The principle of symmetry–extremity

Applied Mathematics and Computation 137 (2003) 293–301 www.elsevier.com/locate/amc

The principle of symmetry–extremity Laszlo Antal Veress

1

Department of Algebra, Mathematical Institute, Budapest University of Technology and Economics, 1111 Budapest, Muegyetem Rkp 1-3, Hungary

Abstract There are several examples which show us a certain relationship between the notion of symmetry and extremity that is when some ‘‘object’’ has a higher symmetry certain quantity (concerning a given property of the object) takes extreme value. It is enough to think about the well-known isoperimetric problem. So by above mentioned quantity we mean a measure defined on some r-ring. In this paper we would like to study that relationship. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Group action; Measure; Symmetry group

1. Introduction As we have already mentioned, we would like to establish some relationship between symmetry and extremity and to conclude our study into a stated principle that we will call the principle of symmetry–extremity. Before all these we make a post-remark concerning the principle of symmetry of states (see [1] and the end of [2] and the radioactive disintegration). Remarks. (a) We have conjected at the end of [2] that an intermediate nucleus of radioactive chain has higher symmetry of states than the anterior. An experimental confirmation of that fact can be found if we take the curve of the

E-mail address: [email protected] (L.A. Veress). Present address: Baktay Ervin Grammar School and Technical School, 2330 Dunaharaszti, Baktay ter 1 sz., Hungary. 1

0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 2 4 - 8

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binding energy per nucleon viewed like a function of the mass number A ([3], Fig. 6, p. 1081). So if we consider the portion A 2 ½200; 240 of the radioactive nucleus we can see that if A is decreasing (through a decay A ! A  4) then the curve (viewed from right to left) is rising that means at the same time a higher symmetry of states. For b decay the neutron number N ! N  1 while the proton number Z ! Z þ 1 that means also a step to more stability ([3], Fig. 4, p. 1079) meaning a higher symmetry of states. (b) Since Oð3Þ ffi SOð3Þ Z2 [4, p. 54] and since we have parity violation through b decay we conclude that Gb contains Z2 and Ga contains a subgroup of SOð3Þ ðGa \ Gb ¼ fegÞ which means that we can make a conjecture about rotational-symmetry violation through a decay (see [1,2]).

2. Relationships between symmetry and extremity Consider ðX ; S; lÞ where X is a separable locally compact Hausdroff measure space [5, p. 73] S the r-ring of the Baire sets which is identical to the r-ring of Borel sets [5, Theorem 50.E] and l a Baire measure defined on S. Let G be a compact topological group such that X is a G-set. (We consider G being a subgroup of F is the whole group of topological automorphisms of X i.e. the action of g 2 G on X, is a topological map from X on X.) Suppose that S and l are left invariant that is if E 2 S and g 2 G then gE ¼ fgx: x 2 Eg 2 S and lðEÞ ¼ lðgEÞ ð8 E 2 S and 8 g 2 G). First we show that the actions of the elements of G on the subsets of X are group actions too. Indeed eE ¼ fex: x 2 Eg ¼ f x: x 2 Eg ¼ E ðg1 g2 ÞE ¼ fðg1 g2 Þx: x 2 Eg ¼ fg1 ðg2 xÞ: x 2 Eg ¼ g1 f g2 x: x 2 Eg ¼ g1 ðg2 EÞ If X is a locally compact topological group l can be the Haar meausre, G a compact topological subgroup of X itself; if X is an Euclidean space l can be the area in two dimensions or the volume in three dimensions and G a compact topological group of isometries. First of all we turn to the topology of F. Let g 2 F and x 2 X . Let Vx a neighborhood of x. Then gVx is a neighborhood of gx that we denote by Vgx . We associate to Vgx a subset of Vg of F in the following way: 
Vg ¼ g0 : g0 x 2 Vgx : ¼ gVx Since x is arbitrarily chosen we get that if g0 2 Vg then g0 ðgxÞ 2 gVgx ¼ gðgVx Þ for some g 2 F . So Vg is independent of x 2 X . We show that the sets Vg forms a complete system of neighborhoods for F. If g1 6¼ g2 then there exists x 2 X such that g1 x 6¼ g2 x. Therefore there exists Vg1 x such that g2 x 62 Vg1 x . Let Vg1 the subset of F associated to Vg1 x . Thus g2 62 Vg1 because if g2 2 Vg1 then g2 x 2 Vg1 x which

