XE,...,xl+l,y,...yt) ) >

tlk t/2 > kzr(G"(XE,...xt+Dy2...,yt) n

since by t > 4kl/2nU2, 1 > 2k 1/2n 1/2 _ tl_ > _8kn _ > t+ l 6kl/2n 1/2

kl/2nl/2

and rc(G"(x2"" Xt+ 1, Y2"'" Y,)) = 1 + t - 1. From (4) we obtain by a theorem of Gallai and myself [5] that G"(x2,... xl+ 1, Yt"" Y 3 has a path of length 2k - 2 {x2, Y2,'", Yk, Xk+ 1}. By our construction Xx is joined to every y of our path and Yl to every x of it. Thus finally G~(xl,... xt+ a, Y l , ' " Y k ) satisfies the requirements of our Theorem. The constant 10 could clearly be reduced, but I made no attempt in doing so since I am not sure if the factor k 1/2 is of the right order of magnitude.

116

P.

ERDOS

REFERENCES 1. P. Erd6s, Extremal problems in graph theory, Proc. Symposium on Graph theory, Smolenice, Acad. C.S.S.R. (1963), 29-36. 2. P. Erd6s, On sequences of integers no one of which divides the product of two others and on some related problems, Irv. Inst. Math. i Mech. Tomsk, 2 (1938), 79-82. 3. P. Erd6s, On an extremal problem in graph theory, Coll. Math. 13 (1965), 251-254. 4. P. Erd6s, On the structure of linear graphs, In. J. Math. 1 (1963), 156-160, see Lemma 1, p. 157-158. 5. P. ErdtSs and P. Gallai, On the maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hung. 10 (1959), 337-357. 6. P. Turin, On the theory of graphs, Coll. Math. 3 (1955), 19-30. TECHNION-ISRAELINSTITUTEOF TECHNOLOGY, HAIFA