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Volumen XXII Numero 3

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SUMARIO

PAG.

An exact diagonalization of the quadrupole and palrmg forces in the nuclear d~s shell, pOI' R. BROGLIA, E. MAQUEDA

Y

D. R. BBS ...• '

107

A predicative extension of elementary logic, Part I, pOI' L. E. SANCHIS 123

On the extension of currents, pOI' M. HERRERA ...........•........•. 139

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D. V. Lindley, Introduction to probability and Statistics, from a Bayesian view-point (L. A. Santal6) ...••....•......••.. 138

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AN EXACT DIAGONALIZATION OF THE

--------t-IQfHU*ADHRwUPOLE AND PAIRING FORCES

IN THE NUCLEAR d-s SHELL

by

R. BROGLIA, E. MAQUEDA and D.

R. BES

Depto. de Fisica, Facultad de Ciencias Exactas y Naturales,

Buenos Aires, Argentina

Abstract: The pairing plus quadruple interaction have been diagonalized for four identical particles in the

N

=

2 shell of the harmonic oscillator. The states of the basic representation are classified accQrding to the SU3 coupling scheme. The energy levels and wave functions are calculated for different values of the relative intensity of the two forces. The results are compared with the predictions of the vibrational and asymmetric rotor models.

1. INTRODUCTION

Many properties of nuclei at low energies can be explained with a model which considers the particles out of closed shell as moving independently in a single-particle potential and interacting through an effective residual force.

The residual interaction considered in this paper has' been proposed by Bohr and Mottelson (1, 2, 3, 4). It has a long range component (quadrupole interaction) which produces the level scheme corresponding to ellipsoidal deformations, and a short range one

(pairing interaction) which gives rise to the "seniority" scheme.

There are different approximations used un order to deal with this Hamiltonian. The more ,usual of them are: a) The quasi-particle formalism which approximately diagonalizes the pairing interaction. b) The quasi-boson method. It is used in order to treat the effects of the quadrupole force which has not been incorporated into the single particle field.

-108c) The deformation of the central field, which is produced by the long range component and which gives rise to collective degrees of freedom. The corresponding parameters are calculated through the "crancking model".

Although these approximations are very usefull, it seems convenient to perform an exact calculation in order to obtain a more detailed insight on the effects of the residual forces and corresponding collective level schemes. This is done in the present paper.

We have solved .the Hamiltonian for the case of four identical particles in the N

=

2 shell of the harmonic oscillator.

Pairing and quadrupole forces have different symmetries. The first is invariant with respect to the symplectic group and the se-· cond one is the Casimir operator of the SU3 group.

We use the SU3 coupling scheme because the pairing force has simpler matrix elements than the quadrupole force.

In order to classify the states of an harmonic oscillator shell,

Elliott (6) introduces the chain of subgroups

SUs::> SU3

::>

R3

::>

R2

where s is the shell degeneracy.

Corresponding to the SU3, R3 and R2 representations, the states are labelled by the pair of numbers

(Ap..) , the angular momentum

L

and its projection

M,

respectively.

Weyl's theorem states that if we label the states according to irreducible representations of SUs we are labelling at the same time according to the re:presentations of the symmetric group IIn:

(n

= number of particles). This is done through the partitions [f].

The wave functions must be totally antisymmetric: this fact determines the relation between the permutation symmetries of the spatial and the spin wave components. Therefore, once S is fixed, we know which irreducible representations of

IIn are possible.

Weare interested then in the possible values of

A and p.. which are compatible with a given [f], and in

L

values which are compatible with each (Ap..). Elliott has' obtained these values of (Ap..) for the

N

=

2 shell and most of the

N

=

3 shell.

The representations of R3, which occur in the representation

(Ap..) of SU3 are given by:

L

=

K, K

+

1,

K

+

2,

K

+ max (A,p..) (la)

-109where the integer K is

K

= min(Ap.) -2, ... , 1 or 0 except when

K

=

0 in which case

L

= max (AP.) , max(Ap.) -2, ... ,1 or 0

(11)

The structure of the Casimir operator of the SU3 group is

(lc)

G(AP.)

=

4

Q.Q

+

T

L.L

(2) with eigenvalues

(2a.)

From here we obtain an expression of the quadrupole interaction

A A A A A

Q . Q =

4 G (AP.) 3 L . L with eigenfunctions having definite values of (AP.) and L.

2. CALCULATIONS

We have constructed the wave functions of four identical particles, which have total spin

S

=

0, using the Hill-Wheeler projection integrals

'\It

«AP.) L M K)

=

C

«AP.)

t,

K)

J

DLMK

(0)

Xu

«AP.» do

(3) where

Xu is the intrinsic wave function referred to a rotated frame and

C

«AP.) L,K) is a normalization factor.

For the present case the

SU3

:::> R3 chain is not sufficient to give a complete classification of the states and we connot use fractional parentage coefficient techniques. In order to perform a complete classification, the parameter K is introduced by (3). This parameter

K

is an approximately good quantum number and can be interpreted as the projection of the angular momentum on the z-axis of the rotated frame.

-110-

The complete set of wave functions is given in table 1.

The intrinsic wave function introduced in (3) must satisfy the system of equations

A",y

X

=

A z",

X

=

A zy

X

=

0

(A"",-Ayy)

X

=

P.X

(2A zz -

Amm-Ayy)

x=

(2A+p.)

X

(4) where

(4a) and the operator

ai+

=

/2

(Xi

+ i

Pi)

creates one quantum of the harmonic oscillator in the the i-direction (i

=

X,

Y or

z).

We can obtain functions which satisfy the conditions (4) using the fact that the

SU3

group clasifies the quanta of the harmonic oscillator and that for each irreducible representation

(Ap.)

correspons a Young tableaux according to Weyl's theorem. For example, the wave function

X

(42) is given by

X

( 42)

=

I

a z

+ (

1 )

a z

+(2)

(5)

Young tableaux corresponding to the representation (42) of the SU3 group. where

I

0

> indicates the state in which the four particles are in the ground state of the harmonic oscillator. The Young tableaux associated with the representation (42) is given in fig. 1.

-111-

In

the Schrodinger representation X «AP.») is expresed as a linear combination of products of four harmonic oscillator functions. In the spherical basis it becomes

XfJ (A f-t)

=

Z;

a) cpfJ

(11

v

1) cpfJ

(l2

v

2)

v,l

cpfJ

(l3 v3) cpQ

(14

v 4)

=

(6) by evaluating the Hill-Wheeler integral we obtain tP

((A f-t) L M K)

=

2L+

1

c ((A f-t) L, K)

f

L

DMJ(

(D)

XfJ (A

/-t) d

Q

= c

2L+1

((A f-t) L, K)

~

I

DMKL

D y1m1

11*

D y2m2

12*

D y3m3 l'J*

D y4m414*]

cp(llm1) CP(l2

Tn

2)CP(l3

Tn

3) cp(14

Tn

4)

(7)

The second equation is deduced from the symmetry properties of the rotation matrices

D.

The following step is to reduce the obtained set of functions with respect to the symmetric group lIn.

We use the Young operators in order to perform this calculation.

The totally antisymetric wave functions can be expressed by

'l1 «Ap.),L,M=O,K,S=O,Ms = 0)

=

=

-V~

{'l1([f] (Ap.),L,M=O,Klrl)cp([f]S=O,Ms=O!rl)

(8) where

ri

states for a particular standard tableaux.

-112-

The nonorthogonality of the wave functions labelled by different

K

in the (42) representation has been overcome through a Schmidt orthogonalization process.

We then diagonalize the proposed Hamiltonian

(9) in the basis given by the complete set of functions (8).

The quadrupole component of the Hamiltonian is given by

HQ=-K

~ ~

(_)mri2rj2 Y2m ((};<pj)Y2- m «()j<pj) i,j m

(9a) and the pairing component

H

p is defined by the non-vanishing matrix elements

<

(li m;) (lj 'YItj)

I

V ij

I

(l'i m'i) (l'j m/j)

>

=

-

-

G8 mi,-rnj

8"

Tn j

a

li lj

8" (

)li+l'i+mi+m'i

(9b)

The quadrupole interaction is diagonal in the chosen basis, meanwhile the pairing force mixes the states with the same

L

but belonging to different prepresentations

(i\p.~.

We obtain the energies and the corresponding eigenfunctions for different values of the ratio

~

K

forces. The results are given in Table 2 in which we have indicated with (42)

K

=

2 the eigenvector coefficient corresponding to the wave function which is orthogonal to the (42)

K

=

0 one.

We know that for

-~

K

=

00 we have the seniority scheme and that for

~

0 we obtain the rotational extreme. With the re-

K

sults corresponding to intermediate values it is possible to determine the regions in which the nuclear motion can be interpreted in terms of the different collective models.

In fig. 2 we give the resulting energies as a function of

~.

The error is estimated to be 0.2

% for each level because the

K

obtained trace for the

L

=

0 matrix in the pairing extreme is

13.97

G

and it must be

-14

G(2).

-113-

3. DISCUSSION

The levels obtained in the quadrupole extreme correspond to a rigid rotator as can be seen in fig. 2. Applying expressions of reference (7), the three inertial parameters result equal. We may assume that for a sufficiently small departure from rigid rotator, the rotational behaviour will remain valid with a perturbation term of the form

H'= bL4

In references (7, 8), the coefficient

b

has been given the same value to all the levels of the same rotational band. We have found that a unique value of

b

does not give agreement with our results, however small is the departure from the extreme -

G

K

=

O.

In no case an axially symetric rotation can be obtained for the ground state.

The triplet predicted by the vibrational model appears for the value of

!i

=

25, the center of gravity of the 0, 2, 4 levels being

K

12

% lower than twice the energy of the first 2 the presence of a 3

+ level. However

+ level at this energy disturbs the vibrational picture.

The overlaps of the functions belonging to the (42) representation are

<

(42) 200 (42) 202

>

=

0.06285

<

(42) 400 (42) 402

>

=

0.29786

To realize the influence of the nonorthogollality in the label [{ we perfomed the diagonalization with the nonorthogonal wave functions and the results are compared with the correct ones in fig. 3.

As Elliott states, t.he lowest states have the main contribution from the (42) representation. However, there is a rather important part corresponding to higher representations, which increases

G

as does.

K

We want to thank Dr. Rebeca Ch. de Guber for kindly making available to us the computing facilities of the Instituto de Calculo

(Universidad de Buenos Aires).

-114-

• c>

Ie e

~ ~ ~ e til

Ii _ ... tit.

J J

.'.J..I. d.

NI

wI-I.!.

~I J

2 r-l

N--r-r-J

I-----'J"---..

.I.'---'-'-Ill-'J

.

,~\o

JJ J .LIJ

Ni:IIJ

JJJJJ

J g

10

J

J

J.

[J

I J;U w

~.

Jill J J

~i

~

:10

"'

J

1 Jll!

l~Jlwc>~JIII

J

!

_'1'--0

---r-J

---:r-J

~--'-----I

w

J

~lr-;; ",~II

0 " ' - - - - - - - - ' ' ' o-rr-rr-UJl-'----'l

J

~

L

~

10

~

L slo

J

NJI

J J

NU

~lllJIr]

J

.J

J wJ

I

J

J

J

.JI

J

Jll

ED

Jl1

J

J

.Jlol

wJ.~ollw]mlJ

~

10

N.N..!

~I Nw.e~~

..

JIUI

:~ l.~J 5::~

Ver leyenda en la pag. nO 122

-115-

4',5,6

D,

~3

Z

0.9

0.1

0.7

0.,4

0.6

0.5 I

2 r :

15

2

2

0

4

4

4

4

0,1

0.

I

1-0,3

~O.2

,-'3']

3

Ii)

U

LJU

0

G-4

I~l ca)

(~)

6-2. flOure 3

0. la)

(b)

~8S

0

2

1

1

0.

(0)

(b)

pairing e1Ctrenle

0.

Plot of the results of the diagonalization for the nonorthogonal wave functions

(b), compared with the correct ones (a).

[I]

[22]

TABLE 1

L

(42)

(31)

0

- - -

2

- - -

1

0, 2, 4

2, 3, 4, 5, 6

1, 2, 3, 4

(04)

(20)

0

0

0, 2, 4

0, 2

II

I

L

-116-

TABLE

2

Energy

(42)K=0

(42)K=2

(04) (31) (20)

Quadrupole extreme

0

0

0

1

2

2

2

2

4

4

4

5

6

2

3

3

4

0.8889

0.4444

0

0.5555 .

1

0.6296

0.5555

0.1111

0.1111

0.7407

0.2222

0.8889

0.8184

0.3703

0.3703

0.5555

0.7778

G

=4

K

0

0

0

1

2

2

2

2

2

3

3

4

5

6

4

4

4

0

0

0

1

0

0

0

0

1

0

0

1

0

0

1

0

0

0

1

1

1

0

0

0

1

0

0

1

0

0

0

1

0

0

0

1

0

0

1

0

0

0

0

1

1

0

1

0

0

1

0

0

0

0

1

0

0

0.8737 -0.026

0.4559 -0.047

0 -0.999

0.5826

1

0.6351

0.1369

0.7569

0.2690

-0.017

-0.999

1

-0.005 -0.002 -0.003

0.002

0.019 -0.172

0.000

-0.985

0.5725 -0.017

0.1601 -0.391

0.920

0.011

0.048

0.920 -0.D13

0.391

0

1

0.985 -0.172

0.013

0.019

0.007

1

0

0.8923

0.8241

0.000

0.000

-0.019

0.018

0.4083 -1.000 -0.002

0.3849

0.5826

0.7917

0.002 -1.000

1

1

0.045 -0.999

0.999

0.000

0.018

0.045

0.000

0.020

1.000

-0.D18

-0.025

1.000

0.000

0.003

0.000

0.005

L

-117-

TABLE

2 (Cont.)

