Isabelle Sivignon
email: isabelle.sivignon@gipsa-lab.grenoble-inp.fr

Exact and Optimal Conversion of a Hole-free 2d Digital Object into a Union of Balls in Polynomial Time

This paper addresses the problem of converting a 2d digital object, i.e. a set 𝑆 of points in ℤ 2 , into a finite union of balls ℬ centered on ℝ 2 , such that the digitization of ℬ is exactly 𝑆 and the cardinality of ℬ is minimum. We prove that, for the specific case of 2d hole-free digital objects, there exists a greedy polynomial-time algorithm. The algorithm is based on the same principle as the simple greedy optimal algorithm for the interval cover problem. After bringing to light under which conditions the latter algorithm can be extended to tree-like structures, we show that such a structure can be defined for any hole-free 2d digital object, so that the extended algorithm applies.

Introduction

Computer representation of shapes is a basic component to digitize, create, visualize or exchange models of physical objects. Different geometric models exist, either to represent the surface (B-rep, point clouds, triangle meshes) or the volume (tetrahedral meshes, digital objects, CSG models) of a solid shape. However, the model used to create or register a shape is not always the one tailored for subsequent processings or applications. Thus, the problem of converting one geometric model into another has been widely studied, for a variety of models. In particular, many provably good conversion algorithms have been designed to output a finite union of balls from other models, including point clouds, polygonal meshes or digital shapes. Indeed, being composed of very simple geometric shapes, finite union of balls are useful in a number of applications, for instance detection of collisions in computer graphics [START_REF] Choi | Performance improvement of haptic collision detection using subdivision surface and sphere clustering[END_REF], or simulation of physical processes [START_REF] Mede | A medial axis based method for irregular grain shape representation in dem simulations[END_REF]. Various metrics can be used to measure the quality of the conversion such as the number of balls, or the difference in volume between the original model and the union of balls.

In this article, we consider the following problem: Problem 1. Given a 2d digital object 𝑆, compute a finite union of balls ℬ such that: ℬ covers exactly the points of 𝑆 (and no point of ℤ 2 \𝑆), and the cardinality of ℬ is minimum.

This problem is closely related to the more constrained problem where the balls of ℬ must be centered in ℤ 2 , which is NP-hard [START_REF] Coeurjolly | Finding a minimum medial axis of a discrete shape is np-hard[END_REF]. It is also very close to the class of well-studied set cover problems that are also NP-hard [START_REF] Cormen | Introduction to Algorithms, Third Edition[END_REF]. The input of the set cover problem is a pair (𝑋, ℛ), where 𝑋 is a set of points (generally in ℝ 𝑛 ) and ℛ is a family of subsets of 𝑋 called ranges. The problem is to find a minimum subset of ℛ that covers all the points of 𝑋. In our problem, 𝑋 = 𝑆 is a subset of ℤ 2 . However, the set of ranges ℛ is not part of the input, but is constrained to be a set of balls centered on ℝ 2 .

We show that, when 𝑆 is a 4-connected digital object and ℤ 2 \𝑆 = 𝑆 𝑐 has exactly one 8-connected component, the problem can be seen as a variant of the interval covering problem (1d set cover problem) for which an optimal greedy algorithm exists. The idea was introduced in [START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF] in the specific case of (𝛿, 𝜀)ball approximation problem: given a shape 𝑆, compute a finite union of balls included in the 𝛿-dilation of 𝑆 while covering its 𝜀-erosion. It was shown that, while the general problem is NP-hard [START_REF] Attali | 𝛿, 𝜀)-ball approximation of a shape: Definition and complexity[END_REF], a greedy optimal algorithm exists when the 𝛿-dilation of 𝑆 has a cycle-free medial axis [START_REF] Nguyen | Epsilon-covering: a greedy optimal algorithm for simple shapes[END_REF].

In Section 2, we revisit the results of [START_REF] Nguyen | Epsilon-covering: a greedy optimal algorithm for simple shapes[END_REF][START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF] in a more general context. We consider the case where the input is a generic set of ranges and exhibit sufficient conditions on this set to ensure that the greedy algorithm is optimal in this setting. Once the good tools and conditions have been defined, the proofs of termination and optimality unfold as in [START_REF] Nguyen | Epsilon-covering: a greedy optimal algorithm for simple shapes[END_REF][START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF]. In Section 3, we show how to implement this algorithm to compute an exact and optimal conversion of a 2d hole-free digital object into a finite union of balls.

General optimal greedy algorithm

Algorithm specification

For the sake of simplicity, given a subset of ranges 𝑅 we denote ⋃ 𝑅 = ⋃ 𝑟∈𝑅 𝑟. We use the same vocabulary as in [START_REF] Nguyen | Epsilon-covering: a greedy optimal algorithm for simple shapes[END_REF][START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF] in the broader context of sets of ranges. A covering of a set of ranges ℛ is a subset of ℛ that covers all the points in ⋃ ℛ. More formally, Definition 1 (Covering). Let ℛ be a set of ranges, and 𝑅 be a subset of ℛ. We say that 𝑅 is a covering of ℛ if ⋃ 𝑅 = ⋃ ℛ.

A covering 𝑅 is said to be minimal if no range can be removed from 𝑅 while keeping the covering property, and minimum if its cardinality is minimum among all possible coverings. In the following, we assume that ℛ can be endowed with a partial order ⪯ such that the poset (ℛ, ⪯) is anti-arborescent: Definition 2 (Anti-arborescence [START_REF] Fauvet | Explicit linearization of one-dimensional germs through tree-expansions[END_REF]). A poset (𝑉 , ⪯) is anti-arborescent if:

• for all 𝑣 ∈ 𝑉, the set of its successors {𝑣 ′ ∈ 𝑉 , 𝑣 ≺ 𝑣 ′ } is totally ordered.

• for any two incomparable elements 𝑣, 𝑣 ′ ∈ 𝑉, the predecessors of 𝑣 and the predecessors of 𝑣 ′ are pairwise incomparable.

A range 𝑟 ∈ ℛ is said to be maximal (resp. minimal) in 𝑅 ⊆ ℛ if for all 𝑟 ′ ∈ 𝑅, either 𝑟 ′ ⪯ 𝑟 (resp. 𝑟 ′ ⪰ 𝑟) or 𝑟 ′ and 𝑟 are incomparable. Given a range 𝑟 ∈ 𝑅, we define the domain covered by ranges smaller than 𝑟 : 𝐶(ℛ, ⪯ 𝑟) = (⋃ 𝑟 ′ ∈ℛ,𝑟 ′ ≺𝑟 𝑟 ′ )\𝑟. Similarly, we define the domain covered by ranges larger than or incomparable to 𝑟 :

𝐶(ℛ, 𝑟) = (⋃ 𝑟 ′ ∈ℛ,𝑟 ′ 𝑟 𝑟 ′ )\𝑟. Remark that by definition, if 𝑟 1 ⪯ 𝑟 2 then 𝐶(ℛ, ⪯ 𝑟 1 ) ∪ 𝑟 1 ⊆ 𝐶(ℛ, ⪯ 𝑟 2 ) ∪ 𝑟 2 and 𝐶(ℛ, 𝑟 1 ) ∪ 𝑟 1 ⊇ 𝐶(ℛ, 𝑟 2 ) ∪ 𝑟 2 .
It will also be useful later to extend these definitions to a set of ranges 𝑅 ⊆ ℛ: 𝐶(ℛ, ⪯ 𝑅) = ⋃ 𝑟∈𝑅 𝐶(ℛ, ⪯ 𝑟) and 𝐶(ℛ, 𝑅) = ⋂ 𝑟∈𝑅 𝐶(ℛ, 𝑟).

Figure 1(c) illustrates these notations in the case of ranges being balls -ℛ = ℬ: a partial order ⪯ on the balls of ℬ is depicted using arrows on the set of centers of the balls (in red). (ℬ, ⪯) being an anti-arborescence, it has a root, indicated with a cross. The sets 𝐶(ℬ, ⪯ 𝒷) and 𝐶(ℬ, 𝒷) are depicted respectively in green and orange for a specific ball 𝒷 outlined in dashed gray. Definition 3 (Partial covering). Let ℛ be a set of ranges, and 𝑅 be a subset of ℛ. We say that 𝑅 is a partial covering of ℛ if it is a covering of 𝐶(ℛ, ⪯ 𝑅), i.e. 𝐶(ℛ, ⪯ 𝑅) ⊆ ⋃ 𝑅. Definition 4 (Candidate range). Let 𝑅 ⊂ ℛ be a partial covering of ℛ. A range 𝑟 ∉ 𝑅 is candidate to 𝑅 if 𝑅 ′ = 𝑅 ∪ {𝑟} is also a partial covering of ℛ and ⋃ 𝑅 ⊊ ⋃ 𝑅 ′ .

A candidate range 𝑟 with respect to 𝑅 is said to be maximal if it is maximal in the set of candidate ranges. Algorithm 1 describes a greedy algorithm that computes a covering given a finite set of ranges ℛ. It uses the fact that, if (ℛ, ⪯) is anti-arborescent, a topological ordering of the elements of ℛ can be defined. The idea is pretty natural: considering ranges in topological order, if a range is critical for the set of uncovered points, then it is added to the covering.

