Journal of Inequalities in Pure and
Applied Mathematics
A NUMERICAL METHOD IN TERMS OF THE THIRD
DERIVATIVE FOR A DELAY INTEGRAL EQUATION
FROM BIOMATHEMATICS
volume 6, issue 3, article 87,
2005.
MARKUS SIGG
Department of Mathematics and Statistics
University of Konstanz
Germany.
Received 16 July, 2004;
accepted 29 June, 2005.
Communicated by: F. Hansen
EMail: mail@MarkusSigg.de
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
Let F be a Schatten p-operator and R, S positive operators. We show that the
1 c
1 c
1 c
inequality |F (R + S) c |p ≤ |F R c |p + |F S c |p for the Schatten p-norm | · |p is true
for p ≥ c = 1 and for p ≥ c = 2, conjecture it to be true for p ≥ c ∈ [1, 2], give
counterexamples for the other cases, and present a numerical study for 2 × 2
matrices. Furthermore, we have a look at a generalisation of the inequality
which involves an additional factor σ(c, p).
2000 Mathematics Subject Classification: 47A30, 47B10
Key words: Schatten class, Schatten norm, Norm inequality, Minkowski inequality,
Triangle inequality, Powers of operators, Schatten-Minkowski constant.
This work was supported by the Dr. Helmut Manfred Riedl Foundation.
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
Title Page
Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
The Case p < c and the Case c 6∈ [1, 2] . . . . . . . . . . . . . . . . . . . 6
4
Some Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5
Generalisation of (MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
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1.
Introduction
Let H and K be complex Hilbert spaces and 0 < p ≤ ∞. Following [1], we
denote by cp (H, K) the space of Schatten p-operators T : H −→ K, equipped
with the Schatten p-norm or quasi-norm | · |p . Note that [1] deals only with the
spaces cp (H) := cp (H, H). The generalisations cp (H, K) can be found in textbooks like [2] and [3] (there written as Bp (H, K) and Sp (H, K) respectively).
By L(H) we denote the space of bounded linear operators on H, and by
L(H)+ the subset of positive operators. With |T | := (T ∗ T )1/2 ∈ L(H)+ for
T ∈ L(H, K) we have for p < ∞
|T |pp = tr |T |p
for T ∈ cp (H, K), and consequently
|T |pp = tr T p
for T ∈ cp (H)+ := cp (H) ∩ L(H)+ .
Applying |T |p = |T ∗ |p for T ∈ cp (H, K), this shows in case of p < ∞
1 2 2
1 2
2 ∗
∗ p2 p
2
= |F U F ∗ | p
F U = U F = tr (F U F )
p
2
p
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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for F ∈ cp (H, K) and U ∈ L(H)+ . Because | · |∞ is the usual operator norm,
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1 2
2
F
U
= |F U F ∗ | p
Close
is also true for p = ∞, with the common convention ∞
:= ∞.
2
Our question, which arose while studying the integration of Schatten operator valued functions in [4], is: For what values of p ∈ (0, ∞] and c ∈ (0, ∞) is
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the Minkowski-like inequality
1 c
1 c
1 c
c
c
c
(MS)
F
(R
+
S)
≤
F
R
+
F
S
p
p
p
true for all F ∈ cp (H, K) and R, S ∈ L(H)+ ?
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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2.
The Conjecture
Let H, K, p, c, F, R, S be as above.
Theorem 2.1. Inequality (MS) is true for p ≥ c = 1 and for p ≥ c = 2.
Proof. For p ≥ c = 1, the triangle inequality for | · |p shows
|F (R + S)|p = |F R + F S|p ≤ |F R|p + |F S|p .
A Minkowski-Type Inequality for
the Schatten Norm
For p ≥ c = 2, the triangle inequality for | · | p shows
2
1 2
1 2
1 2
∗
∗
∗
2
2
2
F (R + S) = |F (R + S)F | p ≤ |F RF | p +|F SF | p = F R +F S .
p
2
2
2
p
Markus Sigg
p
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Theorem 2.1 suggests the following conjecture.
Conjecture 1. Inequality (MS) is true for p ≥ c ∈ [1, 2].
For c ∈ (1, 2) we have at the present time no proof of this conjecture for
other than trivial situations, not even for the special case of 2 × 2 matrices.