En61'gy

(42)K

=0 (42)K =2 (04) (31) (20)

2

2

2

1

2

2

3

3

0

0

0

4

5

6

4

4

4

~=7

K

0.8645 -0.044

0.4655 -0.077

0 -0.996

-0.031

-0.996

0,078

0.999

-0.035

-0.042

0.6016

1

0.6422

0.5822

0.1943

0.009

0.007

-0.029

0.003

0.024

0.391

-0.920

0.005

1

0.000 -1.000

0.030 -0.309 -0.950 -0.001

0.950

0.024

-0.309

-0.034

0.005

0.001

0.1556 -0.920 -0.391 -0.022 -0.012

-0.009

0.7682

0 1

0.3018

1

0

0.8952 0.000 -0.031

0.078 -0.997

0.8306

0.000 0.032

0.997 0.077

0.4350 -1.000 -0.001

0.000 0.000

0.3953

0.001 0.999

0.030 0.034

0.6016

1

0.8015

1

2

3

3

4

4

4

4

5

6

2

2

2

~=10

"

0

0.8568 -0.062

0

0

1

2

0.4753 -0.101

-0.047

-0.993

0.997

-0.053

0

0.6194

1

0.2262

-0.993

0.012

0.6515

0.014

0.5892

-0.040

0.004

0.042

0.390 -0.919

0.104

0.007

0.039 -0.429 -0.902 -0.003

0.902

0.036

1

0.000

-0.428

-0.051

-0.057

-1.000

0.006

0.001

0.013 0.1736

0.7788

0.3326

0.920 0.391

0

1

0.030 0.017

1

0

0.8982

0.8368

0.4600

0.4053

0.000 -0.042

0.000

-1.000

-0.001

0.046

-0.001

0.108

0.993

0.000

-0.993

0.106

0.000

0.998

-0.041 -0.047

0.6194 1

0.8106 1

L

-:-118-

TABLE

2 (Cont.)

Energy (42)K=0 (42)K=2

(04) (31) (20)

3

4

4

2

2

3

4

4

0

0

.0

1

2

2

2

5

6

~=25

K

0.8363 -0.142

0.5197

-0.170

0

0.6921

-0.975

-0.135

-0.973

0.190

0.981

-0.159

-0.115

1

0.7057

0.027

0.050

0.6125 -0.091

1

0.010 0.016

0.000 -:- 0.999

0.059 -0.693 -0.717 -0.009

0.159 0.710

-0.680

0.107 -0.151

0.010

0.003 0.3529

0.2504

0.8218

0.4587

0.382

0.918

-0.906

0.389

0

1

0.9129 -0.001 -0.080

0.063

0.231

0.035

0

-0.970

0.030

0.8625

0.000 0.110 0.969 0.222

0.5622 -1.000 -0.001

0.4466 -0.001 0.991

0.000

-0.089

0.001

-0.103

0.6921

0.8477

1

1

3

4

4

2

2

3

4

4

5

6

2

2

2

0

1

0

0

G

-=<10

K

0.8336

-0.208

0.5503 0.188

0

0.7436

0.960

-0.225

-0.946

-0.234

0.952

-0.265

0.154

1

0.7509

0.039

0.077

0.014 0.022

1

0.001 -0.999

0.070 -0.757 -0.646 -0.013

0.6342 -0.150 0.308

0

0.618 -0.710

1

0.011

0.4347 -0.363

0.868 -0.194 0.278 -0.006

0.3070

0.916 -0.387 0.087 -0.048 -0.043

0.8521

0.5484

0.9254 -0.001

1

-0.100

0

0.314

-0.944

0.9080

0.000

0.234

0.898 0.373

0.6349 -1.000 -0.001

0.4757 -0.001 0.982

0.000

-0.123

0.001

-0.145

0.7436

0.8738

1

1

L

-119-'-

TABLE

2 (Cont.)

Energy (42)K =0 (42)K =2 (04)

(31) (20)

0

1

2

2

0

"

0

2

2

5

6

2

3

3

4

4

4

4

G

= 55

0.8392 -0.258

0.5694 -0.183

0

0.7812

1

0.7854

0.949

-0.303

-0.917

-0.259

0.917

-0.354

0.181

-0.049 -0.018

0.096 0.077

1

-0.027 -0.002 0.998

0.781 -0.612 -0.016

0.6597

0.4851

0.210

-0.332

-0.448 -0.547

0.803 -0.281

0.675

-0.011

0.408

-0.009

0.3491 -0.931 -0.386 -0.104 -0.058 -0.054

0.8742 0

1

0.6139

1

0.9354 -0.001

-0.112

0.7812

0.8928

1

1

0

0.369 -0.923

0.8964

0.000 0.202 0.918

0.6879

-1.000 -0.001 -0.001

0.4964

0.001 -0.973

0.148

0.343

0.001

0.177

4

4

5

6

2

2

2

G

= 70

0

"

0

0

1

2

2

3

3

4

4

0.8482

0.5811

0

0.8096

0.296

-0.170

-0.940

0.364

-0.890

0.275

1

-0.883

-0.423

-0.201

1

0.8908

0.057 0.021

0.8119

-0.110 -0.083

0.6872

0.5152

0.257

-0.563

0.793

-0.484

0.592 0.016

0.618 -0.011

0.297

-0.726 -0.348 -0.513

0.011

0.3812

0.911 0.384

0.117 0.065 0.064

0

0.031 0.002 -0.998

1

0.6635

1 0

0.9432 -0.002 -0.119

0.9080

0.000 0.234

0.408

0.898

-0.905

0.373

0.7280

-1.000 -0.001 -0.001

0.5116 -0.001

0.9u5

0.002

-0.167 -0.202

0.8096

0.9070

1

1

-120-

TABLE

2 (Cont.)

Energy (42)K=0 (42)K=2

(04)

L

(31) (20)

K

G

=

85

0

0

0

1

0.8581 -0.324

0.5884 -U.155

0

0.8317

1

-0.410

-0.866

-0.933

-

0.286

1

-0.064 -0.023

-0.034 -0.003

0.853

0.475

-0.217

2

0.997

2

2

2

2

3

0.8327 -0.122 -0.088

0.7139

0.801

0.288 -0.641 -0.434

0.5336 0.267

-0.659

0.520

0.563

0.392 -0.584

-

0.019

0.010

0.013

0.4062

0.9036

3

0.7021

0.909 0.383

0

1

0.128 0.071

1

0

0.072

4

4

4

4

5

0.9495

0.9173

0.7593

0.5231

0.002

0.000

-1.000

-0.001

0.124 -0.436

0.259

-0.001

0.881

-0.001

0.892

0.395

0.002

0.958

-0.182 -0.222

0.8317 1

6

0.9180 1

0

2

2

2

2

1

2

0

0

3

3

4

4

4

4

5

6

G

K

=

100

0.8677 -0.345

0.5931 -0.141

0

0.8317

-0.928

1

0.8495

0.444

-0.847

0.293

0.827

-0.513

-0.229

1

-0.064 -0.023 -0.034 -0.003

0.131

0.997

0.091 -0.805 -0.571

-0.020

0.519 -0.009

0.631 0.015

0.075 0.078

0.7381

0.5454

0.4263

0.9139

0.308

0.244

0.907

-0.691

-0.606

0.382

-0.396

0.418

0.136

0.7329

0.9546

0.9250

0.003

0.000

0

1

0.127

0.280

0.457

0.868

0.7842 -1.000 -0.001 -0.001

0.5321 0.001 -0.952 0.194

1

0

0.880

0.410

0.003

0.238

0.8492

0.9268

1

1

-121-

TABLE

2 (Cont.)

Energy (42)K=0 (42)K=2

(04)

L

(31)

(20)

G

=

550

IC~~

2

2

3

3

4

4

4

4

5

6

1

2

2

2

0

0

0

4

4

5

6

3

3

4

4

2

2

2

2

0

0

0

1

2

0.9627 -0.449

0.6040 -0.031

0

0.9645

-0.893

1 0.124 0.046

-0.599

-0.731

0.326

0.046

1

0.663

-0.681

-0.309

0.000 -0.990

0.9624 0.191 -0.122

0.9331 0.366 -0.835

0.817

-0.250

0.5908

0.5587

0.147

0.890

-0.382

0.374

0.480

0.193

0.530

-0.776

0.108

0.009

0.326 0.005

0.023

0.138

0.9802

0.9362

0

1

1

0

0.9891 -0.015 -0.141

0.9812 0.003

0.413

0.9482 -1.000

0.002

0.568

0.778

-0.007

-0.810

0.474

0.014

0.5855 -0.001

0.900 -0.268 -0.345

0.9645

1

0.9833

1

Pairing extreme

0.9996 -0.471

-0.625 0.623

0.6023

0

0.9996

0.9996

0.003

-0.882

-0.025

0.707

0.331

-

-0.707

0.335

1

0.602 -0.209 -0.414 -0.650

0.9996

0.9971

-0.150

0.5996

0.426

-0.618 -0.333

0.635

0.012

-0.054

0.754

0.120 -0.558

0.413

0.489

0.513 -0.563

0.119

0.5996

-0.626 0.503 0.121 0.572 -0.116

0.9996

0

1

0.9996

1

1

1

0

1 0.229

0.459 0.623 0.591

0.9996

0.9983

0.689 -0.029

-0.609

I

0.691

0.5996

0.001

-0.123

0.880

0.397 -0.591

-0.289 -

0.398

0.378

0.9996

0.9996

-122-

REFERENCES

(1) A. BOHR,

Comptes Bendus du

Congrcs International de Physique NuoZ6aire,

Paris, July 1958 (Dunod, Paris 1959) p. 204.

(2) B. MOTTELSON,

The Many-Body Problem.

Lectures given at ":EJcole d'Eta de Physique TMorique", Les Houches, France (Dunod, Paris, 1959) p. 283.

(3) B. MOTTLESON,

Prooeedings of the International Conferenoe on Nuclear

Struoture,

Kingston, Canada, September 1960 (University of Toronto Press,

Canada, 1960) p. 525.

(4) B. MOTTELSON,

Bendioonti della Souola Internazionale di F1,sioa,

Corso

XV, Varenna, 1960 (Zanichelli, Bologna, 1962) p. 44.

(5) G. RACAH,

Proo. of t"ke Behovoth Conference on Nuolear Struoture,

Rehovoth, Israel, September 1957 (North-Holland Publishing Co., Amsterdam

1958) p. 155.

(6) J. P. EI.LIOTT,

Proo. Boyal Society A,

245 (1958) 128; 245 (1958) 562.

J. P. ELLIOTT andM. HARVEY,

Proc. Boyal Sooiety A,

272 (1963) 557.

(7) C. A. MALLMANN,

Nuolear Physios

24 (1961) 535

(8) A. S. DAVYDOV and V. S. ROSTOVSKY,

Nuclear Physios

12 (1959) 58.

(9) J. P. ELLIOT, Lectures given at "Eseuela Latinoamerieana de Fisiea-

Mexico 1962" (Universidad de Mexico, Mexico, D. F.)

Explicaci6n de la Fig. 2

Plot of the energy levels as a function of the ratio between the strength of

G

. .

III

.

0 f

1

(M W

)-2 F h lue of

-~ the energy unit is the difference between the lowest and the highest level. The states are characterized by the value of the angular momentum L.

When two angular momenta are written in the Bame line, the corresponding levels are degenerated.

A PREDICATIVE EXTENSION OF

ELEMENTARY LOGIC PART I

by LUIS ELPIDIO SANCHIS

We can extend elementary logic by allowing quantification over class variables and introducing a comprehension axiom. In this paper we shall use Gentzen's methods to study the predicative extension obtained by restricting the comprehension axiom to propositional functions containing only quantification over individuals.

We prove a theorem that permits to pass from a deduction in the extended logic to a deduction in elementary logic; from this theorem several non-derivability and relative consistency results can be obtained.

This simple predicative extension has been studied before, mainly in connection with the Bernays-Godel formalization of set theory. We have not followed the customary procedure of identifying the set of individuals with a subset of the universe of classes. This procedure seems to be natural in dealing with set theory; for a more general investigation the identification is rather artificial.

In this way we return to the original method of Bernays who seems to be the first in studying an extension of this kind.

In his paper [3] Maehara has considered a system very similar to that studied in this paper. He uses it as an auxiliary system to obtain a result on Hilbert's €-symbol. Our investigation is more akin to that of [6]. Leaving aside the identification of indivIduals with classes the result proved in this paper seems to generalize the result obtained by Shoenfield using an extension of the first €-theorem.

1. THE SYSTEM:

LKP. We introduce a system that extends

Gentzen's system LK. This system is called LKP. It is constructed as a syntactial system; i. e. we assume that the formal entities are strings of symbols. Since these symbols are never exhibited

it

-124is clear that the assumption is irrelevant; we might as well assume that the formal entities are n-tuples of some given primitive atoms.

