The measure for a family of hypersurfaces with multiple components

Int. J. Contemp. Math. Sci., Vol. 1, 2006, no. 10, 463-467
The measure for a family of hypersurfaces
with multiple components
Giuseppe Caristi
University of Messina
Faculty of Economics, 98122 (Me) Italy.
E-mail: [email protected]
Giovanni Molica Bisci
University of Reggio Calabria
Faculty of Engineering, DIMET. Via Graziella (Feo di Vito)
I-89100 Reggio Calabria Italy.
E-mail: [email protected]
Abstract
In the real affine 5-dimensional space A5 , we show the measurability
of the family of reducible hypersurfaces of type S = pn1 1 · ... · pn5 5 , where
the components pi are hyperplanes passing trough a fixed point and
multiplicity ni ∈ N.
Mathematics Subject Classification: 60D05, 52A22.
Keywords: Geometric probability; stochastic geometry; random sets; random convex sets and integral geometry.
1
Introduction
Problems of measurability in the affine spaces have been studied by many
authors, for various families of varieties and different kinds of geometric configurations (see [1],[2] and [5],[6]). Let Xn be a space of dimension n, for
example the affine or the projective space over the real field and
Gr : xi = fi (x1 , ..., xn ; a1 , ..., ar ),
(i = 1, ..., n),
(1)
a Lie group of transformations on Xn with a1 , ..., ar independent set of essential
parameters for Gr .
464
G. Caristi and G. Molica Bisci
We assume that the identity of the group Gr is determined by a1 = ... =
ar = 0. A function φ, solution of the system of partial differential equations
n
∂ i
(ξj (x1 , ..., xn )φ(x1 , ..., xn )) = 0,
i=1 ∂xi
where
ξji (x1 , ..., xn ) =
∂xi ,
∂aj a=0
(2)
(j = 1, ..., r),
is called invariant integral function for the group Gr . M. Stoka [8] defines
measurable a group Gr with a unique invariant integral function φ, up to a
constant factor. Let Vq be a family of p-dimensional varieties defined by Vq :
F1 (x1 , ..., xn ; α1 , ..., αq ) = 0, ..., Fn−q (x1 , ..., xn ; α1 , ..., αq ) = 0 where αs ∈ R,
s = 1, ..., q are essential independent parameters and Fk , k = 1, ..., (n − p) are
analytic functions on its domains. Following the definitions and the proof in
[8], if Gr is the maximal invariant group of Vq one has an associated group of
transformations
Hr : αh = ϕh (α1 , ..., αq ; a1 , ..., ar ),
(h = 1, ..., q),
and Gr and Hr are isomorphic.
If we suppose that Hr is measurable of invariant integral function φ(α1 , ..., αq ),
the expressions
μGr (Vq ) =
F (α)
φ(α1 , ..., αq )dα1 ∧ ... ∧ dαq ,
where F (α) = {(α1 , ..., αq ) ∈ Rq | Fk (x; α1 , ..., αq ) = 0, k = 1, ..., n − p} and
|φ(α1 , ..., αq )|dα1 ∧ ... ∧ dαq ,
are called respectively the measure of the family Vq and the invariant density
respect to the group Gr of the family Vq . Hence, by definition, Vq is measurable
if there exists a unique non-zero function φ. Now let A5 be the five dimensional
affine space (over the real field) of coordinates x1 , ..., x5 . The goal of this paper
is to show the measurability of the family of reducible hypersurfaces in A5 of
type
S = pn1 1 · ... · pn5 5 ,
where the components pi are hyperplanes passing trough a fixed point and
multiplicity ni ∈ N.
465
The measure for a family of hypersurfaces with...
2
Main Results
Without loss of generality, we can suppose that the fixed point for the components of the surface S is the origin of coordinates. Hence the “generic” element
of the family is the surface S of equation
S:
5 j=1
(1)
(4)
x1 + Aj x2 + ... + Aj x5
nj
(nj ∈ N),
= 0,
()
(0)
where det(Aj ) = 0, with = 0, ..., 4; j = 1, ..., 5 and Aj := 1, for 1 ≤ j ≤ 5.
The parameters space of this family is of dimension 20 and coordinates
(i)
Aj ∈ R with 1 ≤ i ≤ 4 and j = 1, ..., 5. The maximal group of invariance of
the family is
G52 :
xr =
5
s=1
αrs xs , r = 1, 2, ..., 5
(3)
with αrs ∈ R and det(αrs ) = 0.