However, some justification will be given in Section 4.
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3.
The Case p < c and the Case c 6∈ [1, 2]
In this section we will demonstrate, by providing counterexamples, that inequality (MS) is not necessarily true for other values of (c, p) than those stated in
Conjecture 1. We will offer one example for 0 < p < c < ∞, and one for
arbitrary p when c < 1 or c > 2, both examples using 2 × 2 matrices. The
power U t for t > 0 of a non-negative matrix U can be calculated easily with
help of the spectral decomposition of U .
Example 3.1. Inequality (MS) is violated for 0 < p < c < ∞ by
1 0
1 0
0 0
F :=
, R :=
, S :=
.
0 1
0 0
0 1
Proof. From U t = U for U ∈ {R, S, R + S} and t ∈ (0, ∞) we get
1
1
1
c
F U = |U |p = (tr U p ) p = (tr U ) p ,
p
yielding
1
c
F R = 1,
p
1
c
F S = 1,
p
1
1
c
F (R + S) = 2 p ,
p
and using p < c,
c
1 c
1 c
1 c
p
c
c
c
F R + F S = 2 < 2 = F (R + S) .
p
p
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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p
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The second example makes use of an inequality which is interesting in its
own right. Seeming simple, it is surprisingly fiddly to prove:
Lemma 3.1. For x ∈ (0, 1) ∪ (2, ∞) we have
√ !x 1
1
3+ 5
+ 1− √
1+ √
2
5
5
Proof. Setting r :=
show
√
√ !x
3− 5
< 1 + 3x .
2
5, α1 := 1 + 1r , α2 := 1 − 1r , and ω :=
3+r
,
2
we have to
α1 ω x + α2 ω −x < 1 + 3x .
The case x ∈ (2, ∞): Set f (x) := α1 ω x , g(x) := α2 ω −x , h(x) := 1 + 3x
for x ∈ (0, ∞). Because α2 > 0 and ω > 1, g is strictly decreasing, thus
f (x) + g(x) < f (x) + g(2) for x > 2. We will show f (x) + g(2) < h(x) for
x > 2. Because f (2) + g(2) = h(2), this is done if we prove f 0 (x) < h0 (x)
for x > 2, which is equivalent to α1 ( ω3 )x ln ω < ln 3. This inequality is true
for x = 2. All factors of its left side are positive, and ω < 3, so the left side is
strictly decreasing for x ≥ 2. Hence the inequality is true for x > 2 as well.
ln 3
The case x ∈ (0, 1): After substituting s := ω x and setting δ := ln
, we
ω
have to prove the equivalent inequality
1
1 1
s+ +
s−
< 1 + sδ
s r
s
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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for s ∈ (1, ω), which can be done by building a sandwich with a suitable polyJ. Ineq. Pure and Appl. Math. 6(3) Art. 87, 2005
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nomial function inside: Set
1
s−1
s−
, p(s) := 2 1 +
,
s
ω−1
(s − 1)(s − ω)
q(s) :=
(ϕ(3) − p(3))
(3 − 1)(3 − ω)
1 1
ϕ(s) := s + +
s r
for s > 0. The claim is
ϕ(s) < p(s) + q(s) < 1 + sδ
A Minkowski-Type Inequality for
the Schatten Norm
for s ∈ (1, ω). The left inequality is verified by the fact that s · (p(s) + q(s) −
ϕ(s)) defines a polynomial of degree 3 with three zeros {1, ω, 3}, where 1 <
ω < 3, and with positive leading coefficient λ := 12 (ϕ(3) − p(3))/(3 − ω). To
prove the second inequality, we inspect
Markus Sigg
δ
ψ(s) := 1 + s − p(s) − q(s)
for s > 0 and get ψ 00 (s) = δ(δ − 1)sδ−2 − 2λ. Because 1 < δ < 2, ψ 00 has a
unique zero
1
δ(δ − 1) 2−δ
, 1 < s0 < ω,
s0 :=
2λ
with ψ 00 (s) > 0 for s ∈ (0, s0 ) and ψ 00 (s) < 0 for s ∈ (s0 , ∞). Now ψ(1) = 0,
ψ 0 (1) > 0, and ψ 00 (s) > 0 for s ∈ (1, s0 ) show ψ(s) > 0 for s ∈ (1, s0 ], while
ψ(s0 ) > 0, ψ(ω) = 0, ψ 0 (ω) < 0, and ψ 00 (s) < 0 for s ∈ (s0 , ω) show ψ(s) > 0
for s ∈ [s0 , ω).