The primitive symbols are classified in several groups:

1. A denumerable list of free individual variables ..

2. A denumerable list of bound individual variables.

3. Individual constants and functors; each functor whit a fixed number of arguments.

4. A denumerable list of free class variables.

5. A denumerable list of bound class variables.

6. Class constants and predicate constants; each predicate with a fixed number of arguments.

7. Special symbols:

£, ,},

::l,

V ,

1\

,V,

[if,

A,

=, (, ),

i-·

Let

A

and

B

be expressions (i. e. finite strings of primitive symbols) and let

u

be a primitive symbol. We denote with

[B/u]A

the expression obtained by substitution of

B

for u in

A.

We shall use the following notation: letters

x, y, z,..

for free

-individual variables; letters

a, b,

c, d ... for bound individual variables; letters

X,

Y, Z, .. , for free class variables; letters H, J, .. for bound class variables.

Definition of individual terms. 1. Each free individual variable and individual constant is an individual term. 2. If h, ... ,

t"

are individual terms,

k

> 1, and

f

is a functor with

k

arguments, then

f(h, ... ,

t k )

is an individual term. We shall use letters

t, h, ..

for individual terms.

Definition of class terms and formulas. 1. If

t, h,

rh, ... ,

t"

are individual terms,

k

> 1, and

F

is a predicate constant with

k

arguments, then F(h, ... ,

t k ), t

£

h, t

=

h

are formulas. 2. Every free ciass variable and class constant is a class term. 3. If t is an individual term and

U

is a class term then

t

£

U

is a formula. 4. If A and

B

are formulas then

"lA, (A =>B), (A VA), (A

1\

B)

are for· mulas. 5. If

A is a formula which does not contain the bound variable

b,

then

(Vb[b/x]A)

and

(a

b[b/x]A)

are formulas. 6. If A is a formula which does not contain the bound variable

H

then

(v

H[H/X]A)

and

(:![

H[H/X]A)

are formulas. 7. If A is a formula which does not contain the bound variable

b,

and in which there is no occurrence of bound class variables, then

Ab [b/x]A

is a class term.

We shall use letters

A, B, C, ..

for formulas, and letters

U, V, ..

for class terms. Parentheses that are not necessary for the understanding of a formula will be omitted.

-125-

Let

A

be a formula; by

Gradel (A)

we understand the number of occurrences of

V and

3: preceding a bound class variable;

Grade2(A)

is the number of occurrences of symbols

I,

::>,

V, /\,

V,

3:,

A, where the quantifiers V and

3: are counted only if they precede a bound individual variable. Now

Grade (A)=Grade1(A)+

+ Grade2(A).

In the same way can define

Grade(U)

for a class term

U.

A prime formula is a formula by classes 1 or 3 of the definition, with

U

a class variable or a class constant. A

A prime formula is a formula of the form

t

£

Ab [b/x]A.

Lemma 1.

If

A

is a formula,

U

a class term and

t

an individual term, then [t / x]

A

is a formula and [t / x]

U

is a class term.

Lemma 2. If

A

is a formula,

U

is a class term and

V

is a class term such that no bound variable of

V

occurs in

A

or

U,

then

[V/X]A

is a formula and

[V/X]U

is a class term.

Corollary. An expression of the form V

b [b/x]A

where

A

does not contain the bound individual variable b is a formula if and only if

A

is a formula. The same property holds for

3:

b [b /

x] A,

V

H[H/X]A,

3:

H[H/X]A

and

Ab[b/x]A

with the restriction in the last case that

A

does not contain bound class variables.

Lemma 1 and Lemma 2 can be easily proved by induction on

Grade (A)

and

Grade(U).

The corollary follows inmediately.

A finite sequence of formulas of the form AI>' .. , Ak (k

>

0) is called an L-sequence. Letters

lJf,

N, S, T . .. are used for I.-sequences. The formulas A 1, . .. , Ak are called the components of the I.-sequence. The notation

M

<

N

means that every component of

M

is also a component of

N.

An expression of the form

M

1-

N

is called an L-formula. We shall use also the obvious notation

[B/u]M.

Definition of thesis. Some I.-formulas are called thesis according with the inductive definition given by the following rules:

Rule

(Ax)

For each prime formula

A, A

Rule

(=1)

For every individual term

t,

1-

A

is a thesis.

1-

t

= t is a thesis.

Rule (=2) If

M I-N,[t/x]A

and M I-N,t =

h

are thesis and

A

Ml is a prime formula then

M I-N, [h/x]A

is a thesis.

Rule (U) If M

1-

N1

is a thesis.

1-

N

is a thesis,

M

<

Mb and

N

<

Nl

then

Rule (1*) If

M, A

1-

N

is a thesis then

M

1-

N,IA

is a thesis

Rule (*1) If

M I-N,A

is a thesis then

M,IA I-N

is a thesis.

-126-

Bule (::>*)

If M,AI-N,B is a thesis then

MI-N,A::>B

is a thesis.

Rule (*::» If

M

M,A

::>

B

1-

1-

N

is a thesis.

N, A

and

M, B

1-

N

are thesis then

,

M

Rule (v*) If MI-N,A is a thesis then

MI-N,AvB

and

1-

N, B

V

A

are thesis. ,

Rule (* V )

If

M, A

1-

Nand

M, B

1-

N

are thesis then

M,A

V

B I-N

is a thesis.

M

Rule

(f\"

*) If

M

1-

N, A

and

M

1-

N, B are

thesis then

1-

N, A /\ B is a thesis.

Rule (* /\) If M, A

M, B

6.

A

1-

N

is a thesis then

M, A/\

13

1-

Nand

1-

N

are thesis.

Rule (V *) If M

1-

N, A

is a thesis,

A

does not contain the bound variable

b,

and the free variable then

M

1-

N,

Vb

[b/x]A

is a thesis.

x

does not occur in

M

or

N,

Rule (* V) If M, [t/x]A I:.:.

N

is a thesis and

A

does not contain the bound variable

b,

then

M,

V

b[b/x]A I-N

is a thesis.

Rule (

3:

*) If

M

1-

N, [t /

x] A

is a thesis and

A

does not contain the bound variable

b,

then

M I-N,

3:

b[b/x]A

is a thesis.

Rule (*

3:)

If M, A

1-

N

is a thesis,

A

does not contain the bound variable

b,

and the free variable then

M,3: b [b/x 1A

1-

N

1;,; a thesis.

Rule (V

1

*) If

Jjl x

does not occur in

M

or

N,

1-

N, A

is a thesis,

A

does not contain the bound variable

H

and the free variable

X

does not occur in

M

or

N,

then

M

1-

N,

V

H[H/X]A

is a thesis.

Rule (*V d

If M,

[U /X]A

1-

N

is a thesis and

A

does not contain the bound variable

H,

then

M,

V

H [H / X] A

1-

N

is a thesis

Rule

(3:1*)

If MI-N,

[U/X] A

is a thesis and

A

does not contain the bound variable

H,

then

M I-N, 3:H[H/X]A

is a thesis.

Rule (*

3: 1)

If JJ:1, A

1-

N

is a thesis,

A

does not contain the bound variable

H,

and the free variable

X

does not occur in

M

or

N,

then

M,

3:

H[H/X]A

1-

N

is a thesis.

Rule

(t\

*)

If M

1-

N,

[t / x] A is a thesis,

A

does not contain bound class variables, and the bound variable in

A, then M b

does not occur

1-

N, tdb [b/x]A

is a thesis.

Rule (*A)

If M,[t/x]A

I-N is a thesis,

A

does not contain bound class variables, and the bound variable

b

does not occur in

A,

then

M, t€t\b[b/x]A I-N

is a thesis.

The variable

x

in rules ( V *) and (*

3: ), and the variable

X

in rules (V 1*) and (* 3: 1) are called the proper variable of that

-127application of the rule. We note that the variable

x in rule

(*V ) is used just to indicate a substitution and can be taken arbitrarily.

The same is true for rules

(3:*), (*Vl), ( :[h*) , (A*) and

('lOA).

We define the relation

M

1-

N

is a thesis with order n by the following rules: i) If M

1-

N

is case of rule

(Ax)

or of rule (=1) then it is a thesis with order

O.

ii) If M1

1-

N1

is a thesis with order

n,

and

M

1-

N

results of the application of some rule to

M1

1-

N1

then

M

1-

N

is a thesis with order

n

+

1. iii) If M1

1-

Nl

and

M21- N2

are thesis with order nland

n2

respectively, and

M

1-

N

results of the application of some rule to

Ml

1-

Nl

and

M2

1-

N2

then

M

1-

N

is a thesis with order

Max(nI, n2)

+

1.

We can define also the relation

M

1-

N

is a thesis with general ordeI!

n

by the same rules with the only exception of rule

(U)

in which case the general order does not increase.

We state now several lemmas that can be easily proved by induction on

n)'

they hold also if we replace order by general order.

Lemma 3. If M

Lemma 4. If M

1-

N

is a thesis with order n, then

[t/x]M

1-

[t/x]N

is a thesis with order

n.

1-

N

is a thesis with order

n, M1

1-

N1

is an

L-formula obtained by changing all occurrences of a bound variable

b (H)

in

M

1-

N

by another bound variable

c (J)

then

Ml

1-

N1

is a thesis with order

n.

Lemma 5. If M

1-

N

is a thesis with order n then [Y

/X]M

I-

I-

[Y /

X] N

is a thesis with order

n.

It is easy to show that for every formula A, A \- A is a thesis.

In order to generalize rule (=2) for an arbitrary formula

A

we need the following theorem.

Theorem 1. Let

S

1-

T, t

=

h

be a thesis. For every formula

A

and number n the following hold: If M order

n, N

1-

N

is a thesis with

<

NI,

[t/x]A

and

111

<

M

1 ,

[t/x]A

then

(1) 8,MI-T,Nl , [h/x]A

(2) 8,M l ,

[h/x]A I-T,N

are both thesis.

Note that from

81-

T, t

= hand

1-

t

=

t

we get by rule

..

-128-

(=2) the thesis

S \- T,h =

t.

Note also that if

A

is a prime formula (1) follows inmediately by rule (=2)' Furthermore (1) is trivial if

[t/x]A

is not

a

component of Nand (2) is trivial if

(t/x]A

is not a component of

111.

The proof of the theorem is by induction. We assume the property holds for every formula

A'

and number

m

if one at least of the three following conditions is satisfied.

(i)

Grade1(A')

(ii)

Grade1(A')

(iii)

Grade1(A')

<

Grade1(A)

=

Grade1(A)

and

Grade2(A')

<

Grade2(A).

=

Grade1(A), Grade2(A')

=

Grade2(A), m<n.

In every application of part (iii) of the induction hypothesis the formula

A'

will be the same formula

A.

For the proof of (1) we assume that

[t/x]A

is a component of

N,

and for the proof of (2) that it is a component of

M. a)

M \-

N

is a case of rule

(Ax).

Hence

111\-

N

is the thesis

[t/x]A \- [t/x]A with

A

a prime formula. We have remarked above that (1) follows by rule (=2); (2) also follows by rule (=2) from the thesis S \- T,

h

=

t

and [h/x]A \- [h/x]A.

b) M \- N is a case of rule (=1) is clear by the remark we have made above.

c)

M \- N is obtained by some rule from a premise or premises with smaller order and

[t/x]A"

is not the formula introduced by the rule. Hence that formula is a component of

N1

if it is a right rule, or a component of

M1

if it is a left rule. We show with one example how this case is handled. Suppose we have a derivation by rule (:::J * ) in this way

M, B \-N', C

M \-N',B:::J C

Note that

M, B

<

Mb B,

[t/x]A

and

N', C

<

N

1,

C,

[t/x]A.

Using part (iii) of the induction hypothesis we get

(1')

S,

M, B \- T, Nb C,

[h/x]A

(2')

S,1J,h B,

[h/x] A \- T, N', C and now we get (1)

from

(1') by rule

(U),

rule (:::J *) and again rule

(U)

since B:c

C

is a component of

N

1 ;

from (2') we obtain

(2) with rule

(U)

and rule ( :::J*).

-129-

To handle any rule with restriction on a proper variable we use Lemma 3 or Lemma 5.

d) M

1-

N

is obtained by some rule and

[t/x]A

is the formula introduced by the rule. Here we apply part (iii) of the induction hypothesis and afterward part (i) or part (ii) of the induction hypothesis. We show this in several examples.

The formula

A

is IB and we have an application of rule (i*)

M

M,

[t/x]B

I-N'

1-

N',

[t/x]

I

B

Using part (iii) of the induction hypothesis as in c) we get

(1')

8,

M, [t/x]B

1-

T,

Nb

[h/x]

IB

(2') 8,Mb [t/x]B, [h/x]IB

I-T,N'

Now (2) follows from (2') with rule

(U)

and rule (1*). Since

Gradc2(B)

<

Gradc2{IB)

we apply part (ii) of the induction hypothesis to (1') to obtain

8, M, [h/x]B

1-

T,

Nb

[h/x]

I

B

and from this (1) is obtained by rule

(U)

and rule (1*).

The formula

A

is

g

>..b [b/y]B

and rule (>..*) is applied. We can assume that

y

is distinet from

x and does not occur in

t

or

h.

Hence

[t/x]A

is the formula

[t/x]g€ >..b[b/y] [t/x]A

and

[h/x]A

is the formula

[h/x]g

>..b [b/y] [h/x]B.