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means a contradiction. Let Ug and Vg be two neighborhoods of g. It means that Ug is associated to some Ugx and Vg to some Vgx . But there exists Wgx  Ugx \ Vgx . Let Wg  F , Wg ¼ fg0 : g0 x 2 Wgx g. So g0 x 2 Ugx that is g0 2 Ug and g0 x 2 Vgx that is g0 2 Vg . So Wg  Ug \ Vg . We show that F is a topological group relative to the above defined topology. Let g1 g2 ¼ g and Vg an arbitrary neighborhood of g. We have to show that there exists Vg1 and Vg2 neighborhoods of g1 and g2 respectively, such that Vg1 Vg2  Vg . Since g1 g2 ¼ g therefore ðg1 g2 Þx ¼ gx 8 x 2 X . Let x be arbitrarily chosen from X, Vgx an arbitrary neighborhood of gx and Vg the neighborhood of g associated to Vgx . SoVg is indeed an arbitrary neighborhood of g. If g0 2 Vg then g0 x 2 Vgx ¼ gVx ¼ ðg1 g2 ÞVx ¼ fðg1 g2 Þx0 : x0 2 Vx g ¼ Vðg1 g2 Þx Let Vg2 ¼ fg20 : g20 x 2 Vg2 x g. But Vg2 x ¼ Vg20 x . Consider now Vg1 that neighborhood of g1 which is associated to Vg2 x ¼ Vg20 x such that Vg1 ¼ fg10 : g10 ðg20 xÞ 2 Vg1 ðg20 xÞ ¼ Vg1 ðg2 xÞ ¼ Vðg1 g2 Þx g since g1 Vg2 x ¼ Vðg1 g2 Þx contains g1 ðg20 xÞ and so we look at Vgx as a neighborhood of g1 ðg20 xÞ. Thus Vg1 Vg2  Vg . Finally, we show that if Vg1 is an arbitrary neighborhood of g1 2 F then there exists Vg a neighborhood of g 2 F such that Vg1  Vg1 . Let x 2 X and g 2 F be arbitrarily chosen. Let Vx be a neighborhood of x and Vg1 ¼ fg00 : g00 ðgxÞ 2 g1 Vgx ¼ Vg1 ðgxÞ ¼ Vx g and let Vg ¼ fg0 : g0 x 2 gVx ¼ Vgx g Thus Vg1 ¼ fg01 : g0 2 Vg g. Since g1 is a topological map therefore g1 ðg0 xÞ 2 g1 ðgVx Þ ¼ Vx for all g0 2 Vg . So ðg1 g0 Þx 2 Vx that means that g1 g0 2 the stabilizer of Vx . Since the stabilizer of any ‘‘object’’ (and so of Vx ) is a group, 1 we get that ðg1 g0 Þ ¼ g01 g 2 the stabilizer of Vx and so g01 ðgxÞ 2 Vx (for all 0 1 g 2 Vg Þ. Thus Vg  Vg1 . 1 Let us have G1 ¼ feg  G2      Gi  Giþ1     H ¼ Ui¼1 Gi  G where G1 ; G2 ; . . . are finite groups, Gi ði P 1Þ is a proper nontrivial subgroup of Giþ1 such that jðGiþ1 : Gi Þj ¼ k P 2, 8 i ¼ 1; 2; . . . (For instance Giþ1 ¼ D2n , Gi ¼ Dn and k ¼ 2). We assert that H is a group. If g1 ; g2 2 H then g1 2 Gs , g2 2 Gq where s 6 q that is g1 2 Gq and g1 g2 2 Gq  H . If g1 ; g2 ; g3 2 H then g1 2 Gs , g2 2 Gq , g3 2 Gp where s 6 q 6 p, that is g1 ; g2 ; g3 2 Gp  H and so the associative law holds because Gp is a group. Finally if g 2 H , then g 2 Gs therefore g1 2 Gs  H . It is obvious that e 2 H . Consider A1 2 S such that 0 < lðA1 Þ < 1. We construct the following set A2 ¼ [g2G2 gA1 . Since S is left invariant and since S is a r-ring we have that A2 2 S. We observe that A1  A2 and so lðA1 Þ 6 lðA2 Þ. If g0 2 G2 then g0 A2 ¼ A2