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Example 3.2. Inequality (MS) is violated for 0 < p ≤ ∞ and c < 1 as well as
c > 2 by
0 0
0 0
2 −1
F :=
, R :=
, S :=
.
1 0
0 1
−1 1
Proof. Evaluation of the matrix powers for t ∈ (0, ∞) gives
Rt = R, S t =
(R
1
(α1 ω t + α2 ω −t )
2
1
(ω −t − ω t )
r
t
1 1+3
t
+ S) =
t
2
1−3
1
(ω −t
r
− ωt)
!
ω t + α1 ω −t )
!
1 − 3t
1
(α2
2
,
1 + 3t
Markus Sigg
√
with r := 5, α1 := 1 + 1r , α2 := 1 − 1r , ω := 3+r
. For U ∈ {R, S, R + S} we
2
get in case of p < ∞
1 √
t
2t ∗ p2 p
= ut
|F U |p = tr (F U F )
2
with ut being the top left entry of U 2t . Using |F U t |∞ = |F U 2t F ∗ |∞ , the case
p = ∞ yields the same result, thus for all p:
r
1
1
1
(α1 ω 2/c + α2 ω −2/c ),
F R c = 0, F S c =
2
p
p
r
1
1
(1 + 32/c ).
F (R + S) c =
2
p
2
c
x
−x
A Minkowski-Type Inequality for
the Schatten Norm
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x
Substituting by x, we have to prove α1 ω + α2 ω < 1 + 3 for x ∈ (2, ∞)
and for x ∈ (0, 1), which is the statement of Lemma 3.1.
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4.
Some Numerical Evidence
To justify Conjecture 1, we present the results of a numerical study performed
with 2 × 2 matrices.
From functional calculus it is known: For an operator T ≥ 0 on a complex
Hilbert space the powers T α , T β for α, β ∈ (0, ∞) obey the rule T α T β =
T α+β . If T is invertible, then T α can be defined for α ≤ 0 as well, and T α T β =
T α+β is true for all α, β ∈ R.
Before turning to the matrix case, we note the following general lemma.
Lemma 4.1. Let H, K, F be as above and α ∈ (0, ∞).
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
α
(a) Let T ∈ L(H)+ . Then F T = 0 if and only if F T = 0.
(b) Let R, S ∈ L(H)+ . Then F (R + S)α = 0 if and only if F Rα = 0 and
F S α = 0.
2
Proof. (a) Suppose F T α = 0. Then |F T α/2 | = |F T α F ∗ | = 0, hence F T α/2 =
0. Repeated application yields β ∈ (0, 1) with F T β = 0, thus F T = F T β T 1−β
= 0.
Now suppose F T = 0. There is nothing to prove in the case of α = 1,
so assume α 6= 1. If T is invertible, then F T α = F T T α−1 = 0. If T is not
invertible, then we have 0 ∈ σ(T ), the spectrum of T . Choose polynomials fn ∈
R[t] for n ∈ N such that fn (x) → xα for n → ∞ uniformly for x ∈ σ(T ). Then
fn (T ) → T α and F fn (T ) → F T α for n → ∞, hence F fn (T ) = fn (0)F → 0
for n → ∞, thus F T α = 0.
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(b) Part (a) shows:
F Rα = 0 ∧ F S α = 0
⇐⇒ F R = 0 ∧ F S = 0
=⇒ F (R + S) = 0
⇐⇒ F (R + S)α = 0.
To prove the missing implication, suppose F (R + S) = 0. Then F RF ∗ +
F SF ∗ = 0. Because F RF ∗ ≥ 0 and F SF ∗ ≥ 0, we get F RF ∗ = 0, thus
|F R1/2 |2 = |F RF ∗ | = 0 and F R1/2 = 0. Applying (a) again gives F R = 0.
Symmetry shows F S = 0.