M

M

I-N', [

[t/x]g/y] [t/x]B

1-

N', [t/x]g

Ab [b/y] [t/x]B

Using part (iii) of the induction hypothesis we get

(1')

8,

M

1-

T, N

1,

[[It/x]g/y] [t/x]B, [h/x]g

>..b[b/y] [h/xJB

(2') .8, Mb

[h/x]g

Ab [b/y] [h/x]B

1-

T, N', [[t/x]g/y] [t/x]B

Now (2) follows from (2') using rule (>..*); to get (1) note that the formula

[[t/x]g/y] [t/x]B

is identical with the formula

[t/x] [g/y] B

and since

Gradc2( [g/y]B)

<

Gradc2(A)

we apply part

(ii) of the induction hypothesis to obtain from (1')

-130-

S, M \- T, N1 [[h/x]g/y] [h/x]B, [h/x]g

>..b [b/y] [h/x]B

and from this (1) follows using rule (>.. *) .

The formula

A

is V

H[H/Y]B

and rule (* V

1) is applied.

Now

[t/x]A

is the formula V

H[H/Y] [t/x]B

and

[h/x]A

is the formula V

H[H/Y] [h/x]B.

M', [U /Y][t/x]B \- N

M',

V

H[H/Y] [t/x]B \- N

Using part (iii) of the induction hypothesis we get

(1') S,M', [U/Y] [t/x]B \- T,N!,

V

H[H/Y] [h/x]B

(.2') C,M!, [U/Y] [t/x]B,

V

H[H/Y] [h/x]B \- T,N

Now (1) follows from (1') using rule (* V 1)' To obtain (2) note that by changing the formula B we can assume that the variable

x

does not occur in U; this change consists just in replacing a free variable by another free variable. Hence the formula

[U /Y]

[t / x]

B

is also the formula [t /

x] [U /Y] B;

since Gradel (

[U /Y] B)

<

Gra-

del

(A)

we can apply part (i) of the induction hipothesis to obtain

S, M1 [U/Y][h/x]B,

V

H[H/Y][h/x]B \- T, N and from this (.2) follows by rule

(*V1)'

The other cases can be handled in the same way; if there is a restriction on a proper variable we use Lemma 3 or Lemma 5.

Corollary. If

M \-N,t

= hand

M \-N,[t/x]A

are thesis then

M \- N,[h/x]A is a thesis.

Theorem 2. If M \- N is a thesis, and [U/Y]M \- [U/Y]N is an

L-formula, then [U /Y] M \- [U /Y] N is also a thesis.

Proof by induction on the order of

M \- N; the Corollary to

Theorem 1 is used to handle the case in which rule (=2) is applied.

Theorem 3. If M \- N is a thesis with order

n,

and

M1

is obtained from

M by eliminating components of the form

t

=

t,

then

M1 \- N is a thesis with order

n.

The proof by induction on

n is completely trivial.

Theorem 4. For every formula A and numbers nand m the following hold: given L-sequences

S, T, M, N, T1, M1

such that

then

-131-

(1)

8

1-

T

is a thesis with order

11,

(2) M

1-

N

is a thesis with order

m

(3) T

<

TbA

and

M< M1,A

(4)

8,

M1

1-

T1,N

is also a thesis.

The proof is by induction. We assume the theorem is true for a formula

A'

and number

11,1

and

m1

if one of the three following conditions is satisfied

(i)

Gradel (A')

<

Gradel (A)

nl

(ii)

Grade1(A')

=

Grade1(A)

and

Grade2(A')

(iii)

Gradel (A')

+

m1

<

n

+

m.

Grade1(A), Grade2(A')

<

Grade2(A)

=

Grade2(A),

Note that the theorem is trivial if

A

is not a component of

T

or if it is not a component of

M.

The cases

n

=

0 or

m

=

0 follow easily using rule

(U)

or Theorem 3. In the following we shall assume

n

+

m

>

O.

a)

(1) is obtained by a left rule. We show how this case is handled with an example. Suppose rule (*:::» is applied.

8'

1-

T,B

S',B:::>

8',0

°

1-

T

1-

T

Using part (iii) of the induction hypothesis we get

S',Ml

1-

Tl,B,N

8',0,M1

1-

Tl,N

Now we get (4) using rule (*::> ).

b) (1) is obtained by a right rule and the formula introduced by the rule is a component of

T

1 •

We proceed as in case

a)

using rule

(U)

to eliminate the formula introduced by the rule.

c)

(2) is obtained by a right rule or by a left rule such that the formula introduced by the rule is a component of MI. We use here the same method as in cases

a)

or

b). d)

(1) is obtained by a right rule such that the formula introduced by the rule is not a component of T

I , and (2) is obtained by a

-132left rule such that the formula introduced by the rule is not

It component of

MI.

In both cases this formula must be the formula

Aj

hence

A

is not a prime formula and the case in which (1) is obtained by rule (=2) cannot occur. We shall show in one example the procedure used to handle this case. Suppose A is the formula V

H[H/X]B

with the following derivations

8

8

1-

T',R

1-

T',

V

H[H/X]B

M', [U/X]B

1-

N

M',

V

H[H/X]BT-N

Using Lemma 5 we can assume that the variable X does not occur in

M

1, T1

or

N.

First we apply part (iii) of the induction hypothesis to obtain

8,M

1

1-

ThB,N

8,M h

[U/X]B

1-

T

1

,N

By Theorem 2 we have also the thesis

8,M

1

1-

Th

[U /X]B,N

and since Gradel ( [U/X]B)

<

Grade1(A)

we can apply part (i) of the induction hypothesis to obtain (4).

Corollary. If 8

1-

T,A

and

M,A

1-

N

are thesis then

8,M

1-

T,N

is also a thesis.

(k

Theorem 5. If M

1-

N

is a thesis with general order n, which does not contain bound class variables,

N

>

0) where foreach i furthermore the symbol

<

N

1, it

=

hI, ... , tk

=

hk

=

1, .. .

,k, ti

is not identical with

hi,

and

= does not occur in

M

and occurs in

N

1 only in prime formulas, then

M

1-

N1 is also a thesis with general order

m

<

n.

The proof by induction on in which

M n

is easy. We consider only the case

1-

N

is obtained by rule (=2). Suppose we have the derivation

M

1-

N', t

=

h M

1-

N', [t/x]A

M

1-

N', [h/x]A

If

t

and

h

are the same term the right premise and the con-

-133elusion are the same L-formula and we apply the induction hypothesis. If this is not the case it follows that hence by the induction hypothesis applied to be left premise it follows that

M

1-

N1 is a thesis.

We say that a thesis is independent of some rules of the system LKP if we can show it is a thesis without using those rules.

Corollary. If M [- N is a thesis with general order

n,

which does not contain bound class variables, and the symbol

= does not occur in it, then M [- N is independent of rules (=1) and (=2) .

. Proof by induction on n using Theorem 5.

2. THE SYSTEM LK*. It does not seem possible to prove the

Herbrand-Gentzen Theorem in the system LKP. For this reason we shall study a system LK*, which is a subsystem of LKP. The new system is obtained from LKP if we drop rules (A*) and (*A) and furthermore we allow in rules

(Ax)

and

(=2) that the formula

A

be a prime formula or a A-prime formula. A thesis in the system

LK*

will be called an elementary thesis. We define order and general order as before.

The system LK* is essentially the system LK of first order logic and the A-prime formulas behave as prime formulas since no rule allows the introduction of such a formula. Theorem 4 can be proved by any of the standard methods of proving the elimination theorem; our proof can be applied replacing parts (i) and (ii) of the induction hypothesis by a condition on the number of quantifiers and propositional symbols occurring in the formula i1 not counting those occurring in a A-term. In this way we do not need the restriction on the occurrence of bound class variables in a A-term.

A A-axiom is a formula which is either of the form

tt!

Ab/b[x]B=> [t/x]B

or of the form [t/x]B=> tfAb[b/x]B. It is easy to show that M [- N is a thesis if and only if for some

T

consisting of A-axioms is

M,T [- N an elementary thesis.

A Q-free formula is a formula in which every quantifier occurs inside a A-term. Now every Q-free formula is a formula in prenex normal form. If

A

is in prenex normal form then V b [b /

x] A,

a

b[b/x]A,

v

H[H/X]A

and

aH[H/X]A

are formulas in prenex normal form, provided they are formulas.

-134-

An application of rule

(U)

from M \- N to

Ml \-

Nl

is called restricted if Ml

<

M, N1

<

N

and furthermore no component is repeated in

M1

or

N1.

We say that M \- N is Q-derivable from S \- T if it can be obtained starting with S \- T and applying rule

(U)

or quantifier rules. If every application of. rule

(U)

is restricted we~ay that

M \- N is QR-derivable from S \- T.

Lemma 6. If M \- N is Q-derivable from S \- T then there is

M1 \- N1 which is QR-derivable from S \-

T

and Ml

<

M,N1

<

N.

Lemma 7. If M \- N is Q-derivable from S \- T, the formulas

B

and Care Q-free,

S' and M' are obtained by eliminating all components

B

in Sand

M.

respectively,

T'

and

N'

are obtained by eliminating all components

C

in

T

and

N

respectively, then

Ai' \- N' is

Q-derivable from S' \- T'.

Now suppose M \- N is QR-derivable from S \- T and

x

is a proper variable in the derivation. Let

y

be a variable not occurring in

S \- T; then

M \- N is QR-derivable from

[y/x]S \- [y/x]T and

x

is not a proper variable in the new derivation. The same procedure can be used for a class variable

x.

Let A be a formula in prenex normal form; then for some Q-free formula

B

the formula

A

is of the form where Ri is a bound variable, T i is. a free variable of the same kind, and

Qi

is a quantifier followed by

R i,

i

=

1, ... ,

n. Now let

U

11 ••• ,

Un be terms such that U i is of the same kind as

T

i, and if

· Qi

is a universal quantifier then

U i

is a variable not occurring in .

· A,

U1, ... , Ui-1, i

=

1, ... , n. Then if is a formula it is called a right reduced form of A. In this definition the substitution prefix must be understood in the sense of a

· simultaneous substitution of

U

11 ••• ,

Un for T

1, ••• ,

Tn in B. If

Qi

is an univers.al quantifier we say that

U i

is a proper variable in

· the

ith

place of the right reduced form.

We define a left reduced form of A in a similar way,. but we require that

U i

be a variable when

Qi

is an existential quantifier, and in this case is also called a proper variable in the

ith

place of the left reduced form.

-135-

Let Bl, ... , Bk be right reduced forms of a formula

A,

and suppose B i is determined by Ui

1, ••• ,

Ui n,

i

=

1, ... ,

n.

'Ve say they are compatible right reduced forms if whenever a variable is proper in

Bi

and

B j

then it is proper in the same place, say the

sth

place, and for every m

<

s

the terms Ui

m

and Uj

m

are identical, i=l, ...

,n, j=l, ...

,n.

In the same way are defined compatible left reduced forms of a formula

A.

Our next result is the so called Herbrand-Gentzen Theorem.

We give a new proof of this Theorem which seems to be more convenient than the proof given by Gentzen.

Theorem 6. Let

M

1-

N

be an elementary thesis with all the components in prenex normal form. Then there are elementary thesis 8

1-

T

and Ml

1-

Nl

such that: i) The components of

QR-derivable from

8

8

1-

Tare Q-free formulas, Ml

1-

T

and

Ml

<

M, Nl

<

N.

1-

Nl

is ii) With each component of

M1

we can associate formulas

B

1 , . . . ,

Bk

(k> 0) in S that are compatible left reduced forms of that component. Under this association every formula in

8 corresponds to some formula in

MI.

The same proverty holds for

N1

and

T

with right reduced forms. iii)

If under the correspondence of ii) a variable is proper in two forms in of

Ml

8

1-

T,

then both corresponds to the same component

1-

Nl,

and the variable is proper in the QR-derivation from

81-

T

to

M

1

1-N

1 •

Suppose

M

1-

N

is an elementary thesis with order

n.

It is easy to show by induction on exists and

M n

that the elementary thesis 8

1-

T

1-

N

is Q-derivable from it. This is done using Lemmas

6 and 7. We show in one example a standard procedure which can be applied in any case. Suppose

M

1-

N

is obtained by rule (=2) in this way

M

1-

N', t

=

111 h M

1-

N', [t/x]A

1-

N', [h/x]A

8

1

By the induction hypothesis

M

1-

Tl

and

M

1-

N', [t/x]A

is Q-derivable from 8

2

1-

T

2•

From

Lemma 7 follows that

M

1-

N', t

=

h

is Q-derivable from

1-

N'

is Q-derivable from 8

1

1-

T'1 where

T\

is obtained by eliminating all

components t

=

h

in T

I .

We may assume no proper variable in this derivation occurs in

M, N'

or

-136-

[h/x]A.

8

2

Also from Lemma 7 follows that

M

1-

N'

is Q-derivable from

1-

T'2

where T'2 is obtained by eliminating all components

[,t/x]A

in

T

2 ;

we may assume no proper variable in this derivation occurs in 81,

T\, M, N'

or

[h/x]A.

Now from the elementary thesis

8

1

1-

T\, t

=

t

and 8

2

1-

T'2, [t/x]A

we get using rule (=2) the thesis 8

1

,8

2

1-

T\, T'2, [hjx]A

and from this by Q-derivation is obtained

M

to obtain

M1

1-

N', [h/x]A.

Once it is proved the existence of 8

1-

T

we apply Lemma 6

1-

N1.which is QR-derivable from 8

1-

T.