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(that is A2 is G2 -invariant even if e is eventualy the only element of H which leave A1 fixed). Indeed g0 A2 ¼ g0 ð[g2G2 gA1 Þ ¼ [g2G2 g0 gA1 ¼ A2 . Similarly A3 ¼ [g2G3 gA2 and if g0 2 G3 then g0 A3 ¼ g0 ð[g2G3 gA2 Þ ¼ [g2G3 g0 gA2 ¼ A3 . So A3 is G3 -invariant. If Aiþ1 ¼ [g2Giþ1 gAi , i P 1 then is evident that Aiþ1 is Giþ1 invariant. Thus we obtain an increasing sequence of measurable sets: A1  A2  A3      Ai  Aiþ1      B ¼ [1 j¼1 Aj ¼ lim ðAj Þ j!1

Since Aj 2 S, j ¼ 1; 2; . . . and since S is a r-ring we get that B 2 S. l being a measure we have that lðA1 Þ 6 lðA2 Þ 6 lðA3 Þ 6    6 lðAi Þ 6 lðAiþ1 Þ 6    6 lðBÞ and lðBÞ ¼ lðlimj!1 Aj Þ ¼ limj!1 lðAj Þ [5, Theorem 9.D]. The set B ¼ [1 j¼1 Aj is H ð¼ [1 G Þ-invariant. Indeed let g be any element of H. Thus there exists i i¼1 Gi  H such that g 2 Gi . But B ¼ Ai [1 j¼iþ1 Aj . We have seen that Ai is Gi invariant and Aj , j > i, being Gj -invariant, is Gi -invariant too (because Gi  Gj ). Therefore 1 1 gB ¼ gðAi [1 j¼iþ1 Aj Þ ¼ gAi [j¼iþ1 gAj ¼ Ai [j¼iþ1 Aj ¼ B

In the similar way it can be constructed an other sequence using intersection \ instead of the union [: A1 ¼ A01 ; A02 ¼ \g2G2 gA1 ; . . . ; A0iþ1 ¼ \g2Giþ1 gA0i ; . . . It is obvious that A0i is Gi invariant 8 i ¼ 1; 2; . . . and we define again B0 ¼ 0 0 \1 j¼1 Aj ¼ limj!1 Aj since B0      A0jþ1  A0j  A0j1      A01 ¼ A1 Thus 0 6 lðB0 Þ 6    6 lðA0jþ1 Þ 6 lðA0j Þ 6    6 lðA01 Þ ¼ lðA01 Þ ¼ finite So lðB0 Þ is finite. We note that B0  A1 ¼ A01  B where, B0 and B are H-invariant. We are interested in the finiteness of lðBÞ which is finite if and only if the sequence lðAj Þ is a Cauchy sequence, and so if and only if lim ½lðAjþ1 Þ  lðAj Þ ¼ 0

j!1

that is if l½ðAjþ1  Aj Þ [ ðAj  Ajþ1 Þ ¼ dðAjþ1 ; Aj Þ ! 0ðj ! 1Þ where dðE; F Þ ¼ lðEDF Þ ¼ l½ðE  F Þ [ ðF  EÞ is the distance defined in the metric space of measurable sets of finite measure. Thus lðAjþ1  Aj Þ ¼ lð[g2Gjþ1 gAj  Aj Þ ¼ lð[g2Gjþ1 ðgAj  Aj ÞÞ.

