Post on 14-Jun-2020
Three deformations of generalized Leibniz triangles
Gabriele Sicuroin collaboration with P. Tempesta (UCM), A. Rodríguez (UPM) and C. Tsallis (CBPF)
Centro Brasileiro de Pesquisas FísicasRio de Janeiro, Brazil
19 October 2015
An elementary model
Consider a set of N identical binary random variables {xi}i=1,...,N, xi ∈ {0, 1}
N = 30
Let us call rN,n the probability of having∑
i xi = n in a certain configuration. Due toindistinguishability, the probability of having
∑i xi = n is
pN,n =
(Nn
)rN,n.
Pascal triangleOr the Meru–prastaara, the Staircase of Mount Meru
It is well known that the binomial coefficients can be arranged in a triangle:
1
1 1
1(2
1
)1
1(3
1
) (32
)1
1(4
1
) (42
) (43
)1
1(5
1
) (52
) (53
) (54
)1
1(6
1
) (62
) (63
) (64
) (65
)1
Pingala, II century bC, India
Varahamirhira, 505 aD, India
Mahavira, 850 aD, India
Halayudha, 975 aD, India
Bhat.t.otpala, 1086 aD, India
Al-Karaji, XI century aD, Iran
Omar Khayyám, XI–XII century aD, Iran
Jia Xian, XI century aD, China
Yang Hui, XIII century aD, China
Levi ben Gershon, XIV century aD, France
Peter Apian, 1527 aD, Germany
Michael Stifel, 1544 aD, Germany
Niccolò Tartaglia, 1556 aD, Italy
Gerolamo Cardano, 1570 aD, Italy
Blaise Pascal, 1665 aD, France
Independent variables
For independent variables, rN,n = ρn(1− ρ)N−n for ρ ∈ (0, 1) and therefore pN,n is thebinomial distribution.
1
r1,0 r1,1
r2,0 r2,1 r2,2
r3,0 r3,1 r3,2 r3,3
r4,0 r4,1 r4,2 r4,3 r4,4
r5,0 r5,1 r5,2 r5,3 r5,4 r5,5
=
1
ρ1(1−ρ)0 ρ0(1−ρ)1
ρ2(1−ρ)0 ρ1(1−ρ)1 ρ0(1−ρ)2
ρ3(1−ρ)0 ρ2(1−ρ)1 ρ1(1−ρ)2 ρ0(1−ρ)3
ρ4(1−ρ)0 ρ3(1−ρ)1 ρ2(1−ρ)2 ρ1(1−ρ)3 ρ0(1−ρ)4
ρ5(1−ρ)0 ρ4(1−ρ)1 ρ3(1−ρ)2 ρ2(1−ρ)3 ρ1(1−ρ)4 ρ0(1−ρ)5
pN,n =N�1∼ 1√
2πNρ(1− ρ)e−
N2ρ(1−ρ) ( n
N−ρ)2
.
Introducing correlations: Leibniz triangle
Leibniz considered the set of triangles satisfying the following scale–invarianceproperty:
rN,n+1 + rN,n = rN−1,n
Moreover, this constraint uniquely determines the entries of the triangle if thevalues rN,0 are given.
1
r1,0 r1,1
r2,0 r2,1 r2,2
r3,0 r3,1 r3,2 r3,3
r4,0 r4,1 r4,2 r4,3 r4,4
r5,0 r5,1 r5,2 r5,3 r5,4 r5,5
Leibniz choser(1)
N,0 =1
N + 1=⇒ r(1)
N,n =1
N + 11(Nn
) .Obviously, the limiting distribution is the uniform distribution.
A model for q–GaussiansRodríguez, Schwämmle, Tsallis – J. Stat. Mech., 2008(09), P09006 (2008).
1 N = 0r(1)N,n
r(1)1,0 r(1)
1,1 N = 1
r(1)2,0 r(1)
2,1 r(1)2,2 N = 2
r(1)3,0 r(1)
3,1 r(1)3,2 r(1)
3,3 N = 3
r(1)4,0 r(1)
4,1 r(1)4,2 r(1)
4,3 r(1)4,4 N = 4
r(1)5,0 r(1)
5,1 r(1)5,2 r(1)
5,3 r(1)5,4 r(1)
5,5 N = 5
r(1)6,0 r(1)
6,1 r(1)6,2 r(1)
6,3 r(1)6,4 r(1)
6,5 r(1)6,6 N = 6
r(1)7,0 r(1)
7,1 r(1)7,2 r(1)
7,3 r(1)7,4 r(1)
7,5 r(1)7,6 r(1)
7,7 N = 7
r(1)8,0 r(1)
8,1 r(1)8,2 r(1)
8,3 r(1)8,4 r(1)
8,5 r(1)8,6 r(1)
8,7 r(1)8,8 N = 8
r(1)9,0 r(1)
9,1 r(1)9,2 r(1)
9,3 r(1)9,4 r(1)
9,5 r(1)9,6 r(1)
9,7 r(1)9,8 r(1)
9,9 N = 9
A model for q–GaussiansRodríguez, Schwämmle, Tsallis – J. Stat. Mech., 2008(09), P09006 (2008).
