The Box-Counting Measure of the Star Product Surface

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.() No.3,pp.- The Box-Counting Measure of the Star Product Surface Tao Peng, Zhigang Feng Faculty of Science, Jiangsu University Zhenjiang, Jiangsu,3,P.R.China (Received June, accepted August ) Abstract: The measure, such as length, area and volume, is one of the most important concepts in geometry. It can be used to indicate the magnitude of the set. The concept of (upper, lower) box-counting measure is given strictly in this paper. We discuss the properties of box-counting measure and measurable sets. Some equivalent box-counting measures are proposed, and anyone of them, if it is deemed convenient, can be chosen as a measure of sets in a given system. Then we calculate the box-counting measures of the Star Product Surfaces (SPS) and the profiles of the SPSs. It is showed that they are determined by the box-counting measures of the cures applied to the constructing of the SPS. Meanwhile, as some corollaries, the box-counting dimensions of the profiles of the SPSs are obtained. Key words: box-counting measure; SPS; profile of SPS Introduction Almost all real surfaces in nature are rough, although many of them seem to be smooth for naked eye. Because they have roughness on many different length scales, it is impossible to apply the classical geometry of smoothness and piecewise smoothness to model these surfaces perfectly. To study such a surface it is necessary to use better methods. Because fractals possess of the properties of the similarity, roughness and complexity, it is more fitting to simulate some natural surfaces such as the rock fracture surfaces. Many literatures have been involved in studying and developing the methods for simulating these surfaces by using mathematical fractal surfaces [ 3]. One of kinds of the most important and simplest fractal surfaces is Cartesian product fractal surfaces. Let A be a fractal curve in R, and B is a straight line segment in R. Moving A along B perpendicularly, we construct a fractal surface, which is Cartesian product fractal surface of A and B, noted A B (Fig.). This kind of surfaces are often used to model natural rough surfaces, whereas they exhibit fractal feature only in one direction and cannot fit natural surface perfectly. In [], we improve this model by supposing that both of A and B are continuous curves. Let curve A move along curve B, then a surface, which is so-called the Star Product Surface (SPS) of two curses A and B (Fig.), can be constructed. SPS has roughness on each direction. And its box-counting dimension dim B A B = max{dim B A, dim B B} +, where A B is SPS of A and B. In [5], we studied the box dimensions of the profile curves of SPS. The relations of the box dimensions between the profile curves of SPS and the curves used for constructing the SPS have been obtained. Beside the dimension, the measure also is a very important parameter of sets. Similar to the length, area and volume, the measure can give the magnitude of the sets. Hausdorff measure is a strict mathematical concept[]. It is a base of the theory of the mathematical fractal. But it is difficult to calculate Hausdorff measure of any sets. Up to the present, only a few of sets Hausdorff measures have been obtained. Corresponding author. address: zgfeng@ujs.edu.cn Copyright c World Academic Press, World Academic Union IJNS...5/93