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If g2 2 g1 Gj ðg1 ; g2 2 Gjþ1 Þ then g2 Aj ¼ ðg1 gj ÞAj ¼ g1 ðgj Aj Þ ¼ g1 Aj . Since Aj is Gj -invariant we get that lð[g2Gjþ1 ðgAj  Aj ÞÞ 6 lðg1 Aj  Aj Þ þ    þ lðgk1 Aj  Aj Þ where g1 ; g2 ; . . . ; gk1 2 Gjþ1  Gj and gm Gj 6¼ gn Gj ðm 6¼ n; m; n ¼ 1; 2; . . . ; k  1Þ. We see that ðlðg1 Aj  Aj Þ þ    þ lðgk1 Aj  Aj ÞÞ ! 0 ðj ! 1Þ suffice for the convergence of lðAj Þ (and therefore for the finiteness of lðBÞ). It follows from LagrangeÕs theorem that jGj j !P1 ðj ! 1Þ ðjGj j P j P 1Þ. k1 But a sufficient condition for m¼1 lðgm Aj  Aj Þ ! 0 is lðgAj  Aj Þ 6 a c=ðjGj j Þ 8 g 2 Gjþ1 , j ¼ 0; 1; 2; . . . ; where a and c are positive real numbers. In that case a

lðAjþ1 Þ  lðAj Þ 6 ðk  1Þc=ðjGj j Þ 6 ðk  1Þc=ðja Þ ! 0 ðj ! 1Þ [6, Theorem 3.20(a)] (we emphasize the fact that k and c are constants which do not depend on j). Remarks. (a) l is left invariant therefore lðgAj  Aj Þ ¼ l½gðAj  g1 Aj Þ ¼ lðAj  g1 Aj Þ Hence if lðgAj  Aj Þ ! 0 so is lðAj  g1 Aj Þ. (b) In the case when Gj is normal in Gjþ1 ; j ¼ 1; 2; . . . (for example when k ¼ 2) k1 X

lðgm Aj  Aj Þ ¼

m¼1

k1 X

lðAj  gm1 Aj Þ ¼

m¼1

k1 X

lðAj  gm Aj Þ

m¼1

1 In fact, since gm 62 gn Gj therefore gm1 62 gn1 Gj (otherwise gm1 ¼ n gj and so Pgk1 1 0 gm ¼ gj gn ¼ gn gj 2 gn GP j which means a contradiction). Thus if m¼1 lðgm Aj  k1 Aj Þ ! 0 ðj ! 1Þ so is m¼1 lðAj  gm Aj Þ and since l is non negative we get that lðgAj  Aj Þ ! 0 ðj ! 1Þ implies that lðAj  gAj Þ ! 0 ðj ! 1Þ. But

dðgAj ; Aj Þ ¼ l½ðgAj  Aj Þ [ ðAj  gAj Þ ¼ lðgAj  Aj Þ þ lðAj  gAj Þ g 2 Gjþ1 and hence dðgAj ; Aj Þ ! 0 ðj ! 1Þ if and only if lðgAj  Aj Þ ! 0 ðj ! 1Þ (c) We get the finiteness of lðBÞ if a

dðgAj ; Aj Þ 6 2c=ðjGj j Þ;

ðg 2 Gjþ1 ; a > 0; j ¼ 1; 2; . . .Þ

We have also that dðAjþ1 ; Aj Þ ¼ lðAjþ1  Aj Þ ¼ lðAjþ1 Þ  lðAj Þ 6 ðk  1Þc=ðja Þ ! 0 ðj ! 1Þ

