Author:
Zhu Yuyang,Huang Jing,Yang Xingxing,Liu Xueling
Abstract
Abstract
Subdivide n-dimensional space En
into disjoint n-dimensional unit cubes, and let the set of these cubes be Λn. The vertices of each cube are called nodes. Let LPT
be the set of allcontinuous curve passing through node P(s1,s2
,…,sn)and nodeT(t1,t2
,…,tn) . ∀Γ ∊ LpT, α∊Λ
n
define the characteristic function of Γ to oc,
χ
Γ
(
α
)
=
{
0
,
when
μ
(
α
∩
Γ
)
=
0
,
1
,
when
μ
(
α
∩
Γ
)
>
0
,
Where μ(α ∩ Γ) is Lebesgue measure of point set α ∩ Γ. Name all its non-empty element as α
Γ(1),α
Γ(2),…,α
Γ(r(Γ)), and let λ
k(α
Γ(k)) be the weighting coefficients of α
Γ(k)(k=1,2…,r(Γ)) on Γ . This paper investigated the value of
Γ
∈
L
P
T
inf
{
Σ
k
=
1
r
(
Γ
)
λ
k
(
α
Γ
(
k
)
)
χ
(
α
Γ
(
k
)
)
}
, and by using discrete mathematics theory and construction method, proved that if all weighting coefficients are equal to 1, then max
{
m
1
,
m
2
,
⋯
m
n
}
≤
Γ
∈
L
P
T
inf
{
Σ
k
=
1
r
(
Γ
)
λ
k
(
α
Γ
(
k
)
)
χ
(
α
Γ
(
k
)
)
}
≤
Σ
k
=
1
n
(
−
1
)
k
−
1
Σ
1
≤
i
1
<
i
2
<
⋯
<
i
k
≤
n
gcd
(
m
i
1
,
m
i
2
,
⋯
m
i
k
)
Where
m
i
=
|
s
i
−
t
i
|
, i = 1,2, …, n. The paper also generalizes the result to a wedge-shaped case.
Subject
General Physics and Astronomy
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