2 International Journal of Nonlinear Science,Vol.(),No.3,pp Figure : the Cartesian product surface A B. Figure : the Star Product Surface (SPS) A B. Borodich[7] proposed the concept of s measure m s, showed some of its properties and applications. The s measure can be calculated more easily than Hausdorff measure. Based on the work of Borodich[7], the concepts of (upper, lower) box-counting measure are strictly given in this paper. We discuss its properties and methods of calculation. Finally, we obtained the boxcounting measures of SPS and its profiles. Box-Counting Measure In this paper, we will consider the subsets of the Euclidean space R n, with the Euclidean distance, i.e., if x = (x, x,, x n ) and y = (y, y,, y n ) are two elements of R n, the distance between x and y is x y = (x y ) + (x y ) + + (x n y n ). Let S, U, U,, U k, be subsets of R n, and δ is a positive real number. If S k U k, the {U k } is called a cover of the set S, and additionally if U k δ, k =,,, the {U k } is called a δ cover of the set S, where U k = sup{ x y : x, y U k } called the diameter of the set U k. Borodich [] proposed the concept of the box-counting measure. Definition. Let S be a bounded set in n-dimensional Euclidean space R n. For δ >, let N δ (S) be the smallest number of n-dimensional sets of diameter at most δ that cover the set S. For s, we call the upper and lower-limit lim sup N δ (S)δ s, lim inf N δ(s)δ s, δ + δ + upper and lower box-counting s-measures of the set S respectively, denoted by m s (S) and m s (S). If limit lim δ + N δ (S)δ s exists (finite or infinite), i.e., m s (S) = m s (S), we call it s-measure of S, denoted by m s (S), that is to say m s (S) = lim N δ(s)δ s. () δ + A set S is called D measurable set if its D measure has a finite non-negative value m D (S), i.e., < m D (S) = lim δ + N δ(s)δ D < +. Remark. Generally, the D-measure is not additive. For example, let S be interval [, ], S and S consist of all rational numbers and all irrational numbers in S respectively. We can obtain δ N δ (S) = N δ (S ) = N δ (S ) δ +, IJNS for contribution: editor@nonlinearscience.org.uk

3 T. Peng, Z. Feng: The Box-Counting Measure of the Star Product Surface 3 where x = max{z z Z, z x}. It is obvious that δ < δ δ, then So m (S ) + m (S ) = m (S S ). m (S ) = m (S ) = m (S) =. Therefore, the box-counting measure is not a strict mathematical measure[]. This deficiency of the box counting measure is compensated by an advantage: it can be efficiently computed. In practice, we do not distinguish a set from its closure, and we also do not let scales tend to. In practical application, we can approximately calculate the box-counting measure by using various appropriate methods. Proposition. Let L be a transformation on set S, and there exist a positive constant c, such that L(x) L(y) = c x y, x, y S. If S is a D-measurable set, then L(S) also is a D-measurable set, and m D (L(S)) = c D m D (S). Proof. Let {U α } be a δ cover of S, and the number of the {U α } is N δ (S), the smallest number of sets of diameter at most δ that cover S, then {L(U α )} is a cδ cover of L(S), and the number of {L(U α )} is equal to N cδ (L(S)), the smallest number of sets of diameter at most cδ that cover L(S). So obtain N cδ (L(S)) = N δ (S). Therefore lim sup δ + lim