The correspondence of parts ii) and iii) of the theorem are related in an abvious way with the steps of the QR-derivation. It can be shown in detail by induction on the number of rules applied.

We shall write

A

=

B

as an abbreviation for the formula

(A:::> B)

1\

(B:::> A).

It is well-known that for every formula

A

there is a formula

B

in prenex normal form, such that

1-

A

=

B

is an elementary thesis. A formula is called elementary if it does not contain bound class variables or class terms. It is called quasi-elementary if it does not contain bound class variables, and every class term occurrin in it is a A-term.

It is easy to show that for every quasi-elementary formula

A

there is an elementary formula

B

such that

1-

A

=

B

is a thesis. This formula

B

is obtained by replacing parts of the form

te b[bjx]C

by

[tjx]C.

An v-formula (a-formula) is a formula of the form

V

H[H/X]C (aH[HjX]C)

where

C is an elementary formula in prenex normal form.

An

elementary form of an

V

-formula

V

H[H jX] C

is either a formula

V

b [bjX] C

or is a formula obtained from the formula

[U/X]A,

where

U

is a quasi-elementary term, by ·prefixing universal quantifiers for all variables in

U,

with the understanding that those quantifiers must not bind occurrences of the variables outside

U.

We define similarly the elementary form of an a -formula but using existential quantifiers instead of universal quantifiers.

Theorem 7. Let

M,8

1-

T, N

be a thesis, where the components of

M

and

N

are elementary formulas in prenex normal form, the components of 8 are

V

-formulas, and the components of

T

area -formulas. Then there is a thesis

M,

8'

1-

T', N

where the components of 8' and

T' are elementary forms of the formulas in 8 and T respectively.

Proof: there is an elementary thesis

M,

8,

M1

1-

T, N

where the components of Ml are A-axioms, and we may assume they have been

-137replaced by formulas in prenex normal form. From Theore m6 follows there is an elementary thesis 8

1 \-

T1

from which the former thesis can be obtained by a Q-derivation. We obtain a thesis

8\ \-

T'l

if we replace every free class variable and class constant by some individual constant. This substitution does not affect those formulas that are reduced forms of the formulas in

M

or

N.

\Ve can repeat the Q-derivation but now the class quantification is replaced by the quantification of all individual variables in the corresponding terms.

In place of

M1

we get formulas

M\

which are again A-axioms, hence can be eliminated. In place of 8 and T we get 8' and T' which are elementary forms of the formulas in Sand T respectively.

BIBLIOGRAPHY

[1] BERNAYs, P.,

.d

system ofaxiomatio set theory.

1. Journal of Symbolic

Logic, 2: 65-77 (1937).

[2] GODEL, K.,

The oonsistenoy of the axiom of ohoioe and of the generalized continuum-hypothesis with the axioms of set theory.

Princeton, N. J., 1940.

[3] MAEHARA, S., Equality axiom on Hilbert's e symbol . . Journal of the Faculty of Science. University of Tokyo, VII: 419-435 (1957).

[4]

MOSTOMSKI, A.,

Some impredicative definitions in the axiomatio set-theory.

Fundamenta Mathematicae, 37: 111-124 (1950).

[5] NOVAK, 1. L., .A

construction for models of oonsistent systems.

Fundamenta

Mathematicae, 37: 87-110 (1950).

[6] SHOENFIELD, J. R., .A

relative consistency proof.

Journal of Symbolic Logic,

19: 21-28 (1954).

BIBLIOGRAFIA

D. V.

LINDLEY,

Introduotion to Probability and Statistios, from a Bayesian viewpoint.

Part. I, Probability (260 paginas). Part. II, Inference, (292 paginas),

Cambridge, University Press, 1965.

El libro esta destinado a estudiantes uuiversitarios de matematica. En opiui6n del autor el contenido de los dos volumenes es el minima que todo matematico, cualquiera que sea su especialidad, deberia conocer acerca de las probabilidades y de la estadistica. Con esta idea directriz, el libro no presupone mas conocimientos que los comunes en los dos primeros cursos de analisis matematico de nuestras facultades de ciencias

0 ingenieria. Sin embargo, con esta base y la habilidad del autor para seleccionar y exponer los temas, es. mucho

10 que se puede dar

0, por

10 menos, son muchas las ideas que se pueden iniciar, en el espacio no excesivo de estos dos volu.menes. En el caso de mayor complicaci6n

0 mayor exigencia de espacio, los temas son referidos a otra literatura. El resultado es un libro €xcelente, original en muchos puntos e interesante por varios motivos, entre otros por su modo de desarrollo "des de un punto de vista. bayesiano" como figura en el titulo, por los ejemplos muy variados y bien elegidos, por los ejercicios al final de cada capitulo (sin la soluci6n) y por el estilo propio del autor que hace la lectura agradable y atractiva.

El volumen primero consta de 4 capitulos, cuyo contenido es el siguiente.

El Cap. 1 contiene 1a axiomatica de la probabilidad y sus primeros teoremas.

La axiomatica no es la de Kolmogorov, sino que esta basada mas bien en 1a de

Rilny

(Wahrsoheinliohkeitsreohnung,

Berlin, 1962), establecida sobre la idea de probabilidad condicionaL Lindley se refiere siempre a la probabilidad de un suceso A dado otro B y no simplemente a la probabilidad de A en absoluto. Gontiene tambien una discusi6n detallada de la probabilidad como grado de verosimilitud

0 creencia (belief) 10 cua1 es muy util en estadistica, por

10 menos tal como esta tratada en el vol. II. Los cap. 2 y 3 tratan de las distribuciones, de una y varias variable .. respectivamente. En e1 cap. 4 considera el autor los procesos estocasticos y cadenas de Markov.

La segunda. parte de 1a obra esta dedicada a 1a inferencia estadistica. En el cap. 5 se. trata la inferencia para la distribuci6n normal; se explica como usar el teorema de Bayes para pasar de la verosimilitud o· creencia a priori a la misma idea a postericri. En el cap. 6 se hace un analogo tratamiento para el caso de varias distribuciones normales y en el cap. 7 se trata la inferencia para otras distribuciones. El cap. 8, Ultimo de la obra, se titula "minimos cuadrados" y se usa para test

0 estimaci6n de hip6tesis lineales.

Como dice el autor, el hecho de tomar un punta de vista no ortodoxo (el bayesiano) obliga a ciertos rodeos y precauciones para incluir en el tratamiento los metodos e ideas usuales en la estadistica de manejo comun. Su preocupaci6n para probar que dicho punto de vista presenta ventajas sobre el clasico, al que inc1uye y amplia, da un tono particular al libro. No hay duda de que el mi!'mo habra de contribuir a ganar adeptos para los partidarios de esa tendencia, tal vez no nueva, pero que el autor presenta renovada y actualizada.

L. A. SantaM

ON THE EXTENSION OF CURRENTS

by M. HERRERA

1. Let

A

be an open set in number space

13 n

and F a closed subset of

A. Let T be a closed o-continuous current on A - F and

,\ >

1 a real number such that the following condition Of.. is satisfied on A: Of.. For each relatively compact open set G such that

G-

c

A

there exists a constant

k(G);::::

0 such that

II

T

IIBr::::;;

:::;; keG)

rf..

for every open ball

Br

of radius

r

contained in

G.

P. Lelong has proved in [2] (theor. 4) that, if F is a submanifold of class O'

(s;::::

2) of

A

of dimension

d

< ,\

-1, then the simple extension of

T

on

A

exists and is closed. This is used in [2] to prove that the integration current on a complex analytic set is closed. Here we give a consequence of the first result and show to apply it to the integration on a semianalytic (orientable) set. The properties of currents we use implicitly here are to be found in

[1] y [2].

2. For each open set

A

in Rn, let

CZ'J (A)

be the space of o

<Xl differential forms on

A

with compact support. The norm

II

</> II of a form </>

CZ'J (A)

is the maximum on

A

of the absolute values of the coefficients of </>. If T is a linear form on

CZ'J (A)

and

G

is a relatively compact open set in Rn, the norm II T lIa of T on

G

is defined by

IITlla=sup

(IT(</»

I

:</>€CZ'J(Gn A)

and

1I</>1I~1).

A o-continuous current

T

on

A

is a linear form on

CZ'J (A)

such that II T lIa is finite for each relatively compact open set G with

G-

c

A. All the currents occurring here are o-continuous. For every open subset

U

of

A, TI U

denotes the restriction of

T

on

U.

If A is open in Rn, F closed in

A

and T is a current on A -

F'

such that II

T

lIa is finite for each relatively compact open set

G

-140with

G-

C

A, then it is proved in [2] that there exists one and only one extension T' of T on A (i.e., T'IA F

II

T'

lie

=

T) that verifies

=

II

T

lie

for such G; T' is called the simple extension of

Ton A.

2.1.

DEFINITION:

Let T be a curre'f}t on the open set A in R" and

F

a subset (not necessarily closed) of A. The norm of T on

F

E

is zero

(II

T

IIF .

0)

if, for each com.pact set K in A and each

>

0,

there exists a relatively compact open set G in A such that

K

n

F

C

G

C

G-

C

A

and

II

T

lie

~

II

T

IIF

=

0 if and only if

II

TIU

IlunF

=

0 for each member

U

of an open covering of

A.

If

(Fi)

is a denumerable family of subsets of

A

such that

II

T

IIFi

=

0 for all i, then

II Til u

Fi

=

O.

2.2.

PROPOSITION:

Let A be an open set in Rn,

F

a closed

subset of A, T a current on A F and T' an extension of T on A.

Then the following statements are equivalent:

(i)

(ii)

T' is the simple extension of T on A.

II Tile = II

T'

lie

for any relatively compact open G with

G-

c

A.

(iii) liT' IIF =

O.

Proof: The equivalence between (i) and (ii) has been proved in [2]. Let us suppose (ii) true but not (iii). Then there exists a compact set

K

C

A

and

E pact open set

G, K

>

0 such that for each relatively comn

F

C

G

C

G-

C

A

implies

II

T'

lie>

€.

Choose one such

G

and, by condition (ii), a form cf>

CD (G F) with support

Kl such that T'

(cp)

> ment choose a CPI

CD

(G (F

E and

II

cP

II

~ n

K l )) such that

T'

(CPl)

>

€ and

II

CPI

II ~ tf

=

cP

+

CPI

+ ... is obtained such that

tf

CD (G),

II

cP

II

~

T'

(tf)

> II

T'

lie,

which is absurd.

Conversely, if (iii) holds but not (ii), let

G

be a relatively compact open set in

A such that

II Tile

=;1=

II

T'

lie;

then

II

T'

Ile-

-II Tile

=

2d>

0 and there exists

cP

CD (G)

such that

II

cP

II ~ and

T' (cp)

> II Tile

+

d; let K be the support 01 cpo By (iii) there exists an open set

U

which verifies

K

n

FeU C U- c A and

IIT'llu<d.

Let

CPI€CD (U)

and

CP2€ CD (G-F)

be forms such that

cP

=

CPI

+

CP2,

II

CPI

II

< 1 and

II

CP2

II

< 1. Then

I

T

(CP2)

I

<

I!

T

lie

and consequently

T'

(CPI)

=

T'

(cp) -

T' (CP2)

>

T' (cp) -

- II Tile>

d,

which contradicts the choice of

U.

2.3.

COROLLARY:

Let A be an open set in

Rn and

Foe

F\

-141-

closed subsets of A. Let To be a current on A Fo such that

II

To

IIFl-1"o

=

0 and let Tl

=

TolA -

Fl. Then if one of To or T1

has a simple extension on A, so does the other, and both simple extensions are equal.

2.4.

LEMMA:

Let A be an open set in Rn and T a current on j1 which verifies condition

C).. (A> 1)

on A. Then

II

T 11M

=

0

for each sub manifold

M of

A of class

(s

>

1)

and dimension d

<

A.

C'

Proof: Choose P

M

and a coordinate map X :

U

~ of a neighborhood

U

of

P

such that

X

(U

n

M)

=

Rd

C

R".

It suffices to prove

1/

TIU lIunu

=

O.

As

X is a this is equivalent to

II

T'

IIRd e

s isomorphism,

=

0, where

T'

=

X

(TjU). As TIU satisfies

C).. on U, so does T' on Rn and, because of an usual reasoning in measure theory (see [2], theor. 3), we have

II

T'

IIRd=O.

2.5. PROPOSITION: Let

A be an open set in

Rn ancl F a closed

subset of A. Let T be a closed current on A-F. Let us suppose ,the following two conditions are verified:

(a). -

F

is contained in a denumerable disjoint union of submanifolds Vi of

A, of class

(s

>

2),

of dimensions d i

<

d

(O<d<n), and such that (V i - -Vi)n A

C U (Vi:j >i)

and

U

(Vi:

j > i) is closed for all i.

(b). -

T verifies condition C), on A with

A>

d

+

1.

Then the simple extension of T on A exists and is closed.

P1'00f:

By (b)

II

T

IIG is finite for each relatively compact open

G with G-

C

A; then the simple extension T of T on A exists

([2], Prop. 1). By (b) and 2.4

II T Ilv -1"

=

0 for all i, therefore

IITII

UV.-F

=

0 and the simple extension

T'

of

T' =

TIA ( U Vi) on

A

exists and is equal to

T

by 2.3. Consequently it suffices to prove that

T'

is closed. For each i let T'i be the restriction of T' on A ( U Vi :

j

> i), where (U Vi : j

> i) is cloRed because of

(a). Let us prove by induction that each T'i is closAd. Since

T

is closed, so is T't

(A -

=

T'. Let us suppose T'i is closed. By 2.3 T'; t l is the simple extension of T'i on

A (

U Vi : j

> i

(U Vi :

j

>

+

1)

=

U Vi and, according to the theorem of IJelong quoted in

§ 1, it is closed. This implies that T' is closed, as was to be shown.