1
r(1)1,0 r(1)
1,1
r(1)2,0 r(1)
2,1 r(1)2,2
r(1)3,0 r(1)
3,1 r(1)3,2 r(1)
3,3
r(1)4,0 r(1)
4,1 r(1)4,2 r(1)
4,3 r(1)4,4
r(1)5,0 r(1)
5,1 r(1)5,2 r(1)
5,3 r(1)5,4 r(1)
5,5
r(1)6,0 r(1)
6,1 r(1)6,2 r(1)
6,3 r(1)6,4 r(1)
6,5 r(1)6,6
r(1)7,0 r(1)
7,1 r(1)7,2 r(1)
7,3 r(1)7,4 r(1)
7,5 r(1)7,6 r(1)
7,7
r(1)8,0 r(1)
8,1 r(1)8,2 r(1)
8,3 r(1)8,4 r(1)
8,5 r(1)8,6 r(1)
8,7 r(1)8,8
r(1)9,0 r(1)
9,1 r(1)9,2 r(1)
9,3 r(1)9,4 r(1)
9,5 r(1)9,6 r(1)
9,7 r(1)9,8 r(1)
9,9
r(3)N,n =
r(1)N+4,n+2
r(1)4,2
and in general
r(ν)N,n =
r(1)N+2(ν−1),n+ν−1
r(1)2(ν−1),ν−1
scale–invariant
p(ν)N,n =
(Nn
)r(ν)
N,n
q–Gaussian Pq1(ν)(x)
q1(ν) = 1 − 1ν − 1
x = 2√1−q1(ν)
( nN −
12
)
The q–Gaussian distribution
The q–Gaussian distribution is defined as follows:
Pq(x) :=
3−q2
√1−qπ
Γ(
3−q2−2q
)Γ(
11−q
) [1− (1− q)x2] 11−q+ for q < 1,
e−x2√π
for q = 1,√q−1π
Γ(
1q−1
)Γ(
3−q2q−2
) [1− (1− q)x2] 11−q for 1 < q < 3.
In the previous definition
[x]+ :=
{x x > 0,0 x ≤ 0.
−6 −4 −2 0 2 4 6
0
0.2
0.4
0.6
x
P q
q = 12
q = 1q = 2
The α–numbers and the β–numbersSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
Let us now deform the generalized triangles above introducing two deformation ofthe natural numbers.
The α–numbersGiven n ∈ N ∪ 0, and α > 0, α 6= 1,
{n}α := (n + 1)(
1− 1− α1− αn+1
).
We define the α–binomial as{Nn
}α
:=
∏Nk=1{k}α∏N−n
k=1 {k}α∏n
k=1{k}α.
limα→1{n}α = n.
The β–numbersGiven n ∈ N ∪ 0, and β > 0, β 6= 1,
[n]β := n(
1− 1− β1− βn
)+ 1.
We define the β–binomial as[Nn
]β
:=
∏Nk=1[k]β∏N−n
k=1 [k]β∏n
k=1[k]β.
limβ→1
[n]β = n.
limn→∞
[n]α{n}α
= 1.
The α–triangles and the β–trianglesSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
The Leibniz α–triangles
r(1)N,n,α :=
1{N + 1}α { N
n }αFor ν ∈ N:
r(ν)N,n,α :=
r(1)N+2(ν−1),n+ν−1,α∑N
k=0
(Nk
)r(1)
N+2(ν−1),k+ν−1,α︸ ︷︷ ︸p(ν)N,k,α
The Leibniz β–triangles
r(1)N,n,β :=
1[N + 1]β [ N
n ]β
For ν ∈ N:
r(ν)N,n,β :=
r(1)N+2(ν−1),n+ν−1,β∑N
k=0
(Nk
)r(1)
N+2(ν−1),k+ν−1,β︸ ︷︷ ︸p(ν)N,k,β
The asymptotic scale invarianceSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
Both the deformed triangles satisfy the scale invariance condition asymptotically!
limN→∞
nN≡η
r(ν)N−1,n,•
r(ν)N,n+1,• + r(ν)
N,n,•︸ ︷︷ ︸t(ν)N,η,•
= 1
0 0.2 0.4 0.6 0.8 11
1.01
η
t(2)
50,η,α
=1 2
0 0.2 0.4 0.6 0.8 11
1.01
η
t(2)
50,η,β
=1 2
The main theoremsSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
Robustness of α–trianglesWe have that
N2√ν − δα,1
p(ν)N,n,α
N�1∼ Pqα(ν)(x), x := 2√ν − δα,1
( nN− 1
2
),
where
qα(ν) :=
{1− 1
νfor α 6= 1,
1− 1ν−1 for α = 1.