Algorithm 1: GreedyCovering(ℛ,⪯) Preconditions: ℛ is finite, (ℛ, ⪯) is an anti-arborescent poset 1 𝑅 ← ∅; 2 𝑈 ← ⋃ ℛ (points of ⋃ ℛ not in ⋃ 𝑅); 3 for 𝑟 ∈ ℛ, in topological order do 4 if 𝑟 is a maximal candidate for 𝑈 then 5 𝑅 ← 𝑅 ∪ {𝑟}; 6 𝑈 ← 𝑈 \𝑟 7 return 𝑅
By definition of candidate range, and since Algorithm 1 only inserts candidate ranges to the computed covering, an invariant of Algorithm 1 is that 𝑅 is always a partial covering of ℛ. The next section is dedicated to the proof of the fact that, provided that ℛ fulfills two extra conditions, candidate ranges to non-empty subsets always exist (proving that Algorithm 1 terminates with a covering), and that Algorithm 1 computes a minimum-cardinal covering.

Correctness, termination and optimality of Algorithm 1

In the following, we prove that if the poset (ℛ, ⪯) fulfills the two conditions below, Algorithm 1 terminates and computes a minimum covering :

Property (1) for any 𝑟 1 , 𝑟 2 ∈ ℛ such that 𝑟 1 ∩ 𝑟 2 ≠ ∅, for all 𝑟 1 ≺ 𝑟 ≺ 𝑟 2 , 𝑟 1 ∩ 𝑟 2 ⊆ 𝑟.
Property (2) for 𝑥 ∈ ⋃ ℛ, let 𝐶𝑜𝑣(𝑥, ℛ) = {𝑟 ∈ ℛ, 𝑥 ∈ 𝑟} ; then ∀𝑥 ∈ ⋃ ℛ, 𝐶𝑜𝑣(𝑥, ℛ) admits a greatest element that is called the critical range of 𝑥 and is denoted Crit(𝑥, ℛ).

The proof of optimality requires several technical lemmas. These lemmas were stated and proven in [START_REF] Nguyen | Epsilon-covering: a greedy optimal algorithm for simple shapes[END_REF][START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF] for a specific family of ranges. We show here that they are still valid when the set of ranges fulfills above properties. The proofs are in general very similar, and simply call properties (1) or (2) when necessary. Space being limited, we only provide the most relevant ones.

The first lemma shows that any range 𝑟 separates the elements of ⋃ ℛ into three disjoint subsets of elements: those before, those in, and those after.

Lemma 1 (Proposition 4.10 [START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF]). Let 𝑟 ∈ ℛ. For any 𝑥 ∈ ⋃ ℛ, 𝑥 belongs to one and only one of the three subsets 𝑟, 𝐶(ℛ, ⪯ 𝑟), 𝐶(ℛ, 𝑟).

Proof. By definition, 𝑟 is disjoint from 𝐶(ℛ, ⪯ 𝑟) and 𝐶(ℛ, 𝑟). Suppose now that there exists an element 𝑥 ∈ ⋃ ℛ such that 𝑥 ∈ 𝐶(ℛ, ⪯ 𝑟) ∩ 𝐶(ℛ, 𝑟). Let 𝑟 -≺ 𝑟 such that 𝑥 ∈ 𝑟 -\𝑟 and 𝑟 + such that 𝑟 ≺ 𝑟 + or 𝑟 + and 𝑟 are incomparable and 𝑥 ∈ 𝑟 + \𝑟. By definition, 𝑥 ∉ 𝑟 but 𝑟 -and 𝑟 + are in 𝐶𝑜𝑣(𝑥, ℛ). By property (2), 𝐶𝑜𝑣(𝑥, ℛ) admits a greatest element 𝑟 𝑀 = Crit(𝑥, ℛ), i.e. 𝑟 -⪯ 𝑟 𝑀 , 𝑟 + ⪯ 𝑟 𝑀 and 𝑥 ∈ 𝑟 𝑀 . If 𝑟 𝑀 = 𝑟 -, then 𝑟 + ⪯ 𝑟 -≺ 𝑟, a contradiction. Thus 𝑟 𝑀 is a strict successor of 𝑟 -, as 𝑟. By Definition 2, they are comparable. If 𝑟 ≺ 𝑟 𝑀 , then by property (1), 𝑟 -∩ 𝑟 𝑀 ⊆ 𝑟, leading to a contradiction since 𝑥 ∈ 𝑟 -∩ 𝑟 𝑀 . If 𝑟 𝑀 ≺ 𝑟, then 𝑟 is a successor of 𝑟 𝑀 which is either a successor of 𝑟 + or 𝑟 + itself. Then 𝑟 + ≺ 𝑟 which is a contradiction with the fact that either 𝑟 ≺ 𝑟 + or 𝑟 and 𝑟 + are incomparable.

The following two lemmas were not stated as such in [START_REF] Nguyen | Epsilon-covering: a greedy optimal algorithm for simple shapes[END_REF][START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF], but used in the proofs. Lemma 2 shows that, given a partial covering, there always exists a candidate.

Lemma 2.

Let 𝑅 ⊆ ℛ be a minimal covering of ℛ. Let 𝑅 -⊊ 𝑅 be a partial covering, and 𝑅 + = 𝑅\𝑅 -. Then any range 𝑟 + minimal in 𝑅 + is candidate to 𝑅 -.

The proof is similar to part of the proof of Lemma 4.27 [START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF] and calls Lemma 1 to assert that the points of 𝐶(ℛ, ⪯ 𝑟 + ) are disjoint from 𝑟 + ∪ 𝐶(ℛ, 𝑟 + ) and thus cannot be covered by ranges in 𝑅 + . Lemma 2 implies in particular that any range 𝑟 = 𝑚𝑖𝑛 𝑥∈(⋃ ℛ)\𝑅 𝐶𝑟𝑖𝑡(𝑥, ℛ) is a candidate to 𝑅 (there may be several incomparable candidates). By definition of Crit(𝑥, ℛ), any range 𝑟 ′ ≻ 𝑟 does not contain the point 𝑝 = arg min 𝑥∈(⋃ ℛ)\𝑅 Crit(𝑥, ℛ), 𝑝 ∈ (⋃ ℛ)\𝑅, so that 𝑟 is actually a maximal candidate to 𝑅. Lemma 3. Let 𝑅 ⊆ ℛ be a minimal covering of ℛ. Let 𝑅 -⊊ 𝑅 be a partial covering, and let 𝑟 be a candidate to 𝑅 -. Then any range 𝑟 ′ ∈ ℛ\𝑅 -such that 𝑟 ′ ≺ 𝑟 is also a candidate to 𝑅 -.

Proof. Suppose by contradiction that there exists a range 𝑟 ′ ≺ 𝑟 that is not a candidate to 𝑅 -. Then there exists a point 𝑥 ∈ 𝐶(ℛ, ⪯ (𝑅 -∪ {𝑟 ′ })) which is not in 𝑅 -∪ {𝑟 ′ }. If 𝑥 were in 𝐶(ℛ, ⪯ 𝑅 -), it would be covered by 𝑅 -since 𝑅 -is a partial covering, a contradiction. So 𝑥 ∉ 𝐶(ℛ, ⪯ 𝑅 -), which implies 𝑥 ∈ 𝐶(ℛ, ⪯ 𝑟 ′ ). By definition of 𝐶, there exists a range 𝑟 ″ ≺ 𝑟 ′ that contains 𝑥. If 𝑥 ∈ 𝑟, then by Property (1), we get 𝑥 ∈ 𝑟 ′ , a contradiction. Thus 𝑥 ∉ 𝑟. By transitivity of ≺, we have 𝑟 ″ ≺ 𝑟. Using the fact that 𝑥 ∉ 𝑟, and by definition of 𝐶, we have 𝑥 ∈ 𝐶(ℛ, ⪯ 𝑟). Again by definition of 𝐶, we have

𝐶(ℛ, ⪯ 𝑟) ⊆ 𝐶(ℛ, ⪯ (𝑅 -∪ 𝑟)). 𝑟 being candidate to 𝑅 -, 𝐶(ℛ, ⪯ (𝑅 -∪ 𝑟)) ⊆ ⋃(𝑅 -∪ {𝑟}), a contradiction.
Combining the previous lemmas, we can prove that, to complete a partial covering 𝑅 -, it is necessary to add a range that is smaller than or equal to a maximal candidate to 𝑅 -. Proposition 1 (Lemma 4.27 [START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF]). Let 𝑅 ⊆ ℛ be a minimal covering of ℛ. Let 𝑅 -⊊ 𝑅 be a partial covering, and let 𝑟 be a maximal candidate to 𝑅 -. Then 𝑅\𝑅 -contains a candidate range that is smaller than or equal to 𝑟.

Theorem 1 (Theorem 10 [START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF]). Let ℛ be a finite set of ranges. Suppose that ℛ can be endowed with a partial order ⪯ such that (ℛ, ⪯) is an anti-arborescent poset, and fulfills Properties ( 1) and [START_REF] Dgtal | Digital Geometry Tools and Algorithms Library[END_REF]. Then, Algorithm 1 outputs a cardinal minimum covering of ℛ.

The proofs of the proposition and of the theorem follow exactly the ones of Lemma 4.27 and Theorem 10 in [START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF]. The proof of Proposition 1 calls Lemmas 2 and 3, and the proof of Theorem 1 appllies Proposition 1 to replace one by one the ranges of any optimal covering by the ranges computed by Algorithm 1.

From a digital set to a set of ranges

In this section, we show how Algorithm 1 can be used to solve Problem 1. Here, ranges are balls. Given a digital object 𝑆, a set of balls fulfilling Theorem 1 hypothesis is defined. Moreover, this set is such that the result of Algorithm 1 is indeed a collection of balls of minimum cardinality that covers 𝑆 exactly.