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2.1. The symmetry-optimum case Let us have 1

A1  A2      Ai  Aiþ1      B ¼ [ Aj ¼ lim ðAj Þ j¼1

ð1Þ

j!1

the above mentioned sequence of measurable sets and suppose that m is a measure defined on S too such that we have satisfied the following conditions: 0 < lðA1 Þ < lðA2 Þ <    < lðAi Þ <    < lðBÞ ¼ finite ¼ lim lðAj Þ

ð2Þ

0 < mðA1 Þ < mðA2 Þ <    < mðAi Þ <    < mðBÞ ¼ finite ¼ lim mðAj Þ

ð3Þ

j!1

j!1

Then limj!1 ½mðAj Þ=lðAj Þ ¼ ½mðBÞ=lðBÞ. Thus if ½mðAj Þ=lðAj Þ is a decreasing sequence (that is if lðAj Þ increases more fast than mðAj Þ) then ½mðBÞ=lðBÞ is minimal (viewed like an optimum). If we look at the sequence Aj as to a ‘‘standard’’ sequence in the sense that if Cj is another sequence of sets of S such that Cj is Gj -invariant and C1  C2      Ci  Ciþ1      B ¼ lim Cj j!1

then limj!1 ½mðCj Þ=lðCj Þ ¼ ½mðBÞ=lðBÞ. Since G is compact, if fgj gj2N  H then fgj gj2N can be considered convergent and G being closed so limj!1 gj ¼ g 2 G. Since H (where the overline denotes the closure) is a closed subset of the compact G therefore H is compact [7, 13.A] and is a topological subgroup of G [7, 18.H]. We claim that B is H -invariant. In fact, let g 2 H and x 2 B. Thus x 2 Aj (for some j 2 N Þ. So gx ¼ limj!1 gj x 2 B (we justify that below). Now if y 2 B, then y ¼ limi!1;xi 2B xi and thus gy ¼ limi!1 gxi 2 B (since the closure of B is B). We would like to justify the above used relation, namely: gx ¼ limj!1 gj x (where g ¼ limj!1 gj ). We have to show that if Vgx is a neighborhood of gx there exists gj x 2 Vgx for some j. Let us have Vx a neighborhood of x so gVx ¼ Vgx is an arbitrary neighborhood of gx. Let us have Vg a neighborhood of g associated with Vgx . Since g ¼ limj!1 gj therefore there exists gj 2 Vg and so gj x 2 Vgx that means that gx ¼ limj!1 gj x. In the particular case when B represets a circle in the Euclidean plane H is the full symmetry group of the circle. 2.2. The ‘‘isoperimetric’’ problem Suppose in the sequel that l and m are measures for which we have satisfied the following two conditions: (a) If fAn gn2N is a sequence mentioned in (1) such that conditions (2) and (3) are satisfied then ½mðAn Þs =lðAn Þ is a decreasing sequence where s is a non negative real number s P 1 (the case s ¼ 0 is trivial since lðAn Þ is increasing).

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(b) Suppose that T is a topological subgroup of F isomorphic to Rþ ðtc1 tc2 ¼ tc1 c2 ; c1 ; c2 2 Rþ Þ such that S is T-invariant too (that is tc P 2 S 8 P 2 S, c 2 Rþ ), T commutes with H, and mðtc P Þ ¼ cmðP Þ and lðtc P Þ ¼ cs lðP Þ. For instance T can be the group of similitudes for the Euclidean plane when s ¼ 2 m and l being the perimeter and the area respectively. We show first that if P is Gj -invariant then tc P so is. For, if gj P ¼ P then gj ðtc P Þ ¼ tc ðgj P Þ ¼ tc P . We show now that if gj ðtc P Þ ¼ tc P then gj P ¼ P : gj P ¼ gj ðtc1 ðtc P ÞÞ ¼ tc1 ðgj ðtc P ÞÞ ¼ tc1 ðtc P Þ ¼ P Thus P and tc P have the same symmetry on H. We show now that ½mðAj Þb =lðAj Þ is descreasing 0 6 b < s (the case b ¼ s by assumption): ½mðAjþ1 Þb =lðAjþ1 Þ : ½mðAj Þb =lðAj Þ ¼ ½mðAjþ1 Þ=mðAj Þb ½lðAj Þ=lðAjþ1 Þ s