By simplicity we have worked in an open set in Rn. All remains valid in a differentiable manifold of class

Cs (s >

2).

-142-

3.

AN APPLICATION.

3.1. Let

X

be a connected real analytic manifold. For each

x e

X,

let

S (x)

be the smallest family of germs of sets in

x

such that: (1)

aeS(x) andbeS(x)

implyaU

beS(x) anda-beS(x);

(2) Sex)

(f (y)

contains all the germs defined in

x

by sets of the form

>

0) , where

f

is a real analytic function defi.ned in a neighborhood of

x. Following S. Lojasiewicz ([6]), a subset

M

of

X

is semianalytic if, for each

x e

X,

the germ of

M

in

x

belongs to

Sex).

Let M be a semianalytic set in X;a point

x

M is p-regular if there exists an open neighborhood

U

of

X

such that

Un

M

is an analytic submanifold of

X

of dimension

p.

The set of regular points

(i.e., p-regular points for some

p)

of

M

is dense in

M.

The dimension

dim (M)

of

M

is

<

P

if there are not q-regular points in

JIll

with

q>

p. We set

dim

CM)

= p if

dim (M)

< p but not

dim (M)

<

P -1. If

dim

(111)

= p, we denote by

Mp

the set of p-regular points of 111 and define a

(M) .111 -1I1p;

a

(JYJ)

is a semianalytic set of dimension

<

p -

1 and we can call it the singular part of

JJ1;

if 111 is closed, so is decompose a

(M).

Now it is possible to

a

(M)

in its (p

-1) regular part and its singular part, and so on.

If M is closed semianalytic of dimension

p,

we (lan then write

M

=

(U Vi : i

=

1, ... ,

p)

where the

Vi

are disjoint analytic submanifolds of

X

of dimension i and

V

i - -

Vi

C

U

(Vi: j

<

i)

for each i =0, ... ,

p (cf.

2.5). The family of the connected components of

Mp (dim (M)

=

p)

is locally finite (1).

3.2. Let 111 be a closed semianalytic set of dimension

p

of an open set

A

of

Rn

such that

Mp

is oriented. The restriction of each form

¢

e CJ)

(A a

111)

to 1I1p is a form

¢*

e CJ)

(Mp)

with compact support. Then the current on A -

1

0M

(¢)

=

fM

¢ p

a

M

is well defined, and is a o-continuous current of di-

(1) All this properties have been proved by Lojasiewicz (unpublished). They were enuuciated in a course given in 1964 at the University of Buenos Aires.

Summaries of results will be given in [4] and [5]. Some of the facts are treated in [3] and [6l..

-143mension

p.

In a forthcoming paper it will be proved tbat

1

0M

satisfies condition C-;.. of § 1 with A

=

p.

Then the simple extension

1M

of

1

0

M

on

A

exists; we call it the integration current on

M.

(cf. [2]).

If we recall the decomposition of

a

(M)

into submanifolds

(3.1) and Proposition 2.5, we see that

1M

is closed when dim

oeM) ::::;; dim(M) 2. Trivial examples (consider the interval [0,1] in

R)

show that

1M

is not closed in general if this restriction is not imposed. Moreover, the extension of this definition and properties to a semianatytic set

M

in a real analytic manifold

X

is inmediate. In the case

X

is a complex analytic manifold and

M

a complex analytic set in X of (complex) dimension

p,

the current

1M

defined in this way by considering the canonical orientation of

JJlp

coincides with the one defined by Lelong in [2]. It is always closed because dimR

a

(M) ::::;; dimR M 2

=

2 p 2.

BIBLIOGRAPHY

[1] DE RAHM,

Variet6s difterentiables

(Actualites Scientifiques et Industrielles n

Q

1222, 1955. Hermann, Paris).

[2] P. LELONG,

Integration sur un ensemble analytique complexe

(Bull. Soc. math. France, 85, 1957, p. 239 - 262).

[3] S. LOJASIEWICZ,

Swr le probleme de la division

(Studia Mathematica, T.

XVIII, 1959, p. 88 - 136).

[4] S. LOJASIEWICZ,

On the triangulation of semianalytic sets

(Univ. of Pisa, to be published).

[5] S. LOJASIEWICZ,

Sobre la divisi6n de distribucwnes y la triangulaci6n de' conjuntos semianaliticos

(Fac. C. Exactas y Naturales de la Universidad de Buenos Aires, por publicar).

[6] S. LOJASIEWICZ,

Une propiet6 topologique des

SOU8 -

ensembles analytiques reels

(Colloques du CNRS, Paris, Juin 1962).

CRONICA

LA REUNION DE LA UMA DE 1964

La Reunion anual de la UMA correspondiente a 1964 se realizo entre los dias 10 a 12 de octubre en Buenos Aires, en los Departamentos de matematica de las Facultades de ciencias exactas y naturales, y de Ingenieria de la

Universidad de Buenos Aires.

Ademas de numerosos miembros de la Capital Federal, asistieron a la

Reunion las siguientes delegaciones:

Universidad N. de Cordoba: E. Zarantonello, J. N. Aguirre, P. L. Checchi,

C. Casas de Kalnay, S. Maur, D. M. Dragone de Checchi, O. Zanardi, C. Loiseau,

G. Jover, G. Prieri, M. T. Vasquez y F. Griinbaum.

Universidad N. de La Plata: J. Bosch, M. Herrera, L. Oubina, S. Salvioli, C. Dubost.

Universidad N. del Litoral: E. Gaspar, V. Rein, M. Castagnino, E. Rofman, E. O. Ferrari, E. Cattaneo y V. Schapira de Erlijman.

Universidad N. del Tucuman: F. Herrera, I. G. de D'Angelo,

.T.

Royo,

M. S. Berraondo, E. E. Plaza,

.T.

Bacca, R. Ovejero y C. A. Sastre.

Universidad N. de Cuyo: .T. I. Delgado, O. S. Borghi, J. R. Moreschi y

E. Marchi.

Universidad N., del Sur: A. Monteiro, R. Cignoli, L. Iturrioz, E. Fernandez Stacco, M. Dickmann y R. Chiappa.

Universidad N. del Nordeste: M. R. Marangunic, R. Martinez y

.T.

E.

Borgna.

Instituto de investigaciones cientificas y tecnicas de las Fuerzas armadas:

.T.

F. Diharce.

Las sesiones se inauguraron el sabado 10 a las 9 en el Departamento de la Facultad de ciencias (Ciudad universitaria, Nui'iez) continuando porIa tarde en el Departamento de la Facultad de Ingenieria, celebrandose las restantes sesiones en Nunez.

Durante las sesiones se pronunciaron cuatro conferencias y se presentaron numerosas comunicaciones cientificas.

El acto inaugural se abrio con una "Conferencia Rey Pastor", en homenaje y recuel'do del fundador y animador de la UMA, que estuv'o a cargo del profesor A. Monteiro quien diserto sobre el tema: Las algebras de Nelson. Las restantes conferencias fueron las siguientes:

E. Zarantonello: La clausura del rango numerico contiene el espectro.

W. Ambrose: La teoria de Morse.

A. Gonzalez Dominguez: Generalizaciones del teorema de Bochner.

-145-

Ademas se presenta,ron las siguientes comunicaciones:

A.

R.

RAGGIO (Universidad de B. Aires)

El teorema f1mdamental de Gentz61l para sus sistemas de deducci6n

nat1~ral

NI y NK.

Se demuestra para la logica cuantificacional intuicionista

NI que toda demostracion puede reducirse a una forma normal en la cual los supuestos utilizados son sub formulas de la formula demostrada. Para la logica cuantificacional clasica NK se demuestra - un teorema mas debil. De las demostraciones se puede reconstruir el camino que siguio Genzen para construir sus sistemas

LI y LK.

LUIS E. SANCHIS (Universidad de Buenos Aires),

Una clase de funcionales que satisfacen el esquema de recu1'si6n primitiva para todos los tipos finitos.

En este trabajo consideramos el problema de la existencia de funcionales que satisfagan el esquema de definicion por recursion primitiva para todos los tipos finitos. El metodo usa do no es constructivo pero es admisible desde el punto de vista de la fundamentacion predicativa.

Los tipos finitos se obtienen -partiendo de un tipo atomico

N

(que representa la clase de los numeros naturales) mediante la aplicacion reiterada de la operacion que forma de los tipos

a

y [3 el tipo

Fa[3

(que corresponde a funcionales con dominio

a

y contradominio (3). F

+lal ..• an+l denota el tipo

Fal(Fna2' .. a'+l(3).

Tomando como fundamento este sistema de tip os construimos un sistema aplicativo en el cual: a) cada termino (combinacion en la terminologia usual de la logica combinatoria) es de un tipo determinado; b) contiene los combinadores S, K, I de Schon finkel-Curry pero en forma tal que combinadores con distinta asignacion de tipo se consideran distintos; c) contiene combinadores

Ra

de tipo

F3(F,Naa) aNN,

y constantes 0 de tipo

N

y de tipo

FNN;

d) con cada combinador se asocia una regIa especial de reduccion; para

ROo

hay dos reglas:

RXYO

red

Y,

y

RXYOn+l

red

Xon(RXyon)

donde 0" es un numeral.

Propiedad

(ll):

to do termino de tipo

N

que no contiene variables reduce mediante las reglas d) a un unico numeral. Para demostrar esta propiedad introducimos mediante una induccion infinita (por 10 tanto no formalizable en aritmetica elemental) una clase de terminos llamados regulares. Que la propiedad

(ll)

vale para los terminos regulares se demuestra facilmente por una induccion que corresponde a la forma de la definicion. Como tambien se demuestra que to do termino es regular la propiedad

(ll)

es valida en general.

U sando nuevamente la regularidad de los terminos del sistema se demuestra que es posible 1Lsociar con cada termino X de tipo

a

un funcional X*; el conjunto de los funcionales asi obtenido 10 denominamos

a*.

El conjunto

N*

consiste de todos los numerales; (F

a(3)

* consiste de funcionales con dominio

a

*

y contradominio en [3*. Estos funcionales son computables: e1 proceso de computacion es dado por las reglas de reduccion en e1 sistema.

-146-

L. ITURRIOZ Y A. MONTEIRO (Universidad N. del Sur). Culmllo Proposioional

implioativo olusico con - n variables proposioionales.

Consideremos el caJculo proporcional construido con el alfabeto -+, ( , ),

gl, g., •.. , gn, los axiomas (esquemas):

T1)

(x-+

(y

-+

T2)

T3)

«x

-+

(y

-+

-+

«x

-+

y)

-+

(x

-+

z»)

«

(x

-+

y)

-+ x) -+ x) y la regIa de MODUS PQNENS. Sea F el conjunto de formulas (bien formadas),

T

el conjunto de tesis (teoremas) de este caJculo. Si iV

x,

y •

T

ponemos:

=

Y

para indicar que

x

-+ y

T

Y

Y

-+

X

E

T.

Te.nemos as! una relacion de equivalencia

= definida sobre

F,

compatible con la operacion -+. Sea

(£, -+) el algebra de Lindembaum as! obtenida, donde £ es el conjunto de las c1ases de equivalencia. Si N(£) designa el numero de elementos de Z, se puede demosirar que:

N

(£)

k=n

=

Z; ( 1)

~+l (~)

2"

n-k k=l

es decir,

N(£)es el numero de formulas logicamente distintas que se pueden obtener con n variables proposicionales en el caIculo proposicional implicativo chisico (positivo), considerado por primera vez por Alfredo Tarski (C.

R.

Soc. des Sciences et des Lettres de Varsovie, 1930, pag. 3050). N(£) es tambien el numero maximo de conjuntos que se pueden obtener a partir de

n

partes G, ... ,

Go

de un conjunto dado, efectuando un numero cualquiera de veces, sobre los conjuntos dad os, la operacion

A

-+ B

Y

B

-+ A.

L. MONTElRiO (Universidad N. del Sur).

Algebras implicativas de valenci& menor

0

igual que n. (Presentada por L. Iturrioz).

Ivo Thomas (Notre Dame J. Formal Logie, 3 (1962), 170-174), proM que 8i a los axiomas del Calculo Proposicional Intuicionista Implicativo se agrega el axioma (donde

n;;;;:

2 :

An

=

fJ n .-

2

-+ (fJ

n

-3

-+( ....... -+ (fJo -+ po) .....

» donde

fJi

Pi

-+

Pi+i)

,-+

Po

para

i

=

=

0,1, ... ,,' n - 2 se obtiene un Calcu-

10 Proposicional implicativo iniuicionista n-valente.

Sea

(A,

-+) un algebra implicativa intuicionista

(A.

Diego, Revista de

la

U.M.A., 20 (1962), 310-311), si en

A

vale la formula

An

=

1 diremos que A. es un algebra implicativa de valencia menor

0 igual que

n.

Indicamos los siguientes resultados:

TEOREMA: Para que un algebra implicativa intuicionista

A

sea de valencia menor

0 igual que n es necesario y suficiente que la familia de los sistemas deductivos que contiene un sistema deductivo irreduciible, forme una cadena que tiene a 10 sumo

n

elementos.