Let 𝑆 ⊂ ℤ 2 be a finite 4-connected digital object such that 𝑆 𝑐 = ℤ 2 \𝑆 has one exactly 8-connected component. A digital ball 𝑏 is a subset of ℤ 2 for which there exists a ball 𝒷 such that b ∩ ℤ 2 = 𝑏, where b denotes the interior of 𝒷. Otherwise said, if 𝐷𝑖𝑔 denotes the Gauss digitization function, we have which 𝐷𝑖𝑔(𝒷) = b ∩ ℤ 2 = 𝑏. In the following, we assume that balls 𝒷 are open, so that b = 𝒷. The preimage of a digital ball 𝑏, denoted 𝐷𝑖𝑔 -1 (𝑏) will be useful later on. A digital ball 𝑏 is said to be valid for a digital object 𝑆 if 𝑏 ⊆ 𝑆. It is said to be maximal if there is no other valid digital ball containing it.

Given a digital object 𝑆, we aim at finding a set of ranges ℬ that are (non empty) valid digital balls and such that ⋃ ℬ = 𝑆. Given a set of ranges as input, Algorithm 1 computes a minimum covering for this set of ranges. In order to obtain the minimum covering of a digital object 𝑆, the input set of ranges ℬ must contain all maximal digital balls valid for 𝑆. For instance, taking the set of balls ouput by a distance transform of 𝑆 is not enough to ensure optimality: indeed, all the balls of this set have a center in ℤ 2 , so that it misses all digital balls for which 𝐷𝑖𝑔 -1 (𝑏) contains only balls of center not in ℤ 2 .

The next sections are dedicated to exhibiting a way to grasp the set of all valid maximal digital balls and showing that this set can be endowed with an anti-arborescent poset structure that fulfills sufficient properties (1) and (2).

Getting a grip on valid maximal digital balls

The center of a ball 𝒷 is denoted by 𝑐(𝒷). For 𝑝 ∈ ℤ 2 , let pixel(𝑝) be the unit square centered on 𝑝. For any ball 𝒷 such that 𝑐(𝒷) ∈ 𝑝𝑖𝑥𝑒𝑙(𝑞), 𝑞 ∈ 𝑆 𝑐 , either 𝐷𝑖𝑔(𝒷) = ∅ or 𝐷𝑖𝑔(𝒷) ∩ 𝑆 𝑐 ≠ ∅. These balls do not contribute to the set of valid maximal digital balls and can be discarded. Consequently we define 𝒮 = ⋃ 𝑝∈𝑆 pixel(𝑝) and restrict the study to this set. For 𝑥 ∈ 𝒮, let 𝒷 𝑆 (𝑥) be the maximal ball centered in 𝑥 such that 𝐷𝑖𝑔(𝒷 𝑆 (𝑥)) ⊆ 𝑆. Note that by maximality, 𝜕𝒷 𝑆 (𝑥) contains at least one point of 𝑆 𝑐 . The following Lemma shows that any valid maximal digital ball has a ball in its preimage with at least two points of 𝑆 𝑐 on its boundary.

Lemma 4.

Let 𝑏 be a valid maximal digital ball for 𝑆. Then there exists 𝒷 such that 𝐷𝑖𝑔(𝒷) = 𝑏 and |𝜕𝒷 ∩ 𝑆 𝑐 | ≥ 2.

Proof. Let 𝒷 ′ be a ball such that b′ ∩ ℤ 2 = 𝑏. If 𝜕𝒷 ′ ∩ 𝑆 𝑐 = ∅, then we increase the radius of 𝒷 ′ until 𝒷 ′ = 𝒷 𝑆 (𝑐(𝒷 ′ )). 𝜕𝒷 ′ contains at least one point of 𝑆 𝑐 . Now we use a classical projection from a set of balls to the balls of the medial axis of a shape [START_REF]Image analysis and mathematical morphology 2: theoretical advances[END_REF][START_REF] Lieutier | Any open bounded subset of ℝ 𝑛 has the same homotopy type as its medial axis[END_REF]. The shape considered here is the whole space ℝ 2 punctured by the discrete set 𝑆 𝑐 . In this simple case, the medial axis is simply the set of edges of the Voronoi diagram of 𝑆 𝑐 , i.e. 𝜕Vor(𝑆 𝑐 ). The projection is illustrated in Figure 1: it associates to any ball 𝒷 a ball 𝜋(𝒷) centered on 𝜕Vor(𝑆 𝑐 ) and such that 𝒷 ⊆ 𝜋(𝒷). This projection is well defined since 𝑆 is finite (in particular, no half-space is void of points of 𝑆 𝑐 ). Consider the ball 𝜋(𝒷 ′ ). If 𝐷𝑖𝑔(𝜋(𝒷 ′ )) ≠ 𝑏, we have a contradiction with the maximality of 𝑏, and otherwise, we have found a ball 𝒷 such that 𝐷𝑖𝑔(𝒷) = 𝑏 and |𝜕𝒷 ∩ 𝑆 𝑐 | ≥ 2. Consequently, for any valid maximal digital ball 𝑏, there exists a ball 𝒷 in 𝐷𝑖𝑔 -1 with 𝑐(𝒷) ∈ 𝜕Vor(𝑆 𝑐 ) ∩ 𝒮. Note that: (i) all the balls 𝒷 with 𝑐(𝒷) in this set are such that 𝐷𝑖𝑔(𝒷) is valid for 𝑆 ; (ii) some balls 𝒷 with 𝑐(𝒷) in this set may however be such that 𝐷𝑖𝑔(𝒷) is not maximal. In the following, we denote Vor ⊓ (𝑆) = 𝜕Vor(𝑆 𝑐 ) ∩ ⋃ pixel(𝑆) (see Figure 2(a)), and we consider the set of balls ℬ = {𝒷 𝑆 (𝑥), 𝑥 ∈ Vor ⊓ (𝑆)}. This set contains all the balls which digitization is a valid maximal digital ball for 𝑆.

Ordering balls of ℬ

By construction, Vor ⊓ (𝑆) is a collection of segments. Lemma 5. Vor ⊓ (𝑆) is a geometric embedding of a tree in ℝ 2 .

Proof. Suppose that Vor ⊓ (𝑆) contains a cycle. This cycle is a Jordan curve, and since it is a subset of 𝜕Vor(𝑆 𝑐 ) it must contain a point of 𝑆 𝑐 in its interior. Moreover, this cycle is included in 𝒮, which is an open polygon containing no point of 𝑆 𝑐 since 𝑆 is 4-connected and 𝑆 𝑐 is 8-connected. A contradiction.

Vor ⊓ (𝑆) being a tree, it can be endowed with a partial order by picking any point on it as a root: indeed, it is enough to orient each edge/segment from the leaves to the root. This results in an oriented tree, denoted by 𝒯, that defines a partial order ≤ 𝒯 on the set (of centers) of balls ℬ (see Figure 1(c)). By construction, (ℬ, ≤ 𝒯 ) is an anti-arborescent poset. Moreover, for any 𝑝 ∈ 𝑆, the set of centers of the balls of 𝐶𝑜𝑣(𝑝, ℬ) = {𝒷 ∈ ℬ, 𝑝 ∈ 𝒷} is a connected subset of Vor ⊓ (𝑆). Lemma 6 (Lemma 4.9 [START_REF] Nguyen | Finite unions of balls with inner and outer margins[END_REF]). Let 𝑝 ∈ 𝑆. If 𝑝 ⊆ b1 ∩ b2 , then 𝑝 ⊆ b for all 𝒷 such that 𝑐(𝒷) is on the unique path Γ(𝒷 1 , 𝒷 2 ) between 𝑐(𝒷 1 ) and 𝑐(𝒷 2 ) in Vor ⊓ (𝑆).

The proof uses projection 𝜋 defined in the previous section, together with the fact that Vor ⊓ (𝑆) is the geometric embedding of a tree.

This lemma implies that Property (1) is true for ℬ. It moreover implies that for all 𝑝, 𝐶𝑜𝑣(𝑝, ℬ) admits a supremum according to the order 𝒯. However, since the balls of ℬ are open, these sets are open too (see illustration in Figure 2(b)), except for points 𝑝 that belong to the balls that are either the root or leaves of 𝒯. A consequence is that, in general, 𝐶𝑜𝑣(𝑝, ℬ) does not admit a greatest element, and 𝑝 ∉ 𝐷𝑖𝑔(sup 𝒯 𝐶𝑜𝑣(𝑝, ℬ)). This results in the following property: Lemma 7. For any 𝑝 ∈ 𝑆 that does not belong to the root of 𝒯, sup 𝒯 𝐶𝑜𝑣(𝑝, ℬ) either belongs to an open segment of Vor ⊓ (𝑆) or, if it is a vertex, the balls of 𝐶𝑜𝑣(𝑝, ℬ) are all in the same subtree of predecessors.

Proof. Suppose that sup 𝒯 𝐶𝑜𝑣(𝑝, ℬ) is a vertex 𝑣 ∈ Vor ⊓ (𝑆), and, by contradiction, pick any ball of 𝐶𝑜𝑣(𝑝, ℬ) in a first subtree, and another one in another subtree. Then the unique path between them goes through 𝑣, and the ball centered on 𝑣 must contain 𝑝 by Lemma 6 and thus be in 𝐶𝑜𝑣(𝑝, ℬ). It cannot be the supremum of 𝐶𝑜𝑣(𝑝, ℬ). the centers of all the balls of 𝐶𝑜𝑣(𝑝, ℬ) are on the blue segment, delimited by 𝑐(𝒷 1 ) and 𝑐(𝒷 2 ), but 𝒷 2 does not belong to 𝐶𝑜𝑣(𝑝, ℬ) since it contains 𝑝 on its boundary.