< ½mðAjþ1 Þ=mðAj Þ ½lðAj Þ=lðAjþ1 Þ <1 Now we formulate an ‘‘isoperimetric’’ problem. If S, m and l satisfy the above mentioned two conditions then if we have a class E  S of sets such that for every set of the class the value of m is the same: mðAÞ ¼ l ¼ const: > 0 then lðAÞ is maximal when A has the maximal symmetry. We take the sequence (1) of sets with conditions (2) and (3). Let us have cj 2 Rþ such that cj mðAj Þ ¼ l 8 j ¼ 1; 2; . . . Thus ½cj mðAj Þ=½csj lðAj Þ ¼ l=½csj lðAj Þ ¼ ½mðtcj Aj Þ=½lðtcj Aj Þ ¼ l=½lðtcj Aj Þ s

¼ ½mðAj Þ =½ls1 lðAj Þ which is a decreasing sequence (by assumption). Thus lðtcj Aj Þ is an increasing sequence and since Aj and tcj Aj 2 E have the same symmetry we get that for the sequence Bj ¼ tcj Aj 2 E (for which mðBj Þ ¼ l ¼ const:), lðBj Þ is maximal in the case of maximal symmetry. Remark. If B ¼ B 2 S then lðBÞ ¼ lðBÞ and so we get that l is maximal in the case of maximal symmetry described by H . Let us see a little application for a sequence of dihedral groups: G1 ¼ D3  G2 ¼ D6  G3 ¼ D12      Gj ¼ D3:2j1  Gjþ1 ¼ D3:2j  Gjþ2 ¼ D3:2jþ1     In that case k ¼ jGjþ1 j=jGj j ¼ 2 and jGj j ¼ 3:2j ,

ð4Þ j P 1. We observe that

lðAjþ1 Þ  lðAj Þ ¼ lðAjþ1  Aj Þ ¼ lðAj [ gAj  Aj Þðg2Gjþ1 Gj Þ ¼ lðgAj  Aj Þ Therefore if the increasing sequence Aj (Aj being Gj invariant: j ¼ 1; 2; . . .) satisfy the established condition (that is if lðgAj  Aj Þ 6 c=ðjGj ja Þ, g 2 Gjþ1 ;

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a > 0; j ¼ 1; 2; . . .) then lðBÞ ¼ lðlimj!1 Aj Þ ¼ limj!1 lðAj Þ will be finite. All these give us the reason to consider for example that A1 is the regular 3––gon (equilateral triangle), A2 the regular 3:21 ––gon; . . . ; Aj the regular 3:2j1 ––gon and so on, inscribed into the circle of radius R (l being the area of our figures in the Euclidean plane). Hence gAj ; g 2 ðGjþ1  Gj Þ is Aj rotated with 2 p=ð3:2j1 Þ. If n ¼ 3:2j1 then lðgAj  Aj Þ ¼ nR2 ð1  cosðp=nÞÞ = tanðp=nÞ which is approximated by nR2 p4 =½ð2!Þ2 n4 tanðp=nÞ 6 nR2 p4 =ð4n4 p=nÞ ¼ p3 R2 =ð2nÞ2 2

¼ p3 R2 =jGj j ða ¼ 2Þ If mðAj Þ is the perimeter of Aj it is obvious that mðAj Þ is also a convergent sequence. In fact, a similar computation shows that (recall n ¼ 3:2j1 ): mðgAj  Aj Þ ¼ 4nR sin½p=ð2nÞf1= cos½p=ð2nÞ  cos½p=ð2nÞg ! 0 ðj ! 1Þ