0 0.5 1 1.5 20
0.5
1
α
q α(2
)
0 0.2 0.4 0.6 0.8 10
1
2×10−4
nN
p(2)
N,n,α
α = 13
α = 23
α = 1
α = 2
α = 3
The main theoremsSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
Robustness of β–trianglesWe have that
N2√ν − χ(β)
p(ν)N,n,β
N�1∼ Pqβ(ν)(x), x :=2
√ν−δβ,1 +
1−min{β, 1}min{β, 1}
( nN− 1
2
),
where
qβ(ν) =
1− 1
νfor β > 1,
1− 1ν−1 for β = 1,
1− ββν+1−β for 0 < β < 1.
0 0.5 1 1.5 20
0.5
1
β
q β(2
)
0 0.2 0.4 0.6 0.8 10
1
2
×10−4
nN
p(2)
N,n,β
β = 13
β = 23
β = 1
β = 2
β = 3
A recurrent algebraSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
α–triangles
11− qα(ν)
=1
1− q1(ν)+ 1, α 6= 1
β–triangles
min{β, 1}1− qβ(ν)
=min{β, 1}1− q1(ν)
+ 1, β 6= 1
β–triangles
∆(ν)N (β) :=
√N∑N
k=0
∣∣∣p(ν)N,k,β − p(ν)
N,k
∣∣∣2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
Nonuniform convergence
β
∆(2
)N
(β)
N = 50N = 400N →∞
A non asymptotically scale–invariant deformationSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
Inspired by the Q–calculus, we considered an alternative deformation of the Leibniztriangle based on the so called Q–numbers:
JnKQ :=1− Qn
1− Q, Q ∈ (0,∞) \ {1}, lim
Q→1JnKQ = n.
We can introduce also the Gauss binomial coefficients
sNn
{
Q
:=
∏Nk=1JkKQ∏N−n
k=1 JkKQ∏n
k=1JkKQ
Q→1−−−→
(Nn
).
Q–triangles
r(1)N,n,Q :=
1JN + 1KQ
1J N
n KQ=⇒ r(ν)
N,n,Q :=r(1)
N+2(ν−1),n+ν−1,Q∑Nk=0
(Nk
)r(1)
N+2(ν−1),k+ν−1,Q︸ ︷︷ ︸p(ν)N,k,Q
A non asymptotically scale–invariant deformationSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
The Q–triangles are not asymptotically scale invariant! Indeed
limN→∞
nN≡η fixed
r(ν)N−1,n,Q
r(ν)N,n,Q + r(ν)
N,n+1,Q
=
{12 for Q < 1,0 for Q > 1.
Robustness of Q–trianglesWe have that
p(ν)N,n,Q
N�1∼
{√2πN e−2N(η− 1
2 )2if 0 < Q < 1,
δn,0+δn,N2 if Q > 1.
−5 0 510−24
10−11
102
1√N
(n− N
2
)
√N
p(2)
N,n,Q
Q = 13
Q = 23
0 0.5 10
0.5
nN
p(2)
N,n,
3 2
N = 5N = 10N = 30
Final remarksSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)
We deformed the generalized Leibniz triangle to evaluate the robustness of thelimiting distributions in three different ways. We obtained the following results
The asymptotically scale invariant deformed triangles have still q–Gaussians aslimiting distribution, but the limiting value of q depends in general on theform of the perturbation, even if two asymptotically equivalent perturbations areconsidered.
Switching from exact scale invariance to asymptotically scale invariance canmake a discontinuity appear in the limiting value of q as function of theperturbation parameter.
The not-asymptotically scale invariant deformed triangle can have a limitingdistribution outside the family of q–Gaussians.
These results suggest that the (asymptotic) scale–invariance property plays a centralrole in the robustness of the set of q–Gaussian distributions as limitingdistributions. More specifically, the set of q–Gaussians appears to be robust underasymptotically scale–invariant deformations.
Thank youfor your attention!