(a) 1 2 ( 1 ) ( 2 ) (b)
The set of ranges ℬ does not fulfill property (2), which is required for Algorithm 1 to be valid. We turn to the set ℬ = {𝑏 ⊆ 𝑆, ∃𝒷 ∈ ℬ 𝐷𝑖𝑔(𝒷) = 𝑏} instead. Since ℬ is finite, the sets 𝐶𝑜𝑣(𝑝, ℬ) = {𝑏 ∈ ℬ, 𝑝 ∈ 𝑏} are also finite and are good candidates to admit a greatest element if equipped with a partial order. We show hereafter how to do this without explicitly computing the set ℬ.

Ordering digital balls of ℬ

Let the representative of 𝑏 be Rep(𝑏) = sup 𝑇 {𝒷 ∈ ℬ, 𝒷 ∈ 𝐷𝑖𝑔 -1 (𝑏)}. As seen before, usually, 𝐷𝑖𝑔(Rep(𝑏)) ≠ 𝑏. From the partial order 𝒯 on ℬ, we define a partial order 𝑇 on ℬ as follows:

Definition 5. Given two digital balls 𝑏 1 and 𝑏 2 of ℬ, 𝑏 1 ≤ 𝑇 𝑏 2 if: (1) either 𝑏 1 = 𝑏 2 (2) or 𝑏 1 ≠ 𝑏 2 and (a) either Rep(𝑏 1 ) < 𝒯 Rep(𝑏 2 ) (b) or Rep(𝑏 1 ) = Rep(𝑏 2 ) and 𝐷𝑖𝑔(Rep(𝑏 2 )) = 𝑏 2 . Lemma 8. (ℬ, ≤ 𝑇 ) is a poset.
Sketch of proof. Reflexivity follows directly from (1). Antisymmetry is shown by contradiction considering two cases: either Rep(𝑏 1 ) ≠ Rep(𝑏 2 ) and we get a contradiction by Definition 5 and definition of 𝒯, or Rep(𝑏 1 ) = Rep(𝑏 2 ) and we have a contradiction with unicity of 𝐷𝑖𝑔(𝒷) using Definition 5 (2)(b). To show transitivity, the case 𝑏 1 = 𝑏 2 or 𝑏 2 = 𝑏 3 is trivial. Otherwise, we distinguish the two cases Rep(𝑏 1 ) ≠ Rep(𝑏 2 ) ≠ 𝑅𝑒𝑝(𝑏 3 ) and Rep(𝑏 1 ) = Rep(𝑏 2 ) and Rep(𝑏 2 ) ≠ Rep(𝑏 3 ) and conclude that Rep(𝑏 1 ) < 𝒯 Rep(𝑏 3 ) using the fact that 𝒯 is a partial order.

In order to prove that the poset (ℬ, ≤ 𝑇 ) is anti-arborescent, we need two extra lemmas that express properties of the sets 𝐷𝑖𝑔 -1 (𝑏). The following lemma, together with Lemma 6, moreover ensures that Property (1) is fulfilled for the set of ranges ℬ. Lemma 9. (𝑖) For any 𝑏 ∈ ℬ, 𝐷𝑖𝑔 -1 (𝑏) is connected. (𝑖𝑖) For any 𝑏, 𝑏 ′ ∈ ℬ, 𝑏 ≠ 𝑏 ′ , 𝐷𝑖𝑔 -1 (𝑏) ∩ 𝐷𝑖𝑔 -1 (𝑏 ′ ) = ∅. As a consequence, we have (𝑖𝑖𝑖): let 𝒷 ∈ 𝐷𝑖𝑔 -1 (𝑏) and 𝒷 ′ ∈ 𝐷𝑖𝑔 -1 (𝑏 ′ ) with 𝑏 ≠ 𝑏 ′ : if 𝒷 < 𝒯 𝒷 ′ , then 𝒷 ≤ 𝒯 Rep(𝑏) ≤ 𝒯 𝒷 ′ ≤ 𝒯 Rep(𝑏 ′ ).

Proof. (𝑖) follows from Lemma 6 since 𝐷𝑖𝑔 -1 (𝑏) = ⋂ 𝑝∈𝑏 𝐶𝑜𝑣(𝑝, ℬ). (𝑖𝑖) is straightforward by unicity of the digitization. To prove (𝑖𝑖𝑖), note that Rep(𝑏) and 𝒷 ′ are comparable since they are both successors of 𝒷 and 𝒯 is an antiarborescence. Suppose by contradiction that 𝒷 ′ ⪇ 𝒯 Rep(𝑏). Then 𝒷 ′ is on the unique path between 𝒷 and Rep(𝑏), a contradiction with (𝑖) and (𝑖𝑖). Lemma 10. Let 𝑏 1 , 𝑏 2 ∈ ℬ, 𝑏 1 ≠ 𝑏 2 , and 𝒷 = Rep(𝑏 1 ) = Rep(𝑏 2 ) such that 𝒷 ∉ 𝐷𝑖𝑔 -1 (𝑏 1 ) and 𝒷 ∉ 𝐷𝑖𝑔 -1 (𝑏 2 ). Then for all 𝒷 1 ∈ 𝐷𝑖𝑔 -1 (𝑏 1 ) and all 𝒷 2 ∈ 𝐷𝑖𝑔 -1 (𝑏 2 ), 𝒷 1 ans 𝒷 2 are incomparable.

Proof. Suppose by contradiction that 𝒷 1 < 𝒯 𝒷 2 . Thus 𝒷 1 ⪇ 𝒯 Rep(𝑏 1 ). Since 𝐷𝑖𝑔 -1 (𝑏 2 ) cannot be empty, Rep(𝑏 2 ) ∉ 𝐷𝑖𝑔 -1 (𝑏 2 ) implies that there exists 𝒷 2 ≠ Rep(𝑏 2 ) such that 𝒷 2 ∈ 𝐷𝑖𝑔 -1 (𝑏 2 ). Using Lemma 9, we get 𝒷 1 ⪇ 𝒯 Rep(𝑏 1 ) ≤ 𝒯 𝒷 2 ⪇ 𝒯 Rep(𝑏 2 ), which is a contradiction with Rep(𝑏 1 ) = Rep(𝑏 2 ). Next, we prove by contradiction that the predecessors of two incomparable balls 𝑏 1 and 𝑏 2 are also incomparable. It remains to prove that 𝐶𝑜𝑣(𝑝, ℬ) admits a greatest element for all 𝑝 ∈ 𝑆. To do so, we remark that the ball sup 𝒯 𝐶𝑜𝑣(𝑝, ℬ) of a point 𝑝 ∈ 𝑆 can be written as the maximum representative ball of 𝐶𝑜𝑣(𝑝, ℬ).

sup 𝒯 𝐶𝑜𝑣(𝑝, ℬ) = sup 𝒯 {𝒷 ∈ ℬ, 𝑝 ∈ 𝒷} = sup 𝒯 {𝒷 ∈ ℬ, 𝑝 ∈ 𝐷𝑖𝑔(𝒷)} = sup 𝒯 𝑏∈𝐶𝑜𝑣(𝑝,ℬ) {𝒷 ∈ ℬ, 𝐷𝑖𝑔(𝒷) = 𝑏} = sup 𝒯 𝑏∈𝐶𝑜𝑣(𝑝,ℬ) Rep(𝑏) = max 𝒯 𝑏∈𝐶𝑜𝑣(𝑝,ℬ) Rep(𝑏) (1)
The digital ball 𝑏 𝑚𝑎𝑥 (𝑝) ∈ ℬ that achieves the maximum in Equation ( 1) is actually the critical ball Crit(𝑝, ℬ). In the last subsection, we show how to compute it. Lemma 11. Let 𝑝 ∈ 𝑆, and 𝑏 𝑚𝑎𝑥 (𝑝) ∈ 𝐶𝑜𝑣(𝑝, ℬ) be such that Rep(𝑏 𝑚𝑎𝑥 (𝑝)) = sup 𝒯 𝐶𝑜𝑣(𝑝, ℬ). Then for any 𝑏 ∈ 𝐶𝑜𝑣(𝑝, ℬ), 𝑏 ≤ 𝑇 𝑏 𝑚𝑎𝑥 (𝑝).

Proof. Let 𝑏 ∈ 𝐶𝑜𝑣(𝑝, ℬ). If Rep(𝑏) < 𝒯 Rep(𝑏 𝑚𝑎𝑥 (𝑝)), by Definition 5, 𝑏 < 𝑇 𝑏 𝑚𝑎𝑥 (𝑝). The case Rep(𝑏) > 𝒯 Rep(𝑏 𝑚𝑎𝑥 (𝑝)) is not possible by definition of 𝑏 𝑚𝑎𝑥 (𝑝). The case Rep(𝑏) = Rep(𝑏 𝑚𝑎𝑥 (𝑝)) remains. Since 𝐷𝑖𝑔 -1 (𝑏) are connected and disjoint (Lemma 9), the only way for two balls 𝑏 1 and 𝑏 2 to have the same representative is when it is a vertex of the anti-arborescence. Then 𝐷𝑖𝑔 -1 (𝑏 1 ) and 𝐷𝑖𝑔 -1 (𝑏 2 ) belong to two different subtrees of this vertex, a contradiction with Lemma 7.

