Some combinatorial aspects of differential operation compositions on
  space $R^n$
arXiv:0704.0750v1  [math.DG]  5 Apr 2007
Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. 9 (1998), 29–33
SOME COMBINATORIAL ASPECTS
OF DIFFERENTIAL OPERATION
COMPOSITION ON THE SPACE R
Branko J. Malešević
In this paper we present a recurrent relation for counting meaningful compositions
of the higher-order differential operations on the space Rn (n=3,4,...) and extract
the non-trivial compositions of order higher than two.
1. DIFFERENTIAL FORMS AND OPERATIONS ON THE SPACE R
It is well known that the first-order differential operations grad, curl and div
on the space R3 can be introduced using the operator of the exterior differentia-
tion d of differential forms [1]:
Ω0(R3)
−→ Ω1(R3)
−→ Ω2(R3)
−→ Ω3(R3),
where Ωi(R3) is the space of differential forms of degree i = 0, 1, 2, 3 on the space
3 over the ring of functions A = {f : R3 → R | f ∈ C∞(R3)}. In the consideration,
which follows, we give definitions of the first-order differential operations.
Let us notice that one-dimensional spaces Ω0(R3) and Ω3(R3) are isomorphic
toA and let ϕ0 : Ω
0(R3) → A, ϕ3 : Ω
3(R3) → A be the corresponding isomorphisms.
Next, the set of vector functionsB = {f =(f1, f2, f3) : R
3 → R3 | f1, f2, f3 ∈ C
∞(R3)},
over the ring A, is three-dimensional. It is isomorphic to Ω1(R3) and Ω2(R3). Let
ϕ1 : Ω
1(R3) → B, ϕ2 : Ω
2(R3) → B be the corresponding isomorphisms. In that
case, the compositions ϕ−10 ◦ϕ3 : Ω
3(R3) → Ω0(R3) and ϕ−11 ◦ϕ2 : Ω
2(R3) → Ω1(R3)
are isomorphisms of the corresponding spaces of differential forms. The first-order
differential operations are defined via the operator of the exterior differentiation d
of differential forms in the following form:
∇1 = ϕ1◦d◦ϕ
0 : A → B, ∇2 = ϕ2◦d◦ϕ
1 : B → B, ∇3 = ϕ3◦d◦ϕ
2 : B → A.
Therefore we obtain explicit expressions for the first order differential operations
∇1, ∇2, ∇3 on the space R
3 in the following form:
(1) gradf = ∇1f =
e3 : A → B,
(2) curlf = ∇2f =
e3 : B → B,
(3) divf = ∇3f =
: B → A.
1991 Mathematics Subject Classification: 26B12, 58A10
http://arxiv.org/abs/0704.0750v1
30 Branko J. Malešević
Let us count meaningful compositions of differential operations ∇1,∇2,∇3.
Consider the set of functions Θ = {∇1,∇2,∇3}. Let us define a binary relation
ρ ”to be in composition” with ∇iρ∇j = ⊤ iff the composition ∇j ◦∇i is meaningful
(∇i,∇j ∈ Θ). The Cayley’s table of this relation reads:
ρ ∇1 ∇2 ∇3
∇1 ⊥ ⊤ ⊤
∇2 ⊥ ⊤ ⊤
∇3 ⊤ ⊥ ⊥ .
We form the graph of relation ρ as follows. If ∇iρ∇j = ⊤ then we put the node ∇j
under the node ∇i. Let us mark ∇0 as nowhere-defined function ϑ, with domain
and range being the empty set [2]. We shall consider ∇0ρ∇i = ⊤ (i = 1, 2, 3). For
the set of functions Θ ∪ {∇0} our graph is the tree with the root in the node ∇0.
∇0 f(0) = 1
∇2 ❳❳
∇3 f(1) = 3
∇1 f(2) = 5
∇3 f(3) = 8
✔✔❚❚ ✔✔❚❚ ✔✔❚❚ ✔✔❚❚ ✔✔q∇2 ❚❚q∇3 q∇1 f(4) = 13
✔✔❚❚ ✔✔❚❚Fig. 1 f(5) = 21
Let fi(k) be a number of meaningful compositions of the k
th-order beginning with
∇i. Let f(k) be a number of meaningful composition of the k
th-order of operations
over Θ. Then f(k) = f1(k) + f2(k) + f3(k). Based on partial self similarity of the
tree (Fig. 1), which is formed according to Cayley’s table (4), we get equalities:
f1(k) = f2(k − 1) + f3(k − 1) ∧ f2(k) = f2(k − 1) + f3(k − 1) ∧ f3(k) = f1(k − 1).