Acting by G52 on S we obtain
S :
5 j=1
(i)
(1)
(4)
x1 + [Aj ] x2 + ... + [Aj ] x5
nj
(nj ∈ N),
= 0,
[Aj ] ∈ R, with
(i) [Aj ]
H52 :
=
α1j +
α11 +
where 1 ≤ i ≤ 4; j = 1, ..., 5 and α11 +
4
(l) j
l=1 Aj αl
,
4
(l) 1
l=1 Aj αl
4
l=1
(4)
(l)
Aj αl1 = 0, ∀j = 1, ..., 5.
Theorem 2.1 The family of reducible hypersurfaces in A5 of “generic”
element of equation
S:
5 j=1
(1)
(4)
x1 + Aj x2 + ... + Aj x5
nj
()
= 0,
(0)
(nj ∈ N),
where det(Aj ) = 0 ( = 0, ..., 4; j = 1, ..., 5, Aj := 1, for 1 ≤ j ≤ 5) is
measurable and its unique invariant integral function is given by
φ= 1
.
() 5
det Ai
466
G. Caristi and G. Molica Bisci
Proof : By direct calculations we give the following Deltheil’s system:
5 4
j=1 i=1
5
∂φ
j=1
∂Aj
(i)
5
j=1
5
j=1
(i)
Aj
(i)
Aj
∂φ
(k)
∂Aj
= −20φ,
(i)
∂Aj
= 0, i = 1, ..., 4
∂φ
(i)
Aj
∂φ
= −5φ,
(i)
∂Aj
= 0, i, k = 1, ..., 4; k = i.
And finally
5 (h) 2
j=1
Aj
5 (k) 2
j=1
Aj
∂φ
(h)
∂Aj
∂φ
(k)
∂Aj
+
5
j=1
+
5
j=1
(h)
∂φ
(k)
Aj Aj
(k)
∂Aj
(h) (k)
Aj Aj
∂φ
(h)
∂Aj
= −5
5
j=1
= −5
5
j=1
(h)
Aj φ,
(k)
Aj φ,
for every pair (h, k) with h = k = 1, ..., 4.
The unique non-zero solution (up to a constant factor) is
φ= 1
.
() 5
det Ai
Remark The three dimensional and reduced case is investigated by Santoro in [5].
References
[1] P. Dulio, M. Petriccione, Sulla misurabilità delle famiglie dei sistemi di
k iperpiani passanti per un punto fisso dello spazio affine A7 , Atti Acc.
Peloritana dei Pericolanti Classe S.M.F.N. Vol.LXXI (1993).
[2] P. Dulio, M. Petriccione, Sulla misurabilità delle famiglie dei sistemi di
due e tre iperpiani passanti per un punto fisso di A4 , Atti Acc. Peloritana
dei Pericolanti Classe S.M.F.N. Vol.LXXI (1993).
The measure for a family of hypersurfaces with...
467
[3] A. Duma, M. Stoka, On the measurability of the conic sections’ family
in the projective space P3 , Rend. Circ. Mat. Palermo, Serie II, Tomo LI
(2002), 537-548.
[4] G. Failla, G. Molica Bisci, On the measurability of the conic sections
family in the projective space P4 , to appear Boll. U.M.I.
[5] G. Santoro, Sulla misurabilità della famiglia di superfici cubiche spezzate
in tre piani passanti per un punto fisso, Atti Acc. Peloritana dei Pericolanti Classe S.M.F.N., Vol.LXVIII, (1990).
[6] G. Santoro, Sulla misurabilità di una famiglia di superfici cubiche dello
spazio affine A3 , Atti Acc. Peloritana dei Pericolanti Classe S.M.F.N.,
Vol.LXVIII, (1990).
[7] M. Stoka, Masura familiior de varietati din spatiul E3 , St. Cerc. Mat., IX,
(1958),547-558.
[8] M. Stoka, Géométrie Intégrale, Mém. Sci. Math., fasc. 165, GauthierVillars, Paris, 1968.
[9] M. Stoka, On the measurability of the conic sections’ family in the projective space Pn , Boll. U.M.I. (8) 7-B (2004), 357-367.
Received: June 6, 2006