Computing critical balls

The first step is to find the edge of Vor ⊓ (𝑆) Rep(Crit(𝑝, ℬ)) belongs to. It is convenient to note that each edge of Vor ⊓ (𝑆) corresponds to balls of ℬ that go through a pair of points of 𝑆 𝑐 . This edge can then be described as a parabolic pencil of circles [START_REF] Boissonnat | Géométrie algorithmique[END_REF][START_REF] Schwerdtfeger | Geometry of complex numbers: circle geometry, Moebius transformation, non-euclidean geometry[END_REF] defined by two points of 𝑆 𝑐 and delimited by its two extremities. Each ball 𝒷 𝜆 of the pencil can be expressed as a convex combination of the two extremities, according to the following relation: ∀𝑝, pow(𝑝, 𝒷 𝜆 ) = (1 -𝜆)pow(𝑝, 𝒷 1 ) + 𝜆pow(𝑝, 𝒷 2 ), where pow denotes the power of a point with respect to a ball 𝒷(𝑐, 𝑟) and is equal to pow(𝑝, 𝒷) = 𝑑(𝑐, 𝑝) -𝑟 2 .

Given a topological order on the edges of Vor ⊓ (𝑆), consider the edges [𝒷 1 , 𝒷 2 ] in increasing order. If 𝑝 belongs to 𝐷𝑖𝑔(𝒷 . For all 0 ≤ 𝜆 ′ < 𝜆, 𝐷𝑖𝑔(𝒷 𝜆 ′ ) contains 𝑝 (see Figure 3(a)). We look for a value 𝜆 c𝑟𝑖𝑡 < 𝜆 such that 𝐷𝑖𝑔(𝒷 𝜆 ) ⊂ 𝐷𝑖𝑔(𝒷 𝜆 c𝑟𝑖𝑡 ). Such a value exist thanks to Lemma 7. For all the points 𝑞 ∈ 𝐷𝑖𝑔(𝒷 𝜆 )\𝐷𝑖𝑔(𝒷 1 ), let 𝒷 𝜆 𝑞 be the ball of [𝒷 1 , 𝒷 2 [ such that 𝑞 ∈ 𝜕𝒷 𝜆 𝑞 . For all values 𝜇 > 𝜆 𝑞 , 𝑞 ∈ 𝐷𝑖𝑔(𝒷 𝜇 ). By setting 𝜇 = max 𝑞 {𝜆 𝑞 }, we have that ∀𝜇 ′ > 𝜇, 𝐷𝑖𝑔(𝒷 𝜆 ) ⊂ 𝐷𝑖𝑔(𝒷 ′ 𝜇 ). By setting 𝜆 c𝑟𝑖𝑡 to any value strictly between 𝜇 and 𝜆, we have 𝐷𝑖𝑔(𝒷 𝜆 c𝑟𝑖𝑡 ) ⊃ {𝑝} ∪ 𝐷𝑖𝑔(𝒷 𝜆 ) as desired. Note that, as mentioned before, 𝐷𝑖𝑔(𝒷 𝜆 c𝑟𝑖𝑡 ) may not be maximal. Indeed, as illustrated in Figure 3(b), by definition of 𝒷 𝜇 there is no point of 𝑆 in the grey region. However, 𝐷𝑖𝑔(𝒷 𝜇 ) may contain of 𝑆 other than 𝑝, for instance point 𝑞 in the figure. Thus, if we consider the two balls 𝒷 and 𝒷 ′ both between 𝒷 𝜇 and 𝒷 𝜆 , 𝐷𝑖𝑔(𝒷 ′ ) ⊂ 𝐷𝑖𝑔(𝒷), so that 𝐷𝑖𝑔(𝒷 ′ ) is not maximal. As proven in the section before, this is not a problem: in the course of the algorithm, either 𝑞 belongs to the subset not covered yet, and then the critical ball of 𝑞, which is equal to 𝐷𝑖𝑔(𝒷), is chosen, or 𝑞 is already covered, and picking 𝐷𝑖𝑔(𝒷 ′ ) instead of 𝐷𝑖𝑔(𝒷) does not change anything.

Results

Algorithm 1 was implemented1 using three open-source libraries: DGtal [START_REF] Dgtal | Digital Geometry Tools and Algorithms Library[END_REF] to handle digital sets, CGAL [1] to compute Vor ⊓ (𝑆), and Boost Graph [START_REF][END_REF] to compute topological order on trees. A kernel with exact predicates and constructions was used to avoid rounding errors. As a conclusion, some results are presented in Figure 4.

Figure 1 :

 1 Figure 1: (a) The projection 𝜋(𝒷) of 𝒷 is defined from the center 𝑐(𝒷) and its closest point 𝑞 in 𝑆 𝑐 . (b) Projection 𝜋 is continuous on any continuous path: the continuous path in green is projected on the bolder dark green continuous subpath of the Voronoi diagram. Grey arrows represent the projection. (c) Illustration of a partial order (in red) on ℬ, and of the sets 𝐶(ℬ, ⪯ 𝒷) and 𝐶(ℬ, 𝒷).

Figure 2 :

 2 Figure 2: (a) Cropped Voronoi diagram Vor ⊓ (𝑆) for a set pixels 𝒮 depicted in grey. (b) 𝐶𝑜𝑣(𝑝, ℬ) is an open set. Part of Vor ⊓ (𝑆) is depicted in red :the centers of all the balls of 𝐶𝑜𝑣(𝑝, ℬ) are on the blue segment, delimited by 𝑐(𝒷 1 ) and 𝑐(𝒷 2 ), but 𝒷 2 does not belong to 𝐶𝑜𝑣(𝑝, ℬ) since it contains 𝑝 on its boundary.

Theorem 2 .

 2 The poset (ℬ, ≤ 𝑇 ) is anti-arborescent. Sketch of proof. We first prove by contradiction that the successors of any 𝑏 ∈ ℬ are comparable, by considering three cases: Rep(𝑏) = Rep(𝑏 1 ) = Rep(𝑏 2 ), or Rep(𝑏) = Rep(𝑏 1 ) and Rep(𝑏) < 𝒯 Rep(𝑏 2 ), or Rep(𝑏) < 𝒯 Rep(𝑏 1 ) and Rep(𝑏) < 𝒯 Rep(𝑏 2 ). In the first two cases, we have a direct contradiction with Definition 5. The third case is a little bit trickier and uses Lemma 10.

1 )

 1 but not to 𝐷𝑖𝑔(𝒷 2 ), then Rep(Crit(𝑝, ℬ)) belongs to the edge [𝒷 1 , 𝒷 2 [. Using the fact that 𝑝 ∈ 𝜕Rep(Crit(𝑝, ℬ)), and the relation above, we can compute the value 𝜆 such that 𝒷 𝜆 = Rep(Crit(𝑝, ℬ)) on the pencil [𝒷 1 , 𝒷 2 [, as 𝜆 = pow(𝑝,𝒷 1 ) pow(𝑝,𝒷 1 )-pow(𝑝,𝒷 2 )

Figure 3 :

 3 Figure 3: (a) Computation of Crit(𝑝, ℬ): 𝒷 𝜆 = Rep(Crit(𝑝, ℬ)); any ball between 𝒷 𝜇 and 𝒷 𝜆 (see for instance the ball in gray) contains 𝑝 and all the points of 𝐷𝑖𝑔(𝒷 𝜆 ) (circled). (b) Illustration of the fact that the critical ball is not always maximal.

Figure 4 :

 4 Figure 4: (a-c) Three different optimal coverings of the same toy example with 4 balls, obtained using different roots ; (d-g) Results on images of the database MPEG7 CE Shape-1 Part B :(d) 9 balls (e) 113 balls (f) 40 balls (g) 36 balls.

https://github.com/isivigno/ConvertDigitalObjectToBalls.git


Mohamed Guedda

Zakia Hammouch
email: zakia.hammouch@u-picardie.fr

Lamfa

Similarity flow solutions of a non-Newtonian power-law fluid

Keywords: Boundary-layer, Power-law fluid, Multiple solutions, Similarity transformation

. Under appropriate assumptions, partial differential equations are transformed into an autonomous third-order nonlinear degenerate ordinary differential equation with boundary conditions. Using a shooting method, we establish the existence of an infinite number of global unbounded solutions. The asymptotic behavior is also discussed. Some properties of those solutions depend on the viscosity power-law index.

Introduction

In view of their wide applications in different industrial processes, and also by the interesting mathematical features presented their equations, boundary-layer flows of non-Newtonian fluids have motivated researchers in many branches of engineering in recent years. The most frequently used model in non-Newtonian fluid mechanics is the Ostwald-de Wael model (with a power-law rheology [START_REF] Astarita | Principles of Non-Newtonian Fluid Mechanics[END_REF][START_REF] Astin | Boundary layers in non-Newtonian fluids[END_REF][START_REF] Bohme | Non-Newotonian Fluid Mechanics[END_REF][START_REF] Denier | On the boundary-layer equations for power-law fluids[END_REF][START_REF] Howell | Momentum and heat transfer on a continuous moving surface in a power-law fluid[END_REF]), which the relationship between the shear stress and the strain rate is given as follows

τ xy = k|u y | n-1 u y , (1)
for n = 1 the fluid is called Newtonian with dynamic coefficient of viscosity k. For n > 1 the behavior of the fluid is dilatant or shear-thickening and for 0 < n < 1 the behavior is shear-thinning, in these cases the fluid is non-Newtonian and k is the fluid consistency. In this work we shall restrict our study to the dilatant fluids, then throughout all the paper, the exponent n will be taken in the range (1, ∞). The problem of laminar flows of power-law non-Newtonian fluids have been studied by several authors. For the sake of brevity, we mention here some examples, Acrivos et al. [START_REF] Acrivos | Momentum and heat transfer in laminar boundary-layer flows Equations of non-Newtonian fluids past external surfaces A[END_REF] and Pakdemirli [START_REF] Pakdemirli | Similarity analysis of boundary layer equations of a class of non-Newtonian fluids[END_REF] derived the boundary layer equations of power-fluids, Mansutti and Rajagobal [START_REF] Mansutti | Flow of a shear thinning fluid between intersecting planes[END_REF] investigated the boundary layer flow of dilatant fluids. Adopting the Crocco variable formulation, Nachman and Talliafero [START_REF] Nachman | Mass transfer into boundary-layers for power-law fluids[END_REF] established existence and uniqueness of similarity solution for a mass transfer problem. Filipussi et al. [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] obtained similarity solutions and their properties using a phase-plane formalism. Recently numerical solutions have been given by Ece and Büyük in [START_REF] Ece | Similarity solutions for free convection to power-law fluids from a heated vertical plate[END_REF] for the steady laminar free convection over a heated flat plate. More recently Guedda [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF] studied the free convection problem of a Newtonian fluid, he showed the existence of an infinite number of solution and studied their asymptotic behavior. In this work we aim to extend the analysis of [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF] to the non-Newtonian case, we are interested also in the effect of the power-law index on the existence and the asymptotic behavior of solutions. The remainder of this work is organized as follows, in the next section, we introduce the mathematical formulation of the problem, section 3 deals with some preliminary tools which will be useful in section 4 and 5 to prove the main results. Finally, we give some concluding remarks in section 6.