We will also use the following well-known property of 2 × 2 matrices:
Lemma 4.2. A complex 2 × 2 matrix M is positive semidefinite if and only if
there exist a, b ∈ [0, ∞) and γ ∈ C with |γ|2 ≤ ab such that
a γ
M=
.
γ b
Lemma 4.1(b) shows that, when checking Conjecture 1, one may assume the
denominator to be non-zero, or setting 00 := 0, in
1 c
F (R + S) c p
.
qc,p (F, R, S) := 1 c
1 c
F R c + F S c p
p
We are searching for the supremum of qc,p (F, R, S) over all complex 2 × 2
matrices F, R, S with R, S ≥ 0. For r ∈ [0, ∞) and x ∈ C define r ∧ x := x
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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if |x| ≤ r and r ∧ x := (r/|x|) x otherwise. Lemma 4.2 shows that R has the
structure
α2
|α β| ∧ γ
R=
=: P (α, β, γ)
|α β| ∧ γ
β2
with α, β ∈ R and γ ∈ C, and a corresponding representation is valid for
the matrix S. This means that we have to deal with six complex and four real
variables, resulting in a 16-dimensional real optimisation problem: For λ =
(λ1 , . . . , λ16 ) ∈ R16 we set
λ1 + λ2 i λ 3 + λ4 i
Fλ :=
,
λ5 + λ6 i λ 7 + λ8 i
Rλ := P (λ9 , λ10 , λ11 + λ12 i),
Sλ := P (λ13 , λ14 , λ15 + λ16 i)
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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and are asking for
σ(c, p) := sup qc,p (Fλ , Rλ , Sλ ).
λ∈R16
To attack this problem, GNU Octave [5], version 2.1.57, was utilised. It
offers a function for determining the singular values of a matrix, which can
be employed for calculating the Schatten norms. For the optimisation task the
implementation [6], version 2002/05/09, with standard parameters of the Downhill Simplex Method of Nelder and Mead ([7], 10.4) was used. The results are
in perfect agreement with Conjecture 1. For visualisation, approximations for
σ(c, p) for c ∈ {1.2, 1.4, 1.6, 1.8, 2.0} have been calculated and plotted with a
step size of 0.01 for p, see Figure 1.
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Figure 1: Experimental approximations of σ(c, p).
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
c = 1.2
c = 1.4
c = 1.6
c = 1.8
c = 2.0
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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1
1.2
1.4
1.6
1.8
2
p
The apparently smooth shape of p 7→ σ(c, p) for p ≤ c, together with the
fact that for each p a new random starting point λ was used for the Nelder-Mead
algorihm, gives some confidence in the validity of the data.
A closer inspection of some of the calculated numerical values suggests
1
1
σ(2, 1) = 2, σ 32 , 1 = σ 95 , 65 = 2 2 , σ 85 , 65 = σ 2, 32 = 2 3 ,
1
1
σ 45 , 1 = σ 32 , 54 = σ 74 , 75 = σ 2, 85 = 2 4 , σ 65 , 1 = σ 95 , 32 = 2 5 ,
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which leads to the idea to look at log2 σ(c, p). It seems there is a linear depen-
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dency of log2 σ(c, p) from c if c ≥ p. This observation will be made precise in
the next section.
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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5.
Generalisation of (MS)
It is natural to generalise (MS) and to ask for the smallest σ(c, p) ∈ [0, ∞] for
c ∈ (0, ∞) and p ∈ (0, ∞] such that
1 c
1 c
1 c
F (R + S) c ≤ σ(c, p) F R c + F S c p
p
p
for all F ∈ cp (H, K) and R, S ∈ L(H)+ (and for all complex Hilbert spaces
H and K). It is tempting to call σ(c, p) the Schatten-Minkowski constant for
(c, p). By choosing F 6= 0 and setting R to be the identity and S := 0 it can be
seen that σ(c, p) ≥ 1. Now Conjecture 1 can be re-phrased using σ(c, p), and,
motivated by the numerical results, we add another conjecture:
Conjecture 2. (a) For 1 ≤ c ≤ 2 and p ≥ c we have σ(c, p) = 1.
c
(b) For 0 ≤ c ≤ 2 and p ≤ c we have σ(c, p) = 2 p −1 .