De este teorema se deduce que toda algebra implicativa intuicionista de valencia men or

0 igual que

n es isomorfa a un sub-producto de cadenas con a

10 sumo n elementos.

Estos resultados valen tambien para las algebras de Heyting.

-147-

L. A. SANTALO (Universidad de Buenos Aires),

Sobre la teoria del campo

~mi­

ticado de Einstein.

Supuesto el espacio-tiempo previsto de una estructura de~ariedad diferen· ciable con una conexi6n afin no simetrica, se busca la expresi6n general de las ecuaciones del campo derivadas de un principio variacional. Se parte de 1a cxpresi6n mas general del integrando dentro de ciertas condiciones naturales de simplicidad y se observan las influencias que sobre las ecuaciones finales tienen las condiciones de pseudo-hermiticidad y lambda-invariancia propuestas pOl'

Einstein.

M. CASTAGNINO (Universidad Nacional del Litoral).

Sobre las t6rmulas de Frenet··Serret para las curvas nulas de una

V. riemanniana a metrica hiperbelica normal.

Con el metodo de las "referencias m6viles'" de Cartan, se hallan las formulas de Frenet-Serret para las curvas nulas de una V.. Se define longitud de pseudu-areo y se hallan las interpretaciones geometricas de las curvaturas.

R. SCARFIELLO (Universidad de Buenos Aires).

Sobre la transtormada de Han7cel del espacio de tunciones indetinidamente diterenciables a decrecimiento rapido.

Se demuestra la siguiente propiedad: Si mada de Hankel de

t

(t)

SR+

entonces la transfor-

t(t2)

pertenece tambien a

SR+.

C. CALDERON (Universidad de Buenos Aires).

Sobre tunciones harm6nicas representables por integrales de Poisson-Lebesgue.

Se dan condiciones suficientes para que una funci6n harm6nica en el interior de la esfera unit aria n-dimensional sea representable por una integral de Poisson-Lebesgue. v.

PEREYRA (Universidad de Stanford, E. Unidos).

La correcci6n diterencial en problemas de contOTno, no lineales y de clase M.

(Presentada pOl' P.

Zadunaisky) .

En 1949 Fox ha introducido en forma sistematica 1a noci6n de correcci6n diferencial. En 1957 y 1962, en dos de sus libros, hace un extenso uso de esta tecnica, en problemas que varian des de ecuaciones diferenciales ordinarias hasta ecuaciones integrales.

El prop6sito de esta comunicaci6n es discutir algunos aspectos de la aplieaci6n de la correcci6n diferencial a la soluci6n numerica de problemas de contorno del tipo,

y"

=

t(x,y)

con divel'sas hip6tesis sobre

t

(x,y).

Yea)

a

y(b) fJ

-148-

Se ha demostrado que la aplicaci6n de una correcci6n diferencial (generalizada) permite mejorar una soluci6n aproximada en al menos dos 6rdenes en el paso

h.

Esta presentaci6n generalizada tambien permite introducir la correcci6n diferencial de diferentes maneras, de las cuales se .han estudiado tres. Se dan resultados numericos obtenidos en la Burroghs-5000 del Stanford

Computation Center, Stanford, Calif. y se hace un esquema de posibles areas de investigaci6n en esta tecnica.

H. FELIOIANGELI (Universidad Nacional de Asunci6n).

Sobre la posibilidad de representar los enteros en

~~n

sistema de base negativa.

ao

Se establece un isomorfismo entre los enteros y las sumas de la forma

+

a,(-q)

+ a.(~q)·

+ ... con

q>

1 y los coeficientes

ai

no negativos y menores que q.

A. MARTESE (Universidad de Buenos Aires).

Estimaci6n de errores en la integraci6n m,merica de sistemas de ecuaciones dife1-enciales ordinarias me-

diante.~~etodos

de paso multiple.

EI metodo (motivado por los estudios realizados en el Seminario superior del Prof. Zadunaisky) tiene la ventaja de eludir los obstaculos que se presentan al usar los resultados de la teoria asint6tica de Peter HenricL

J. C. MERDO Y R. PANZONE (Universidad de Buenos Aires):

Sobre el problema de la extensi6n de medidas.

Sea (0,

Q,

P)

un espacio de probabilidad y

Q'

una ".-algebra conteniendo a la ".-algebra

Q.

EI problema es dar condiciones para la existencia de una probabilidnd P' sobre

Q'

tal que en

Q,

P'

=

P.

E. OKLANDER (Universidad de Buenos Aires).

Interpolaciones tipo Lorentz.

Se introducen interpoladores que dan los espacios de Lorentz cuando se les aplica al par

L\ Loo.

Por medio de estos interpoladores se demuestra el teorema de Marcinkiewicz-Calder6n y se da una :versi6n abstracta del mismo.

P. J. ARANDA, E. CATTANEO Y E. OKLANDER (Universidad Nacional del Litoral,

Universidad de Buenos Aires).

Una demostraci6n de la continuidad de los operadores potenciales.

Se prueba la continuidad de los operadores potenciales (teorema de Cotlar y Panzone, Ilin) por un metodo mas sencillo basado en una idea de R. O'Neil, que consiste en expresar el nucleo como una suma de una funci6n acotada y otra de soporte acotado.

M. M. HERRERA (Universidad Nacional de La Plata):

Sobre la extensi6n de corrientes.

Sea X una variedad compleja y M un conjunto analitico complejo de X de dimensi6n

p;

sea

Mp

la variedad de los puntos regulares de dimensi6n

p

de

M

y

oeM)

=

M -Mp

la parte singular de

M.

-149-

P. Lolong ha demostrado que existe una corriente sobre

X,

o--eontinua, ceo rrada y de dimension

p,

cuya restriccion sobre X -

a(M)

es Ill, integracion sobre

Mp

orientada canonicame:i:tte.

Aqui se extiende el 'resultado al caso en que

M

es unconjunto semi·analitico (de acuerdo con Ill, definicion de S. Loj asiewicz) orientado de una variedad analitica real. La correspondiente corriente de integracion sobre

X

no es cerrada en general. Es cerrada si dim

a(M)

~ M 2.

R. A. RWABARRA (Universidad de Buenos Aires).

Sobre represl3ntaeiones de olases de homologf,a de superfioies oerradas.

Por ausencia del autor se leyo el titulo) .

En J. Fac. Sci. Niigata Univ. Sor. I,3, 131·137 (1963) los senores Kane" ke-Aeki-Kobayashi dan una condicion suficiente para que una combinacioIi. lineal de l-ciclos canonicos sea representable por una curva simple, Aqui se da

Ill, condicion completa (necesaria y suficiente): que los coeficientes sean globalmente primos, resolviendo Ill, cuestion planteada por Kervairo en Math. Rev., vol. 27, 6261 (junio 1964). Hay generalizacion por variedades de cualquier dimension. o.

VILLAMAYOR

Y

D. ZELINSKY (Universidad de Buenos Aires).

Derivados de funtore8 oompue8tos.

(Por ausencia de los autores se leyo el titulo).

Metodo que sustituye las sucesiones espectrales para calcular los funtores derivados de un funtor compriesto bajo condiciones particulares. Se puede conseguir mayor informacion que con Ill, sucesion espectral.

A. LEVINE (Universidad de Buenos Aires).

Extrapolation problem in polynomial regre88ion.

When the variables

Y.l ...•... y ... in polynomial regression are uncorrolated but when the number of observations at each observation point is fixed and/or when the variance of the y's are not equal, it is shown how to select observation points so that the variance of the predicted value of

E(y.)

for t beyond the interval of observations is bounded by a function of

t.

The problem of extrapolation when .the variables are correlated is also treated and corresponding results are obtained.

G. KLIMOVSKY (Universidad de Buenos Aires).

Si8temas inferenoiales para las l6gioas polivalente8 y

8U8 teoremas de eliminaci6n.

Se definen "secuencias" para las logicas polivalentes indicandose su interpretacion semantica y sus teoremas de eliminacion.

Sur).

Computadoras ternarias.

(Por ausencia de los autores Be leyo el titulo) .

Si el costo del equipo requerido para estatizar informacion numerica es proporcional a la base del sistema numerico utilizado, la base entera mas economica es 3. Sin embargo, la mayoria de las computadoras desarrolladas hasta

-150el presente utilizaron base 2 (u otras bases en ·codificacion binaria) debido a las dificultades experiIllentales para desarrollar elementos fisicos confiables y facilmente reproducibles con mas de dos estados discriminables. El re· ciente desarrollo de elementos de ese tipo confiere nuevo interes al desarrollo

. de computadoras tern arias.

En este trabajo se presenta un metodo de analisis y sintesis de sistemas digitales ternarios utilizando operaciones de umbral. Se definen primero dichas operaciones y las formas normales, se expone una tecnica de minimizacion (que resulta de una extension del metodo de Quine) y se describen los metodos de analisis y sintesis de circuitos trivalentes y su implementacion con transistores y fenitas convencionales. Finalmente se comparan dos sumadores sincronicos que procesan informaci6n en base 2 y en base 3 respectivamente, determinandose la relacion entre sus factores de merito en funcion de la relacion entre la cantidad de informacion procesada y el numero de etapas en paralelo.

L.

ITURRIOZ

(Universidad N. del Sud):

Axiomas para el caloulo proposioional trivalente de Lukasiewicz.

En este trabajo se expone otra axiomatica para el calculo proposicional trivalente de Lukasiewicz, diferente a la dada por Gr. Moisil

(Logique Mo-

dale, Disquis. Mathem. et Phys., 2 (1942) p. 3 - 98).

Este calculo proposicional esta construido con el alfabeto 1\ , V , - ,

{Oi}ieI' (,), los axiomas (esquemas):

II) a-)-

(b~a)

12)

(a~(b~o»~«a~b)~(a~o»

C1)

(a

1\ b)

-+

a

C2)

(al\b)~b

C3)

(o~a)~«o~b) ~(o~(aAb»)

D1)

a~

V

b)

D2) b

~ V

b)

D3) «b «a

V

b)

~c»

13)

«a,-*c)~b)~«(b-+a)-*b)~b)~b)

Nl _ _ a~a

N2)

a~

N3)

(a 1\""" a)

~ V ,....,

b)

Y las reglas:

R1)

Modus Ponens

R2)

a~

,....,b~,....,a

Estos axiomas muestran que el calculo proposicional trivalente de Lukasiewicz es una extension del calculo proposicional trivalente de Hilbert-Bernays, al cual se Ie agrega un conectivo, _, y una nueva regIa de deduccion.

-151-

A. MONTEIRO (Universidad

N. del Sur),

Generalizaoi6n de un teorema de Sikorski sobre algebras de Boole.

Sea

A

un algebra de Boole,

C

un algebra de Boole completa,

d

una apIicacion de

A

en

C

tal que:

d(l)

=

1 ,

d(x

V

y)

=

d(x)

V

dey) ,

8 una sub-algebra de

A, h

un homomorfismo de

S

en

C,

tal que:

h(s)

~ entonces existe un homomorfismo

H

de

A

en

C,

tal que:

1

0 )

H(s)

=

h(s) para todo s.S,

2

0 )

H(a)

~d(a) para todo

a.A,

Este resultado generaliza un teorema obtenido por R. Sikorski (Ann. Soc.

Pol. Math., 21 (1948), p. 332-335).

ROBERTO OIGNOLI (U niversidad

N acional del Sur).

Elementos booleanos en las algebras de Lukasiewioz trivalentes.

Un

algebra de Morgan normal

0

algebra de Kleene (A.K.) es un ~istema.

(A,

1,

V,

1\,- ), donde

(A,

1, V,

1\)

es un reticulado distributivo con ultimo elemento y una operacion unaria sobre

A

que satisface los siguientes axiomas:

M

1) -

(x

V

y)

M

2) -

x

=

x

= -

1\ -

Y

K) x

1\ -

x

~

y

V -

y

(Para referencias sobre estas algebras ver A. Monteiro:

Matrioes de Mor-

gan oaraoteristiques pour Ie caloul propositionnel olassique, Anais Acad. Bras. ei.,

32 (1960),

1-7).

A. Monteiro ha probado (en un trabajo no publicado). que Ia nocion de

algebra de Lukasiewicz trivalente (A.L.T.) introducida por Gr. Moisil (.Re-

cherches sur l'algebra de la logique, Ann. Sci. Univ. Jassy,

22 (1935),

1-117) es equivalente a Ia de A. K. sobre Ia que esta definida un operador

V que satisface los postulados siguientes:

L 1)

V

(x

1\ y)

L

2)

x

1\ -

x

=

V

(x

1\ y)

1\

V

x

1\

V

Y

= -

x

1\

V

x

L

3) -

x

V V

x

=

1

Llamaremos boolcanos a los elementos de un A. K. que tienen complemen-

-152to booleano. Sea

A

un A. L. T. Y designemos con

B

la familia de sus elementos booleanos. Entonces

B

foza de las siguientes propiedades:

B 1)

B

2)

b

e

B

si y solo si

b

=

V

b

B e8 relativamente completa inferiormente, es decir, para todo x

3

A,

e1 conjunto

B",

= {

b e B : formula:

x:::;;

b} tiene primer elemento, y ademas vale la

B 3)

VX

B

es separadora: si

= /\

Bx (1) x, yeA

y no

x:::: y,

entonces

0 existe un·

b

e

B

tal que

y:::;;

b Y no

x::::

b, oexiste un b' e B tal que b' :::;;

x

Y no b' ::::

y.