Similarity procedure

The problem is geometrically defined by a semi-infinite power-law stretched rigid plate, over which flows a non-Newtonian fluid obeying to [START_REF] Blasius | Grenzschichten in Flüssigkeiten mit kleiner Reibung[END_REF]. The main hypotheses for the mathematical formulation of this problem are given by:

• Two-dimensional, incompressible and steady-state laminar flow,

• Physical properties are taken as constants,

• Body force, external gradients pressure and viscous dissipation are neglected.

Under these assumptions, and referred to a Cartesian system of coordinates Oxy, where y = 0 is the plate, the x-axis is directed upwards to the plate and the y-axis is normal to it, the continuity and momentum equations can be simplified, within the range of validity of the Boussinesq approximation [START_REF] Acrivos | Momentum and heat transfer in laminar boundary-layer flows Equations of non-Newtonian fluids past external surfaces A[END_REF], to the following equations

   uu x + vu y = ν(|u y | n-1 u y ) y , u x + v y = 0, (2)
The functions u and v are the velocity components in the xand ydirections respectively. The boundary conditions accompanied equation ( 2) are given by

u(x, 0) = U w (x), v(x, 0) = V w (x), u(x, y) → 0 as y → ∞. (3)
The functions : U w (x) = u w x m is called the stretching velocity and u w > 0, the exponent m is negative, and

V w (x) = v w x m(2n-1)-n n+1
is the suction/injection velocity where v w > 0 for suction and v w < 0 for injection. From the incompressibility of the fluid we introduce the dimensionless stream function ψ = ψ(x, y) satisfying (u = ψ y , v = -ψ x ). Hence equations ( 2) are reduced to the single equation

ψ y ψ xy -ψ x ψ yy = ν(|ψ yy | n-1 ψ yy ) y . (4)
The boundary conditions (3) are transformed into

ψ y (x, 0) = u w x m , ψ x (x, 0) = -v w x m(2n-1)-n n+1
, ψ y (x, y) → 0 as y → ∞.

(

) 5
Since the broad goal of this paper is to obtain similarity solutions to (4), [START_REF] Denier | On the boundary-layer equations for power-law fluids[END_REF] we introduce the following similarity transformations ψ(x, y)

:= Ax α f (t), t := B y x β . (6)
Where A, B, α and β are real numbers, f is the transformed dimensionless stream function and t is the similarity variable. In terms of (6), equation (4) can satisfy the ordinary differential equation of the shape function :

f (|f ′′ | n-1 f ′′ ) ′ + αf f ′′ = (α -β)f ′ 2 , (7)
where the primes denote differentiation with respect to t, if and only if the following

α(2 -n) + β(2n -1) = 1,
It follows that

α = 1 + m(2n -1) n + 1 , β = 1 + m(n -2) n + 1
, and p = m(2n -1)n n + 1 .

Consequently, we have

ψ(x, y) := ν 1 n+1 x 1+m(2n-1) n+1 f (t), t := ν -1 n+1 yx -1+m(n-2) n+1 . ( 9
)
The corresponding boundary conditions [START_REF] Denier | On the boundary-layer equations for power-law fluids[END_REF] are expressed as

f (0) = -v w Aα , f ′ (0) = u w A , f ′ (∞) = lim t→∞ f ′ (t) = 0. (10)
In the remainder, we deal with the following problem

   (|f ′′ | n-1 f ′′ ) ′ + αf f ′′ -mf ′ 2 = 0, f (0) = a, f ′ (0) = b, f ′ (∞) = lim t→∞ f ′ (t) = 0. (11)
For Newtonian fluids (n = 1), problem [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] reads

   f ′′′ + αf f ′′ -mf ′ 2 = 0, f (0) = a, f ′ (0) = b, f ′ (∞) = 0. (12)
We notice that this problem arises from two different contexts in fluid mechanics when looking for similarity solutions. First, in natural convection along a vertical heated flat plate, embedded in a saturated porous medium, where the temperature is a power function with the exponent m, for more details, we refer the reader to [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF][START_REF] Belhachmi | On a family of differential equation for boundary layer approximations in porous media[END_REF][START_REF] Cheng | Free-convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike[END_REF] and the references therein. Equation [START_REF] Ece | Similarity solutions for free convection to power-law fluids from a heated vertical plate[END_REF] appears also in the study of the boundary layer flow, of a Newtonian fluid, adjacent to a stretching surface with a power-law velocity (see [START_REF] Brighi | On a similarity boundary layer equation[END_REF][START_REF] Magyari | Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls[END_REF]).

In [START_REF] Belhachmi | On a family of differential equation for boundary layer approximations in porous media[END_REF], the authors proved that ( 12) with a = b = 0, has a solution (which is bounded) if m ≥ -1 3 and this solution is unique for 0 ≤ m ≤ 1 3 . In [START_REF] Brighi | On a similarity boundary layer equation[END_REF] the author gives a complete study about existence and nonexistence of solutions to [START_REF] Ece | Similarity solutions for free convection to power-law fluids from a heated vertical plate[END_REF], where b = 1.

Recently some new results have been obtained in [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF]. The author considered the problem [START_REF] Ece | Similarity solutions for free convection to power-law fluids from a heated vertical plate[END_REF], where m ∈ (-α, 0). He showed that, under some assumptions, problem [START_REF] Ece | Similarity solutions for free convection to power-law fluids from a heated vertical plate[END_REF] has an infinite number of unbounded solutions and these solutions satisfy f (t) ∼ t α α-m , as t goes to infinity. Based on the results of [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF], the interest in this work will be in existence and asymptotic behavior of solutions of problem [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF].

Preliminary results

As it is announced above, the existence of solutions will be established by a shooting method. We replace the boundary condition at infinity by f ′′ (0) = d, where d = 0. Therefore, we consider the initial value problem

   (|f ′′ | n-1 f ′′ ) ′ + αf f ′′ -mf ′ 2 = 0, f (0) = a, f ′ (0) = b, f ′′ (0) = d. (13)
We shall see that for appropriate d problem ( 13) has a global unbounded solution and this solution satisfies the boundary condition at infinity.

Remark 1

We notice that for n = 1, equation ( 11) 1 can be degenerate or singular at the point t 0 where f ′′ (t 0 ) = 0. The existence the t 0 is done for d > 0. We shall see also that f ′′′ is not bounded at t 0 (the solution f is then not classical). By a solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] we will mean a function

f ∈ C 2 (0, ∞) such that |f ′′ | n-1 f ′′ ∈ C 1 (0, ∞), f ′ (∞) = 0 and f ′′ (∞) = 0.
|f ′′ d | n-1 f ′′ d (t) + αf ′ d (t)f d (t) = |d| n-1 d + αab + (m + α) t 0 f ′ d (s) 2 ds, ∀ t < T d . (14)
which will be used later for proving some results.

A solution f d of [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF], is of class C 2 on [0, T d ), and satisfies

|f ′′ d | n-1 f ′′ d ∈ C 1 ([0, T d ))
. We shall investigate whether f d admits an entire extension. First, we give the following result characterizing the existence time T d .

Proposition 2 Let f d be the local solution to [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF], if T d is finite then the functions f d , f ′ d and f ′′ d are unbounded as t approaches T d from below.

Proof. Similar to [START_REF] Brighi | On a similarity boundary layer equation[END_REF][START_REF] Coppel | On a differential equation of boundary layer theory[END_REF].

Let us note also that if we require a classical solution of (13) (ie. f ∈ C 3 (0, ∞)), it is possible that f ceases to exist at some T < ∞ and such that f, f ′ and f ′′ remain bounded on [0, T ). More precisely we have the following result.