inf δ + N δ (L(S))(δ) s = lim sup N cδ (L(S))(cδ) s = lim sup c s N δ (S)(δ) s, cδ + δ + N δ(l(s))(δ) s = lim inf cδ + N cδ(l(s))(cδ) s = lim inf δ + cs N δ (S)(δ) s. If S is D-measurable, namely, < lim inf δ + N δ (S)δ D = lim sup δ + N δ (S)δ D < +, then, obviously, < lim inf δ + N δ (L(S))δ D = lim sup δ + N δ (L(S))δ D < +, it is said L(S) is D- measurable and m D (L(S)) = c D m D (S). Similar to Hausdorff measure and Hausdorff dimension, the (upper and lower) box-counting measure can be use to determine the (upper and lower) box-counting dimension. Proposition. Let S be set, then m s (S) = m s (S) = { +, for s < dimb (S),, for s > dim B (S). { +, for s < dimb (S),, for s > dim B (S). () (3) Proof. Suppose s < dim B (S), there exists ε >, such that s+ε < dim B (S). For this ε, there exists δ >, when < δ < δ, log N δ(s) log δ > s + ε. Then N δ (S)δ s > δ ε. Let δ +, obtain m s (S) = m s (S) = +. For s > dim B (S), let ε >, such that s ε > dim B (S). Then exist a positive sequence {δ n }, lim δ n = and N N + log N, such that if n N, δn (S) n + log δ n s ε, i.e., N δn (S)δn s δn. ε So lim N δ n + n (S)δn s =. Then m s (S) = lim inf N δ(s)δ s =. δ + For Eq.(3), we suppose s > dim B (S) first. There exists ε >, such that s ε > dim B (S). For this ε >, there exists δ >, such that, when < δ < δ, log N δ(s) log δ < s ε, namely, N δ (S)δ s < δ ε. Let δ +, obtain m s (S) =. If s < dim B (S), there exist ε >, such that s + ε < dim B (S). Then exist a positive sequence {δ n }, lim δ n = and N N + log N, such that if n N, δn (S) n + log δ n s + ε, i.e., N δn (S)δn s δn ε. Then lim n + N δn (S)δn s = +. Therefore m s (S) = +. Obviously, if m s (S) = +, then m s (S) = +, and if m s (S) =, then m s (S) =. With Proposition. we have the following corollary. Corollary. If set S is D-measurable, then S has box-counting dimension D, i.e. dim B (S) = dim B (S) = D. IJNS homepage:

4 International Journal of Nonlinear Science,Vol.(),No.3,pp. - Definition. Let m s( ) and m s( ) be two measures, if there exist positive constants c and c, such that the following inequality is true for any set S, c m s(s) m s(s) c m s(s), () then we call m s( ) and m s( ) are equivalent measures, denoted by m s( ) m s( ). Now let F be a subset of the Euclidean space R n (n > ). For any integers i, i, i n, C δ (i, i,, i n ) denotes a column {(x, y) x [(i )δ, i δ] [(i )δ, i δ] [(i n )δ, i n δ], y R}. Let Ñδ(F, i, i,, i n ) be the smallest number of the cubes of side δ in C δ (i, i,, i n ) needed to cover F C δ (i, i,, i n ), and denote Ñ δ (F ) = Ñ δ (F, i, i, i n ). (5) We call Ñδ(F ) the smallest number of δ-collum cubes that cover F. It is obvious that N δ (F ) Ñδ(F ) N δ (F ), then we have the following proposition. Proposition.3 m s (F ) and m s (F ) are equivalent, where m s (F ) = lim δ + Ñ δ (F )δ 5 () According to the equivalence of the measures m s (F ) and m s (F ), we can use any one of them to indicate the magnitudes of the sets in a given system. The SPS and the profile of the SPS will be discussed follow. In these cases, we will calculate their box-counting measure m s. For convenience, the m s is still simply denoted by m s below. 