For that case s ¼ 2 and 2

xn ¼ ½mðAn Þ =lðAn Þ ¼ 4ntgðp=nÞn P 3 Using the Taylor series for tgðxÞ function we get that xnþ1  xn < 0 that is xn is decreasing. Thus if cj mðAj Þ ¼ l ¼ const: then we have that the circle has the maximal area (note that B ¼ B ¼ circle and H is the symmetry of the circle since any rotation is the limit of some sequence in H (see (4)). s

Remarks. (a) ½mðAn Þ =lðAn Þ cannot be decreasing for any s 2 Rþ (that is we have an upper bound for s) since ½mðAjþ1 Þs =lðAjþ1 Þ : ½mðAj Þs =lðAj Þ ¼ ½mðAjþ1 Þ=mðAj Þs ½lðAj Þ=lðAjþ1 Þ s

> ½lðA1 Þ=lðBÞ½mðAjþ1 Þ=mðAj Þ ! 1

ðs ! 1Þ

(b) In the mentioned relations: mðtc P Þ ¼ cmðP Þ and lðtc P Þ ¼ cs lðP Þ; for Rn product spaces s ¼ 2=1 for (perimeter;area) pair s ¼ 3=2 for (area;volume) pair and in general s ¼ n þ 1=n ! l ðn ! 1). In that last case the ‘‘isoperimetric’’ problem loses any sense and we have at most only a symmetry-optimum case. (Note that if we have a solution for an ‘‘isoperimetric’’ problem (m; l) we have always a symmetry-optimum case since if ½mðAj Þs =lðAj Þðs > 1Þ is decreasing then mðAj Þ=lðAj Þ is decreasing too.) We can see from our theory that we can obtain a sequence fAj gj2N of sets––by group actions––for which Ajþ1 has a higher symmetry than Aj , the convergence of mðAj Þ or lðAj Þ is group(symmetry)––depending and if the conditions (a) and (b) of the ‘‘isoperimetric’’ problem are satisfied (which depends on ðX ; S; m; lÞ then we have a close relation between symmetry and extremity: maximal symmetry implies an extreme value. Thus this relation materializes the principle of symmetry–extremity.

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3. Connection between the principle of symmetry of states and the principle of symmetry–extremity Finally we try to find the relation between the two mentioned principles. In both cases we have an ascending symmetry since C G ðxÞ ¼ [g2G gfxg is the orbit of x (see [2]) and similarly Ajþ1 ¼ [g2G gAj ¼ C G ðAj Þ is the orbit of Aj and so C G ðxÞ and C G ðAj Þ are G-invariant. However since the sequence of finite groups that appear in the theory concerning the principle of symmetry–extremity need not be a normal series therefore from this point of view the last principle can be viewed as a generalization of the first. On the other hand, in the theory of the principle of symmetry of states we need not define any measure on the orbits of elements. Therefore, from this point of view the principle of symmetry–extremity can be viewed as a particular case of the principle of symmetry of states. At the end of [1] we have stated that the second law of thermodynamics is a particular case of the principle of symmetry of states since we have a symmetry increasing of the external macroscopical states. But we see that this law can be considered a special case of the principle of symmetry–extremity too since maximal entropy means the maximal number of microscopical states (which depends on the number of particles of the system) whose permutation leaves the external macroscopical state fixed. Sn being a subgroup of Sm if n 6 m (Sn denote the symmetric group of the set of n elements) and since the number of particles which constitute an isolated thermodynamic system is a very big but finite and invariant number, we can interpret the second law of thermodynamics like an ‘‘isoperimetric’’ problem. (For finite sets we consider A  B if and only if jAj 6 jBj and the map A ! jAj is a measure.) Thus the classical isoperimetric problem and the second law of thermodynamics have the same mathematical origin.

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