Proposition 3 Let f d be the local solution to [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF] where n > 1 ans d = 0. Assume that there exists

t 0 ∈ (0, T d ) such that f ′′ d (t 0 ) = 0. Then d > 0, f ′′ d < 0 on (t 0 , T d ) and f ′′′ d is unbounded on (0, t 0 ).
Proof. Assume first that d < 0. Therefore f ′′ d < 0 on [0, ε), ε small, and the following equation

n|f ′′ d | n-1 f ′′ d + αf f ′′ -mf ′2 = 0, (15)
holds on (0, ε). Hence

(f ′′ d e F ) ′ = m n e F |f ′′ d | 1-n f ′ d 2 , on (0, ε), (16)
where

F (t) = α n t 0 f d |f ′′ d | 1-n (s)ds.
Consequently, the function t → f ′′ d e F (t) is decreasing, and then f ′′ d (t) remains negative for all t ∈ [0, T d ). A contradiction. Then d > 0. Actually, we have f ′′ d > 0, f ′ d > b on (0, t 0 ) and equation [START_REF] Brighi | On a similarity boundary layer equation[END_REF] holds on (0, t 0 ). Assume now that f ′′′ d is bounded on (0, t 0 ). Thanks to equation (11

) 1 we deduce that f ′ d (t 0 ) = 0 this is contradiction with f ′ (0) > b.
From the above we can see, in particular, that f ′′ d < 0 on (0, T d ) for any d < 0. Then

f d ∈ C ∞ ([0, T d ))
. While for the case d > 0 the solution f d is not classical.

Proposition 4 Let f d be the local solution to [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF]

for d = 0 and n > 1. If T d < ∞ then lim t→T d f d (t) = -∞.
Proof. First we show that sup [0,T d ) |f d (t)| = ∞. Suppose not and f ′′ d (t 0 ) = 0 holds, for some t 0 ∈ (0, T d ). From ( 14) we get

-(-f ′′ d ) n (t) + αf ′ d (t)f d (t) = αf ′ d (t 0 )f d (t 0 ) + (m + α) t t 0 f ′ d (s) 2 ds, ∀ t 0 < t < T d .
Hence

α 2 f 2 d (t) -αf ′ d (t 0 )f d (t 0 )(t -t 0 ) - α 2 f 2 d (t 0 ) = (m + α) t t 0 τ t 0 f ′ d 2 (s)dsdτ + t t 0 (-f ′′ d ) n (s)ds.
Since the right-hand side of the above is monotonic increasing with respect to t, the function f d has a finite limit as t → T d . Consequently the function (-f ′′ d ) n is integrable on (t 0 , T d ). Since n > 1 we deduce that f ′′ d is also integrable on (t 0 , T d ). Therefore the function f ′ d is bounded. Next we use [START_REF] Belhachmi | On a family of differential equation for boundary layer approximations in porous media[END_REF] to deduce that f ′′ d is also bounded. A contradiction with Proposition 2. It remains to prove that the hypothesis f ′′ d > 0 on (0, T d ) leads also to a contradiction. Actually, in such situation, we know that f d is classical and satisfies [START_REF] Brighi | On a similarity boundary layer equation[END_REF], which yields to

(f ′′ d ) n-2 f ′′′ d ≤ - α n f d ,
and then

(f ′′ d ) n-2 f ′′′ d ≤ α n sup [0,T d ) |f d (t)|.
Therefore f ′′ d and f ′ d are bounded. A contradiction. Becausef d is monotonic on (τ, T d ), for some 0 < τ < T d , we deduce that |f d (t)| goes to infinity as t → T d . Finally, to show that f d (t) → -∞ as t → T d we assume on the contrary that lim t→T d f d (t) = ∞. Hence the functions f d and f ′ d are positive on (τ, T d ). Moreover, using (11) 1 , we can deduce from the Energy-function defined by

E(t) = n n + 1 |f ′′ d (t)| n+1 - m 3 f ′ d 3 , (17)
and satisfies

E ′ (t) = -αf d f ′′ d 2 ≤ 0. That f ′′ d and f ′ d are bounded.
Hence f d is also bounded and this is a contradiction with Proposition.2. We conclude that if T d is finite the function f d goes to minus infinity as t → ∞.

Existence of solutions

In this section we shall obtain a sufficient condition on d such that the local solution f d of ( 13) is global and satisfies the condition f ′ d (∞) = 0. We show that, for each d satisfying |d| n-1 d > -αab, f d exists on the entire positive axis R + and satisfies f ′ d (∞) = 0. We begin by a simple observation that: if m + α > 0 and |d| n-1 d > -αab, (14) yields the important fact that f d cannot have a local maximum. Thus we prove the following result.

Theorem 5 Let a ∈ R, b ≥ 0 and m ∈ (-α, 0). For any d such that |d| n-1 d > -αab, there exists a unique global solution f d , to [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF], which goes to infinity with t, and its first and second derivative tend to 0 as t approaches infinity.

For our analysis, we need to distinguish two cases for the parameter a = f d (0); namely a ≥ 0 and a < 0. First we prove the following lemma. Proof. Because |d| n-1 d + αab > 0, the first assertion of the lemma is immediate from [START_REF] Belhachmi | On a family of differential equation for boundary layer approximations in porous media[END_REF]. To demonstrate that T d = ∞ it suffices to show that f d remains bounded on any bounded interval [0, T ]. Let us consider the Lyapunov function E for f d defined by [START_REF] Magyari | Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls[END_REF]. Since

E ′ (t) = -αf d f ′′ d 2 ≤ 0, thanks to (11) 1 , it is seen that n n + 1 |f ′′ d (t)| n+1 - m 3 f ′ d (t) 3 ≤ n n + 1 |d| n+1 - m 3 b 3 , ∀t < T d .
This Proof. Since f ′ d is monotonic on (t 1 , ∞), t 1 , large enough, and bounded there exists a l ≥ 0 such that

lim t→∞ f ′ d (t) = l.
This implies the existence of a sequence (t n ) tending to infinity with n satisfying lim n→+∞ f ′′ d (t n ) = 0 and then lim t→∞ f ′′ d (t) = 0, with the help of the energy function E. Now we assume that f d is bounded, therefore l = 0. Subsequently

|d| n-1 d + αab + (m + α) ∞ 0 f ′ d (t) 2 dt = 0.
This is impossible. Therefore f d is unbounded and then lim t→+∞ f d (t) = ∞. It remains to prove that l = 0. Assume on the contrary that l > 0. Together with ( 14) we get

|f ′′ d | n-1 f ′′ d (t) = -αl 2 t + (m + α)l 2 t + o(t), |f ′′ d | n-1 f ′′ d (t) = ml 2 t + o(t)
, as t approaches infinity, that This is only possible if m = 0. Consequently l = 0. Next we consider the case a < 0. The first simple consequence is that f d (t) < 0 and f ′ d (t) > 0 for small t > 0. Since f d cannot have a local maximum, we have two possibilities • Either f d (t) vanishes at a some point and remains positive after this point.

• Or f d (t) < 0 for all t > 0. Hence the proof of Theorem 5 is completed by the following lemma. Proof. Assume that the first assertion holds. Since f ′ d is positive we deduce that f d is bounded and then is global. On the other hand, using ( 14) one sees that f ′′ d > 0. Therefore we get lim t→∞ f d (t) ∈ (a, 0] and lim t→∞ f ′ d (t) = 0, since f ′ d is monotonic. This is absurd since f ′ d is positive and increasing function. Hence f d has exactly one zero, say t 0 . To finish the Proof of Lemma 8 and therewith that of Theorem 5 we note that the new function

h(t) = f d (t + t 0 )
satisfies equation ( 11)

1 and h(0) ≥ 0, h ′′ (0) > -αh(0)h ′ (0).
Therefore we use Lemmas 6 and 7 to conclude.

In the next result we complete our analysis on the existence of global solutions by the case b < 0.

Theorem 9 Let b < 0, a > 0 and m ∈ (-α, 0). For any d > 0 satisfying

ad n - 1 2 b 2 d n-1 + αa 2 b > 0. ( 18
)
The unique local solution, f d to ( 13) is global unbounded and satisfies

lim t→∞ f ′ d (t) = lim t→∞ f ′′ d (t) = 0.
Proof. Since a, d > 0 and b < 0, there exists a real t 0 > 0 such that f d is positive, decreasing and convex on (0, t 0 ). Define

T = sup t : f d (s) > 0, f ′ d (s) < 0, f ′′ d (s) > 0, for all s ∈ (0, t) .
The real number T is larger than t 0 and may be infinite. Assume that T = ∞. Then the function f d has a finite limit at infinity and f ′ d (t) (and f ′′ d ) go to zero as t → ∞. Since the function

H = f d |f ′′ d | n-1 f ′′ d - 1 2 f ′ d 2 |f ′′ d | n-1 + αf 2 d f ′ d , satisfies H ′ = f (f ′ d ) 2 m + 2α + α(n -1) 2n - m(n -1) 2n (f ′ d ) 4 (f ′′ d ) -1 ,
thanks to (11) 1 , we deduce that H is increasing on (0, ∞). Hence for t > 0 we have

H(0) < lim t→∞ H(t) = 0,
which yields to

ad n - 1 2 b 2 d n-1 + αa 2 b < 0.
A contradiction. Therefore T is infinite. Next, we assume that f d (T ) = 0 or f ′′ d (T ) = 0. Arguing as above we deduce H(0) < 0 and then we get a contradiction. In conclusion if condition [START_REF] Coppel | On a differential equation of boundary layer theory[END_REF] holds the function f d has a local positive minimum at some t 1 > 0. We use Theorem 5 to deduce that the new function h(t) = f d (t + t 1 ) is global, unbounded and satisfies h ′ (∞) = h ′′ (∞) = 0. The proof is finished.

Remark 10

We notice that we can extend the results of Theorem 5 to the case (-2α, -α) by using the function H defined above, as in the work by Guedda [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF] for the Newtonian case.

Asymptotic behavior

In this section we shall derive the asymptotic behavior of any possible global unbounded solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] for m ∈ (-2α, 0). First we give the following result Lemma 11 Let f be a positive solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] for m ∈ (-2α, 0). Then f ′ goes to zero at infinity and f ′′ is negative.