Again, the cases c = 1 and c = 2 are not too difficult to prove:
Theorem 5.1.(
1
(a) σ(1, p) =
1
2 p −1
(
1
(b) σ(2, p) =
2
2 p −1
for p ≥ 1
for p ≤ 1
for p ≥ 2
.
for p ≤ 2
Proof. σ(1, p) ≤ 1 for p ≥ 1 and σ(2, p) ≤ 1 for p ≥ 2 is the subject of
Theorem 2.1, while σ(c, p) ≥ 1 is noted above. Example 3.1 tells us that
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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σ(c, p) ≥ 2c/p−1 for 0 < p ≤ c < ∞, yielding
1
σ(1, p) ≥ 2 p −1 for p ≤ 1
2
σ(2, p) ≥ 2 p −1 for p ≤ 2.
and
Now for the missing ‘≤’ inequalities. For the case c = 1, recall the inequality
between the power means of degrees p ≤ 1 and 1, see e.g. [8], 8.12, which
reads
p
1
α + βp p
α+β
≤
or equivalently αp + β p ≤ 21−p (α + β)p
2
2
for α, β ∈ [0, ∞). Together with the quasi-norm inequality of | · |p this gives
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
|F (R + S)|pp ≤ |F R|pp + |F S|pp ≤ 21−p (|F R|p + |F S|p )p
1
and thus |F (R + S)|p ≤ 2 p −1 (|F R|p + |F S|p ).
For the case c = 2, start with the power means inequality for the degrees
p ≤ 2 and 2,
p
1 2
1
p
p
α + βp p
α + β2 2
≤
or equivalently αp + β p ≤ 21− 2 (α2 + β 2 ) 2
2
2
for α, β ∈ [0, ∞). Together with the quasi-norm inequality of | · | p this gives
2
p
p
1
F (R + S) 2 = |F (R + S)F ∗ | p2
p
2
∗
p
2
p
2
∗
p
2
p
2
≤ |F RF | + |F SF |
p2
p
1 p
1 p
1 2
1 2
1−
= F R 2 + F S 2 ≤ 2 2 F R 2 + F S 2 p
p
p
p
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and consequently
2
1 2
1 2
1 2
−1
F (R + S) 2 ≤ 2 p
F R 2 + F S 2 .
p
p
p
A Minkowski-Type Inequality for
the Schatten Norm
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6.
Conclusion
Starting with Conjecture 1, which we proved for the cases c = 1 and c = 2
in Theorem 2.1, a numerical study of 2 × 2 matrices led to the generalised
Conjecture 2, which we also proved for c = 1 and c = 2 in Theorem 5.1.
The given proofs make use of the (quasi-) triangle inequality of the Schatten
(quasi-) norm. Another ingredient to Theorem 5.1 is the power means inequality. Presumably, a combination of these inequalities shall also be central when
dealing with the case c 6= 1, 2. However, it is unclear how to apply the triangle inequality in this situation, because there is no obvious way to get from
F (R + S)1/c to an expression where R and S can be separated.
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the Schatten Norm
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References
[1] C.A. MCCARTHY, cp , Israel J. Math., 5 (1967), 249–271.
[2] J. WEIDMANN, Linear Operators in Hilbert Spaces, Springer, New York
1980.
[3] R. MEISE AND D. VOGT, Introduction to Functional Analysis, Clarendon
Press, Oxford 1997.
[4] M. SIGG, Zur sesquilinearen Integration Schatten-operatorwertiger Funktionen, Konstanzer Schriften in Mathematik und Informatik, 200 (2004),
http://www.inf.uni-konstanz.de/Preprints/.
[5] J.W. EATON et al, http://www.octave.org.
[6] E. GROSSMANN, http://octave.sourceforge.net/index/
f/nelder_mead_min.html.
[7] W.H. PRESS, B.P. FLANNERY, S.A. TEUKOLSKY AND W.T. VETTERLING, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press 1992, http://www.nr.com.
[8] J. HERMAN, R. KUČERA
Springer, New York (2000).
AND
J. ŠIMŠA, Equations and Inequalities,
A Minkowski-Type Inequality for
the Schatten Norm
Markus Sigg
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J. Ineq. Pure and Appl. Math. 6(3) Art. 87, 2005
http://jipam.vu.edu.au