(Estos resultados pueden verse en A. Monteiro:

Sur la definition des algebras de Lukasiewicz trivalentes,

a publicarse en el Bull. Soc. Sc. Math. R.

P. Roumaine).

En el presente trabajo probamos que las condiciones

B

2) Y

B

3) son suficientes para que en un A.K., la formula (1) defina un operador que satisfaga

L 1), L

2)

1

L

3). Ma.s precisamente, probamos el siguiente

Teorema:

Para que un algebra de Kleene admita una estructura de algebra de Lukasiewicz trivalente, es necesario y suficienie que la familia de sus elementos booleanos sea relativamente completa inferiormente y separadora.

ANTONW MONTEIRO Y ROBERTO CIGNOLI (Universidad Nacional del Sur).

Construcci6n geometrica de las algebras de Lukasiewicz trivalentes libres.

Un

algebra de Lukasiewicz trivalente

(A.L.T.) es un sistema

(A,

1, V f:,., ,

V) donde

(A,

1, V,I\) es un reticulado distributivo con ultimo elemento y y

V son operaciones unarias definidas sobre

A

que satisfacen los siguienies axiomas:

M 1)

M

2)

L 1)

L

2)

L

3)

(x

V

y)

- - x

=

x

= -

x

1\ - . y

- x V \ 7 x = l x l \ - x = - x l \ \ 7 x

\7 (x 1\ y)

=

\7

x 1\

\7

y

(Para vel' que esta definicion coincide con la original de Gr. Moisil as! como otras referencias sobre estas algebras, consuItar A. Monteiro:

Sur la definition des algebres de Lukasiewicz trivalentes,

a publicarse en Bull. Soc. Sci.

Math.

R.

P. Roumaine).

En esta nota indicamos una construccion geometrica del A.L. T. libre cou un conjunto de generadores libres

G

de potencia arbitraria. La existencia y unicidad de una tal algebra resulta de teoremas generales de algebra universal

(G. Birkhoff).

Sea

B

=

I

Po, P" P2, P3}

un conjunto con cuatro puntos donde esta definida una involuci6n <p de la siguiente forma:

<p (Po)

=

P3 ;

<p (P,)

=

P, ; <p

(P2)

=

P2

<p

(P3) .

=

Po

-153-

Sea I un conjilllto de cardinal g, y sea E = II Bi, donde Bi = B para to~

i e

I

do

i e

I. Indicaremos a los puntos de E con la notacion x =

[xi]'

Sea E' e1 sUbconjunto de

E

formado por los elementos que no tienen simultaneamente his coordenadas

Po Y P31 es decir, E' = E { x e E :

~

j,

k e I tales que

Xj

=

Po

Y

xi

=

P. '} •

Definamos ahora sobre E' la involucion </> por medio de la siguiente formula:

Si

x

=

[x;],

entonses </>

(x)

=

[<j;(x i )]

Para toda parte X

C

E' definimos:

- X

=

C </>(X)

(1)

(C

indica complemento) y

V'X

=X

U</>(X) (2)

Pongamos,paracadaieI,G i

=

{xEE':xi=po 0 xi=P,{ yseaG= {G

i }.

Es claro que potencia de G = potencia de I

=

g.

Sea, finalmente, £ la menor familia de partes de E' que contiene a los G. y que es cerrada con respecto a

U, n, y a las operaciones-

y V' definidas por (1) y (2) respectivamente. Entonces probamos el siguiente

Teorema: £ es el A.L.T. que tiene por generadores libres los G

i .

La tecnica de la demostracion es similar a la que se usa en la construccion de las algebras de Morgan libres. (A. MONTEIRO y O. CHATEAUBRIAND:

Algebras

de Morgan Libres, no publicado).

M. A. DICKMANN, (Universidad N. del Sud).

La independencia del axioma de eleccion en la teoria NBG.

Paul Cohen ha probado en 1963 la indemostrabilidad del axioma de eleccion (en

10 sucesivo denotado por Z) en la teo ria ZF (Zermelo-Fraenkel) sin individuos y con axioma de regularidad (P. COHEN,

The Independence of the

Axiom of Choice. Nota de la exposicion correspondiente en el Simposio de Teoria de Modelos. Stanford Univ. Julio 1963); la prueba de Cohen no satisface los requisitos corrientes de rigor usados por los matematicos; aparte de esto,· presupone la existencia de un modelo standard fuertemente regular; la existencia de tal modelo puede ser justificada usando el teorema de Lowenheim-

Skolem,

0 por ejemplo, asumiendo un axioma que garantice la existencia de ordinales fuertemente inaccesibles.

EI proposito de este trabajo es ofrecer una prueba de la consistencia relativa de la negacion del axioma de eleccion en la teo ria NBG (Von Neumann-Bernays-Godel), en la misma forma que K. Godel ha probado en 1940 la consistencia de tal axioma. Hemos elegido la teoria NBG, pues esta permite justificar ciertas construcciones, que resultan muy problematicas en ZF (cf.

-154-

Journal oy Symbolic Logic, V 116-7)_ La demostracion consiste en adaptar las ideas de un reciente trabajo de P. Vopenka ("La Consistencia de la Hip6tesis del Continuo" (en ruso). Commentationes Mathematicae Universitatis Carolinae. Julio 1964) con el objeto de combinarlas con los metodos clasicos de teoria de grupos desarrollaClos por Fraenkel-Mostowski-Mendelson. Definimos una interpretacion de la teoria NBG-+ - Z en la teoria NBG

+

Z

consistente en tomar como "conjuntos' ciertas funciones con dominio en determinado es~ pacio topologico, a valores ordinales y cuyo contradominio satisface cierta condicion de simetria respecto a extensi9nes de transposiciones de

w,

como "clases", ciertas clases de tales funciones y como relacion de "pertenencia" cierta relacion binaria que definimos a parti!' del concepto de "forzar" introducido por Cohen. La existencia de esta interpretacion resuelve el problema, puesto que si NBG es consistente, 10 es NBG

+

Z,

y por 10 tanto NBG

+ -

Z.

El pun-

'to central de la demostracion consiste en formalizar la nocion de "forzar" y definir la nocion de ,"simetria" involucrada en las funciones que constituyen la interpretacion dada.

E. ROFMAN (Universidad N. del Litoral): Sobre una desigualdad a que sa-

tisfacen las integralos de funciones convexas.

Se presenta la solucion de una inecuacion, haciendo uso reiterado de una desigualdad relativa a funciones convexas. Se agrega una demostracion elemental de tal desigualdad, a la vez que se generaliza la validez de la misma para el caso de integrales multiples.

Al finalizar las sesiones de comunicaciones el lunes por la manana,

la

UMA of recio, en las instalaciones de Nunez, un asado de camaraderia a las delegaciones.

La Union Matematica Argentina deja constancia de su agradecimiento a

, la Facultad de Ciencias Exactas y Naturales, a la Facultad de Ingenieria y al Consejo Nacional de Investigaciones Cientificas y Tecnicas por la ayuda prestada que ha permitido la realizacion de la Reunion.

UNION lVIATElVIATICA ARGENTINA

MIEMBROS HONORARIOS

Tulio Levi-Civita (t) ; Beppo Levi (t) ; Alejandro Terracini; George D. Birkhoff (t); Marshall H. Stone; Georges Valiron (t); Antoni Zyg!ymnd; Godofredo Garcia; Wilhelm Blaschke (t); Laurent Schwartz; Charled Ebresmann;

Jean Dieudonll(3; Alexandre Ostrowski;

REPRESENTANTES EN EL EXTRANJERO

Dr. Godofredo Garcia (Peru). Dr. Leopoldo Nachbin (Brasil). Dr. Roberto

Frucht (Chile). Dr. Mario GonzaJez (Cuba). Dr. Alfonso Napoles Gandara

(Mexico). Alejandro Terracini (Italia).

Este mimero de la Revista de la Union Matematica Argentina y de la

Asociacion Fisica Argentina se ha publicado con 1a contribucion del Consejo

N acional de Investigaciones Cientificas y Tecnicas. Tal contribucion no significa que el Consejo asuma responsabilidad alguna pOI' el contenido del mismo.

PUBLICACIONES DE LA U. M.

A.

Revista.de la U.M.A. -

Vol. I

(1936-1937);

Vol. II

(1938-1939);

Vol. III

(1938-1939); Vol. IV ,(1939); Vol. V (1940); Vol. VI (1940-1941); Vol.

VII (1940-1941); Vol. VIII (1942); Vol. IX (1943); Vol. X (1944-1945).

Revista de la U. M. A. y de la A. F. A. -

Vol. XV

(1951-1953);

Vol. XVI

XII (1946-1947); VoL XIII (1948); Vol. XIV (1949-1950).

Revista de la U. M. A. y de .la A.

F.

A. -

Vol. XV

(1951-1953);

Vol. XVI

(1954-1955);

Vol. XVII

(1955);

Vol. XVIII

(1959);

Vol. XIX

(1960-

1962);

Vol. XX

(1962);

Vol. XXI

(1963).

Los volllmenes III, IV, V y VI comprenden los siguientes fasciculosseparados:

N0 1. GINO LORIA.

GoNZALEZ DOMiNGUEZ.

Le Matematiche in Ispagna e in Argentina. -

NQ 2 A.

Sobre las series de funciones de Hermite. NQ 3. MI-

CHEL PETROVICH.

Remarques arithmetiques sur une equation differentielle du

premier ordre. -

NQ 4. A. GONZALEZ DOMiNGUEZ.

Una nueva demostracion del teorema limite del Calculo de ProbaMlidades. Condiciones necesarias y suficien-'

tes pam que una funcion ilea integral de Laplace. -

N° 5. NIKOLA OBRECHKOFF.

Sur la somm.ation absolue par latransformation d'Euler des ser'ies diverg'entes.

NQ 6. RICARDO SAN JUAN.

Derivacion e integracion de series as'intoticas. -

NQ 7. Resoluci6n adoptada por la U. M. A. en la cuesti6n promovida por el

Sr. Carlos Biggeri. N° 8. F. AMODEO.

Origen y desarrollo de la Geome-

tria Proyectiva. -

N0 9 CLOTILDE A. BULA.

Teoria y calculo de los momentos

dobles. NQ 10. CLOTILDE A. BULA_

Calm!lo de superficies de frecuencia.

N° 11. R. FRUCH'l'_

Z~tZ

Geometria aUf ein81' Flache mit indefiniter Metrile

(Sobre la Geometria de una superficie con metrica 'indefinida). NQ 12. A.

GONZALEZ DOMINGUEZ.

Sobre una memoria del Prof. J,

C.

Vignaux_ -

13.

E. TORANZOS.

BALANZAT.

Sobre las singularidades de las cUI-vas de Jorda.n. -

FOl'mulas integmles de la interseccion de

conj~mtos. -

NQ

14.

15.

M.

G_

KNIE. El problema de varius electTones en la mecanica cuantista. NQ 16.

A. TERRAClNL

Sobre la existencia de superficies cuyas lineas principales son

dadas. -

N° 17 _ L. A. SANTALO.

Valor medio del numero de parte en que una figum con'vexa es dividida por

n

rectas arbitmTias. -

N° 18. A. WINT-

NER.

On the itemtion of distribution functions in the CG,lculus of probability

(SobTe la itemcion de funciones de distribuci6n en el ealculo de probabilida-

des). -

N° 19. E. FERRARI.

Sobre la pamdoja de Bertrand. -

N0 20. J. BA-

BINI.

Sobre algunas propiedades de la derivadas y ciertas primitivas de los

polinomios de Legendl-e. NQ 21. R. SAN JUAN.

Un algoritmo de sumacion

de series divergentes. NQ 22. A. TERRACINI.

Sobre algunos lugares geome-

tricos. -

NQ 23. V. y A. FRAILE Y C. CRESPO.

El lugar geometrico y lugares

de puntos al'eas en el plano. -

NQ

24. R.

FRUCHT.

Coronas de grupos y sus

subgru,pos, con una aplicacion a los determinantes. -

NQ

25. E. R.

RAIMONDI.

Un problema de probabilidades fleometricas sobre los conjuntos de triangulos.

En 1942 la U. M. A. ha iniciado la publicaci6n de una .nueva serie de

"Memorias y monografias" de las que han aparecido hasta ahora las siguientes:

Vol. I; N" 1. GUU,LERMO KNIE,

Mecanica ondulatoria en el espacio curvo.

NQ 2, GUIDO BECK,

El espacio fisico.

NQ 3. .JULIO RE" PASTOR,

Integrales paTciales de las funciones de dos'variables en intm'valo infinito.

NQ 4.

JULIO HEY PASTOR,

Los ultimos tem'e1nas geometricos de PoincaTe y sus aplicaciones.

Homenaje p6stumQ al Prof. G. D. BIRKHOFF.

Vol. II; NQ 1. YANNY FRENI{EL,

Criterios de bicompac'idad y de H-complctidad de

ifn

espcwio topol6gico accesible de Frecht-Riesz.

N° 2. GEOR-

GES VALIR.ON,

Fonctions ent'iel'es.

. Vol. III; N° 1. E. S. BERTOMElJ Y C. A. MALI,MANN,

1m

generadol' encascadas de alta tension.

Funcionamiento de

Ademas han aparecido tres cuadernos de

].iisnelanea Matematica.

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