Proof. Since f is monotonic on [t 0 , ∞), t 0 large enough, we get the positivity of f ′ and f on (t 0 , ∞). In addition we use the Lyapunov function to get the boundedness of f ′ and f ′′ . Arguing as in the previous section we get that f ′ → 0 and f ′′ < 0 for large t.

Proposition 12 Assume that n > 1 and m ∈ (-2α, 0). Let f be a positive solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF]. Then

lim t→∞ f d (t)f ′′ d (t) = lim t→∞ |f ′′ | n-1 f ′′ ′ (t) = 0.
Proof. Thanks to lemma [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] we have f ′ (t) > 0, f ′′ (t) < 0 for all t > t 0 , t 0 large enough and f ′ and f ′′ tend to 0 as t → ∞. Then equation [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] 1 can be written as

f ′′′ + α n f f ′′ |f ′′ | 1-n = m n f ′ 2 |f ′′ | 1-n , ∀t > t 0 .
By differentiation we have

f (iv) + f ′′′ α(2 -n) n |f ′′ | 1-n f - m(1 -n) n |f ′′ | -n-1 f ′′ f ′ 2 = - α -2m n f ′ f ′′ |f ′′ | 1-n . ( 19
)
Then the function f ′′′ e G is monotonic increasing on (t 0 , ∞), where

G ′ = α(2 -n) n |f ′′ | 1-n f - m(1 -n) n |f ′′ | -n-1 f ′′ f ′ 2 .
This indicates that the function f ′′′ has at most one zero. Because f ′′ is negative and goes to 0 at infinity, we deduce that f ′′′ (t) > 0 on (t 1 , ∞), for t 1 large. On the other hand, from (11) 1 we deduce

(|f ′′ | n-1 f ′′ ) ′′ + (α -2m)f ′ f ′′ = -αf f ′′′ . Therefore the function t -→ (|f ′′ | n-1 f ′′ ) ′ + α-2m 2 f ′ 2
is positive and monotonic decreasing on (inf {t 0 , t 1 } , ∞). Together with the fact that f ′ tends to 0 as t → ∞ we deduce that

lim t→+∞ |f ′′ | n-1 f ′′ ′ (t) = 0
and then we conclude that f f ′′ (t) → 0 as t → ∞, thanks to (11) 1 .

Proposition 13

Let f be a solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] where n > 1, m ∈ (-2α, 0). Then

lim t→+∞ f f ′′ (t) =            ∞, if m + α > 0, L ∈ (0, ∞), if m + α = 0, 0, if m + α < 0.
Proof. Let f be a global solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF]. First we claim that there exists t 0 ≥ 0 such that

|f ′′ | n-1 f ′′ (t 0 ) + αf f ′ (t 0 ) > 0.
Suppose not; that is

|f ′′ | n-1 f ′′ (t) + αf f ′ (t) ≤ 0,
for all t ≥ 0. Since f ′′ (t) → 0 as t → ∞ the following

f ′′ + αf f ′ ≤ 0
holds on some (t 1 , ∞), t 1 large. Consequently the function f ′ + α 2 f 2 is decreasing and goes to infinity with t, which is absurd. Now we use the identity

|f ′′ | n-1 f ′′ (t) + αf f ′ (t) = |f ′′ | n-1 f ′′ (t 0 ) + αf f ′ (t 0 ) + (m + α) t t 0 f ′ 2 (s)ds,
to deduce that f f ′ has a limit L ∈ [0, ∞] at infinity. This limit is finite for α + m = 0. Assume that α + m = 0. If L ∈ (0, ∞) we get immediately that f ′ ∼ L η at infinity which implies that f f ′ → ∞. A contradiction. Consequently L ∈ {0, ∞}, we use again the above identity to conclude that L = ∞ if m + α > 0 and L = 0 if m + α < 0.

Remark 14

We stress that the condition

|f ′′ | n-1 f ′′ (t 0 ) + αf f ′ (t 0 ) > 0, f ′ (t 0 ) ≥ 0
is necessary and sufficient to obtain a global solution converging to plus infinity with t in the case m ∈ (-α, 0). Now we are ready to give the result concerning the large t-behavior of solutions to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF].

Theorem 15 Suppose n > 1, -2α < m < 0. Let f be a solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] such that f → ∞. Then there exists a constant, A > 0, such that

f (t) = t α α-m (A + o(1)), (20)
as t → ∞.

Proof. Let f be a global solution to [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF]. First we prove the result for the case m + α > 0.

Let t 0 be a real number such that f ′′ < 0 and f ′′′ > 0 on (t 0 , ∞). Dividing equation [START_REF] Filipussi | The self-similar laminar boundary layer of power-law non-Newtonian fluids[END_REF] 1 by f f ′ gives

(|f ′′ | n-1 f ′′ ) ′ f f ′ = m f ′ f -α f ′′ f ′ .
Integrating over (t 1 , t), for t 1 > t 0 , gives

t 1 t (|f ′′ | n-1 f ′′ ) ′ f f ′ ds = log (f m (t)f ′-α (t) -log (f m (t 1 ) -f ′-α (t 1 )).
According to Proposition 13, f f ′ goes to infinity with t and then the left hand side of the above is integrable, consequently f m f ′-α has a positive finite limit at infinity. The desired asymptotic behavior [START_REF] Afrouzi | Numerical methods for finding multiple solutions of a Dirichlet problem with nonlinear terms[END_REF] follows by a simple integration. Now we deal with the case m + α < 0. For this sake we define

Ψ = ϕ(f )|f ′′ | n-1 f ′′ - 1 2 ϕ ′ (f )(f ′ ) 2 |f ′′ | n-1 + αϕ(f )f f ′ ,
where ϕ is a smooth function. Then if follows from (11) 1

Ψ ′ (t) = f ′ 2 αf ϕ ′ (f ) + (α + m)ϕ - 1 2 ϕ ′′ (f )f ′ 3 |f ′′ | n-1 - n -1 2 ϕ ′ (f )f ′ 2 |f ′′ | n-3 f ′′ f ′′′ .
Let the function ϕ be defined by ϕ(s) = s -m+α α .

It satisfies the following differential equation αsϕ ′ (s) + (α + m)ϕ = 0.

This implies that

Ψ ′ = - 1 2 ϕ ′′ (f )f ′ 3 |f ′′ | n-1 - n -1 2 ϕ ′ (f )f ′ 2 |f ′′ | n-3 f ′′ f ′′′ ≥ 0, Ψ = ϕ(f ) |f ′′ | n-1 f ′′ - α + m 2αf (f ′ ) 2 |f ′′ | n-1 + αf f ′ ,
and then

|Ψ ′ (t)| ≤ ε f -3α+m α f ′ + (n -1)(-f ′′ ) n-2 f ′′′ ,
for all t ≥ t 0 , t 0 large. Therefore Ψ ′ is integrable on [0, ∞) and then Ψ has a finite limit at infinity, say L.

Next we show that L > 0. It will be sufficient to show that Ψ(t 1 ) > 0 for some t 1 large. Suppose not; that is for any t > t 2 , t 2 large we have

|f ′′ | n-1 f ′′ - α + m 2αf f ′ 2 |f ′′ | n-1 + αf f ′ ≤ 0.
Since α + m < 0 then f ′′ + αf f ′ ≤ 0, from which we deduce, as above, that f ′ + α 2 f 2 is a decreasing function going to infinity with t. A contradiction. We conclude that lim t→∞ f -m α f ′ = L α . Finally, a simple integration leads to estimate [START_REF] Afrouzi | Numerical methods for finding multiple solutions of a Dirichlet problem with nonlinear terms[END_REF]. To finish, we pay attention to the case m = -α. Here the identity (14) leads to |f ′′ | n-1 f ′′ + αf f ′ = |γ| n-1 γ + αab.

According to Theorem 5, f is global and satisfies f ∼ t 1 2 at infinity.

Conclusion

The laminar two-dimensional steady boundary layer flow, of a non-Newtonian incompressible fluid, over a stretching surface have been considered. Using the shooting method, existence of global unbounded similarity solutions have been shown, the dependency of those solutions on the power-law index has been investigated, and their asymptotic behavior was also discussed. Coming back to the original problem (2),(3) we find that, for -2α < m < 0, the stream function satisfies ψ(x, y) ∼ y 1+m(2n-1) 1+m(n-2) as yx

(2-n)m-1 n+1 → ∞.

  and αβ = m, holds, and the parameters A, B and ν satisfy νA n-2 B 2n-1 = 1, and m = αβ.

  Note also that any solution is classical on any interval where the second derivative does not change the sign. Consider now the initial value problem (13) with n > 1, a, d ∈ R, b ≥ 0 and m ∈ (-α, 0). By the classical theory of ordinary differential equations the above problem has local (maximal) solutions on some interval (0, T d ), T d ≤ ∞ and they are uniquely determined by d (d = 0). Let us denote this such solution by f d . Integrating (13) 1 to het the following identity

Lemma 6

 6 If a ≥ 0 and |d| n-1 d > -αab the functions f ′ d and f d are positive on (0, T d ) and T d = ∞; that is f d is global. Moreover f ′ d and f ′′ d are bounded.

Lemma 8

 8 Assume a < 0 and |d| n-1 d > -αab. Then f d has exactly one zero, goes to ∞ with t, and the functions f ′ d , f ′′ d converge to 0 as t → ∞.

Acknowledgments

The authors are indebted to the anonymous referees for valuable comments and Prof. Robert Kersner for stimulating discussions. This work is an outcome of a PhD thesis [19], it was partially supported by la Direction des Affaires Internationales (UPJV) Amiens France and by the Research Program: PAI No MA/05/116 (France-Morocco Scientific Cooperation Volubilis).