Now, a recurrent relation for f(k) can be derived as follows:
f(k) = f1(k) + f2(k) + f3(k)
f1(k − 1) + f2(k − 1) + f3(k − 1)
f3(k − 1) + f2(k − 1)
= f(k − 1) +
f1(k − 2) + f2(k − 2) + f3(k − 2)
= f(k − 1) + f(k − 2).
Based on the initial values: f(1) = 3, f(2) = 5, f(3) = 8 we conclude that f(k) =
Fk+3, where is Fibonacci’s number of order k + 3.
Let us note that ∇2 ◦∇1 = 0 and ∇3 ◦∇2 = 0, because d
2 = 0. On the other
hand, the compositions ∇1 ◦∇3, ∇2 ◦∇2 and ∇3 ◦∇1 are not annihilated, because
of ϕ−10 ◦ ϕ3 6= i and ϕ
1 ◦ ϕ2 6= i. Thus, as in the paper [2], we conclude that the
non-trivial compositions are of the following form:
(∇1◦)∇3 ◦ · · · ◦ ∇1 ◦ ∇3 ◦ ∇1,
∇2 ◦ ∇2 ◦ · · · ◦ ∇2 ◦ ∇2 ◦ ∇2,
(∇3◦)∇1 ◦ · · · ◦ ∇3 ◦ ∇1 ◦ ∇3.
As non-trivial compositions we consider those which are not identical to the zero
function. Terms in parentheses are included in for an odd number of terms and are
left out otherwise.
Some combinatorial aspects of differential operation compositions ... 31
2. DIFFERENTIAL FORMS AND OPERATIONS ON THE SPACE R
Let us present a recurrent relation for counting meaningful compositions of
the higher-order differential operations on the space Rn (n = 3, 4, . . .) and extract
the non-trivial compositions of order higher than two. Let us form the following
sets of functions:
Ai = {f : R
)|f1, . . . , f(n
) ∈ C
for i = 0, 1, . . . ,m where m = [n/2]. Let Ωi(Rn) be a set of differential forms of
degree i = 0, 1, . . . , n on the space Rn. Let us notice that Ωi(Rn) and Ωn−i(Rn),
over ring A0, are spaces of the same dimension
, for i = 0, 1, . . . ,m. They can
be identified with Ai, using the corresponding isomorphisms:
ϕi : Ω
i(Rn) → Ai (0 ≤ i ≤ m) and ϕn−i : Ω
n−i(Rn) → Ai (0 ≤ i < n−m).
We define the first-order differential operations on the space Rn
via the operator of the exterior differentiation d as follows:
∇i = ϕi ◦ d ◦ ϕ
i−1 (1 ≤ i ≤ n). (1 ≤ i ≤ m)
Therefore, we obtain the first order differential operations on the space Rn, de-
pending on pairity of dimension n, in the following form:
n = 2m : ∇1 : A0 → A1
∇2 : A1 → A2
∇i : Ai → Ai+1
∇m : Am−1 → Am
∇m+1 : Am → Am−1
∇n−j : Aj+1 → Aj
∇n−1 : A2 → A1
∇n : A1 → A0,
n = 2m+ 1 : ∇1 : A0 → A1
∇2 : A1 → A2
∇i : Ai → Ai+1
∇m : Am−1 → Am
∇m+1 : Am → Am
∇m+2 : Am → Am−1
∇n−j : Aj+1 → Aj
∇n−1 : A2 → A1
∇n : A1 → A0.
Consider the set of functions Θ = {∇1,∇2, . . . ,∇n}. Let us define a binary relation
ρ ”to be in composition” with ∇iρ∇j = ⊤ iff the composition ∇j ◦∇i is meaningful
(∇i,∇j ∈ Θ). It is not difficult to check that Cayley’s table of this relation is
determined with:
(6) ∇iρ∇j =
⊤ : (j = i+ 1) ∨ (i+ j = n+ 1),
⊥ : (j 6= i+ 1) ∧ (i+ j 6= n+ 1).
Let us form an adjacency matrix A = [aij ] ∈ {0, 1}
n×n of the graph, determined
by relation ρ. Let fi(k) be a number of meaningful compositions of the k
th-order
32 Branko J. Malešević
beginning with ∇i (notice that fi(1) = 1 for i= 1, . . . , n). Let f(k) be a number
of meaningful composition of the kth-order of operations over Θ. Then f(k) =
f1(k)+. . .+fn(k). Notice that the following is true:
(7) fi(k) =
aij · fj(k − 1),
for i = 1, . . . , n. Based on (7) we form the system of recurrent equations:
f1(k)
fn(k)
a11 · · · a1n
an1 · · · ann
f1(k − 1)
fn(k − 1)
If vn = [ 1 · · · 1 ]1×n then:
(9) f(k) = vn ·
f1(k)
fn(k)
So, the expression:
(10) f(k) = vn · A
k−1 · vTn .
follows from (8) and (9). Reducing the system of the recurrent equations (8), for
any of the functions fi(k) we have:
(11) α0fi(k) + α1fi(k − 1) + · · ·+ αnfi(k − n) = 0 (k > n),
where α0, . . . , αn are coefficients of the characteristic polynomial Pn(λ) = |A−λI| =
n+ . . .+αn. Thus, we conclude that the function f(k) =
fi(k) also satisfies:
(12) α0f(k) + α1f(k − 1) + · · ·+ αnf(k − n) = 0 (k > n).
Hence, the following theorem holds.
Theorem 1. The number of meaningful differential operations, on the space R
(n = 3, 4, . . .), of the order higher than two, is determined by the formula (10), i.e.
by the recurrent formula (12).
In n-dimensional space Rn, for dimensions n = 3, 4, 5, . . . , 10, using the pre-
vious theorem we form a table of the corresponding recurrent formula:
Dimension: Recurrent relations for the number of meaningful compositions:
n = 3 f(i+ 2) = f(i + 1) + f(i)
n = 4 f(i+ 2) = 2f(i)
n = 5 f(i+ 3) = f(i+ 2) + 2f(i+ 1)− f(i)
n = 6 f(i+ 4) = 3f(i+ 2)− f(i)
n = 7 f(i + 5) = f(i + 3) + 3f(i + 2) − 2f(i + 1)− f(i)
n = 8 f(i + 4) = 4f(i + 3) − 3f(i)
n = 9 f(i+ 5) = f(i+ 4) + 4f(i+ 3)− 3f(i+ 2)− 3f(i + 1) + f(i)
n = 10 f(i + 6) = 5f(i + 4) − 6f(i + 2) + f(i)
Some combinatorial aspects of differential operation compositions ... 33
Let us determine non-trivial higher-order meaningful compositions on the
space Rn. For isomorphisms ϕk we have:
(13) ϕ−1
◦ ϕn−k 6= i,
for k = 1, 2, . . . , n and 2k 6= n. Then, based on (6) and (13), all second-order
compositions are given by the formula:
(14) ∇j ◦ ∇k =
0 : j = k + 1,
gj,k : (k + j = n+ 1) ∧ (2k 6= n),
ϑ : (j 6= k + 1) ∧ (k + j 6= n+ 1);
where 0 is a trivial composition, gj,k is a non-trivial second-order composition and
ϑ is a nowhere-defined function for j, k = 1, . . . , n. Notice that in gj,k = ∇j ◦∇k =
ϕn+1−k ◦ d ◦ ϕ
◦ ϕk ◦ d ◦ ϕ
k−1 (j=n+1−k ∧ 2k 6=n) and switching the terms
is impossible, because in that way we get nowhere-defined function ϑ. Hence, we
conclude that the following theorem holds.
Theorem 2. All meaningful non-trivial differential operations on the space R
(n = 3, 4, . . .), of order higher than, two are given in the form of the following com-
positions:
(∇k) ◦ ∇j ◦ ∇k ◦ · · · ◦ ∇j ◦ ∇k,
(∇j) ◦ ∇k ◦ ∇j ◦ · · · ◦ ∇k ◦ ∇j ,
with to the condition k+ j = n+ 1 and 2k, 2j 6= n for k, j = 1, 2, . . . , n. Terms in
parentheses are included in for an odd number of terms and are left out otherwise.
Acknowledgment. I wish to express my gratitude to ProfessorsM. Merkle and
M. Prvanović who examined the first version of the paper and gave me their
suggestions and some very useful remarks.
REFERENCES
1. R.Bott, L.W.Tu: Differential forms in algebraic topology, Springer, New York 1982.
2. B.J.Malešević: A note on higher-order differential operations, Univ. Beograd, Publ.
Elektrotehn. Fak.,Ser. Mat. 7 (1996), 105-109.
University of Belgrade, (Received September 8, 1997)
Faculty of Electrical Engineering, (Revised October 30, 1998)
P.O.Box 35-54, 11120 Belgrade,
Yugoslavia
malesevic@kiklop.etf.bg.ac.yu