3 Box-Counting Measure of SPS Definition 3. [] In 3-dimension Euclidian space R 3, let A : z = f(x), x [a, b], B : z = g(y), y [c, d] and they are continuous curves on the coordinate planes xoz and yoz respectively. In the space oxyz, the surface F = {(x, y, z) x [a, b], y [c, d], z = f(x) + g(y)} (7) is called star product surface of continuous curves A and B, or simply Star Product Surface (SPS), denoted A B (Fig.). Obviously, the SPS can be obtained by moving the continuous curve A (B) along the continuous curve B (A) orthogonally, if neglecting the height of the surface. Proposition 3. Let A : z = f(x), x [a, b], B : z = g(y), y [c, d] be continuous curves on the coordinate planes xoz and yoz respectively. Then m s (A B) = { +, for s < max{dimb A, dim B B} +,, for s > max{dim B A, dim B B} +. () And additionally, if at least one of sets A and B have box dimension, then m s (A B) = { +, for s < max{dimb A, dim B B} +,, for s > max{dim B A, dim B B} +. (9) Proof. According to the theorem in [], dim B (A B) = max {dim B (A), dim B (B)} +, and if at least one of sets A and B have box dimension, we have also dim B (A B) = max {dim B (A), dim B (B)} +. By Proposition., Eq.() and Eq.(9) are both true. IJNS for contribution: editor@nonlinearscience.org.uk

5 T. Peng, Z. Feng: The Box-Counting Measure of the Star Product Surface 5 Proposition 3. Let A : z = f(x), x [a, b], B : z = g(y), y [c, d] be continuous curves on the coordinate planes xoz and yoz respectively. If A is D measurable, and B is D measurable, max{d, D } >, then A B is (max{d, D } + ) measurable, and (d c)m D A, for D > D, m D (A B) = (b a)m D B, for D < D, () (d c)m D A + (b a)m D B, for D = D >, where D = max{d, D } +. Proof. According to Corollary., the sets A and B both have box-counting dimensions, and dim B A = D and dim B B = D. For a set E R n and a function h on E, the nonnegative real number sup h(x) h(y) is called oscillation of the function h on E, denoted osc(h, E), i.e. For I i = [(i )δ, iδ] and J j = [(j )δ, jδ], obviously, osc(h, E) = sup{ h(x) h(y) : x, y E}. () osc(f(x) + g(y), I i J j ) = osc(f, I i ) + osc(g, J j ), () and the smallest numbers of cubes of side δ in column C δ (i) = I i R, C δ (j) = J j R and C δ (i, j) = I i J j R needed to cover A C δ (i), B C δ (j) and (A B) C δ (i, j) respectively are osc(f, Ii ) osc(g, Jj ) Ñ δ (A, i) =, Ñ δ (B, j) =, (3) δ δ osc(f(x) + g(y), Ii J j ) Ñ δ (A B, i, j) =, () δ where x is equal to the smallest integer not less than x. For example, 5. =, =. It is easy to prove for any α, β R α + β α + β α + β. (5) Therefore, according to Eq.(-5), Ñ δ (A, i) + Ñδ(B, j) Ñδ(A B, i, j) Ñδ(A, i) + Ñδ(B, j). () Because of Eq.(), let i = a/δ, a/δ +,, b/δ, and j = c/δ, c/δ +,, d/δ, calculating the sum of Eq.(), we have ϱ δ (c, d)ñδ(a) + ϱ δ (a, b)ñδ(b) ϱ δ (a, b)ϱ δ (c, d) Ñδ(A B) ϱ δ (c, d)ñδ(a) + ϱ δ (a, b)ñδ(b), where ϱ δ (x, y) = y/δ x/δ +. Obviously, y/δ x/δ ϱ δ (x, y) y/δ x/δ +, for x, y >, then lim δ + ϱ δ (x, y)δ = y x. Therefore, if D > D, and lim δ +(ϱ δ(c, d)ñδ(a) + ϱ δ (a, b)ñδ(b) ϱ δ (a, b)ϱ δ (c, d))δ D+ = m D (A)(d c), lim δ +(ϱ δ(c, d)ñδ(a) + ϱ δ (a, b)ñδ(b))δ D+ = m D (A)(d c). It can be deduced that lim δ +(Ñδ(A B))δ D + = m D (A)(d c), i.e. m D +(A B) = m D (A)(d c). If D > D, lim δ +(ϱ δ(c, d)ñδ(a) + ϱ δ (a, b)ñδ(b) ϱ δ (a, b)ϱ δ (c, d))δ D + = lim δ +(ϱ δ(c, d)ñδ(a) + ϱ δ (a, b)ñδ(b))δ D + = m D (B)(b a). IJNS homepage:

6 International Journal of Nonlinear Science,Vol.(),No.3,pp. - Then lim δ +(Ñδ(A B))δ D + = m D (B)(b a), i.e. m D +(A B) = m D (B)(b a). For D = D, D = D + = D +, lim δ +(ϱ δ(c, d)ñδ(a) + ϱ δ (a, b)ñδ(b) ϱ δ (a, b)ϱ δ (c, d))δ D = lim δ +(ϱ δ(c, d)ñδ(a) + ϱ δ (a, b)ñδ(b))δ D = m D (A)(d c) + m D (B)(b a). Therefore lim δ +(Ñδ(A B))δ D = m D (A)(d c) + m D (B)(b a), i.e. m D (A B) = m D (A)(d c) + m D (B)(b a). Corollary 3. Let A : z = f(x), x [a, b] have box dimension, D = dim B A >, and A is D measurable, then Cartesian product A [c, d] is D + measurable, and m D+ (A [c, d]) = (d c)m D A. (7) Remark 3. When D = D =, maybe Eq.() is not right. For example, A : z = αx, x [, ], and B : z = βy, y [, ], where α and β are two constants. Obviously, when α ( β ), have Ñδ(A) = δ (Ñδ(A) = δ ), hence m (A) = m (B) =. On the other hand, if α = β =, A B : z =, (x, y) [, ] [, ], it can easy be deduced that Ñδ(A B) = δ, hence m (A B) = ; if α = β =, A B : z = x + y, (x, y) [, ] [, ], we can deduce Ñδ(A B) = δ, hence m (A B) =. Box-Counting Measure and dimension of the profile of SPS Let A : z = f(x), x [a, b] and B : z = g(y), y [c, d] be continuous curves on the coordinate planes xoz and yoz respectively. And let A B = {(x, y, z) x [a, b], y [c, d], z = f(x) + g(y)} be a SPS of A and B. Now suppose plane π is perpendicular to the coordinate plane xoy, if (A B) π φ, generally it is a curve. We will study the box-counting measures of these curves. In this paper, we only considered the plane π passing through the vertex (a, c, ). If the plane π intersects segment {(x, d, ) : x [a, b]} (or {(b, y, ) : y [c, d]}) at (x, d, ) (or (b, y, )), the plane π is denoted by π(x, d) (or π(b, y )). Obviously, plane π(b, d) passes through the diagonal of the [a, b] [c, d]. Fig.3 gives one of the profiles of the SPS A B showed in Fig.. For clarity, the figure has been revolved a proper angle. Figure 3: one of the profiles of SPS A B. IJNS for contribution: editor@nonlinearscience.org.uk

7 T. Peng, Z. Feng: The Box-Counting Measure of the Star Product Surface 7 Proposition. Let A : z = f(x), x [a, b] and B : z = g(y), y [c, d] be continuous curves. If b a = d c, then m s (Γ(b, d)) m s (A) + m s (B). () Additionally, if dim B A dim B B, then m s (Γ(b, d)) = max{m s (A), m s (B)}, i.e. +, for s < D, m s (Γ(b, d)) = max{m s (A), m s (B)}, for s = D,, for s > D, where D = max{dim B A, dim B B}. Proof. According to Proposition., transform T : R 3 R 3, (x, y, z) (x a, y b, z) conserves box-counting measure of every set in Euclidean space R 3. Then we could suppose a = c = and hence b = d = l. By the right inequation of Eq.(), noticing that Γ(b, d) = {(x, y, z) : x = y, (x, y, z) A B}, it can be deduce that (9) Ñ δ (Γ(b, d)) n Ñ δ (A B, i, i) Ñδ(A) + Ñδ(B), () i= where n = b/δ. Multiplied by δ s, let δ +, Eq.() can be proved. Without loss of generality, suppose dim B A > dim B B. If x I i = [(i )δ, iδ], osc((f + g)(x), I i ) osc(f(x), I i ) osc(g(x), I i ). Because x x x x, then N δ (Γ(b, d); I i ) N δ (Γ(f, I i )) N δ (Γ(g, I i )). So Ñ δ (Γ(b, d)) Ñδ(A) Ñδ(B) l/δ. () Let dim B B < s < dim B A, meanwhile s >, according to Proposition., lim sup δ + Ñδ(A)δ s = +, lim sup δ + Ñδ(B)δ s = and lim δ + l/δ δ s =, then m s (Γ(b, d)) = +. Because m s (Γ(b, d)) is non-increasing with s, it is proved that m s (Γ(b, d)) = + for s < dim B (A) = D. If s = dim B A > dim B B, have lim sup δ + N δ (A)δ s = m s (A), lim δ + N δ (B)δ s = m s (B) = and lim δ + l/δ δ s =. with Eq.(-), we can prove lim sup δ + Ñδ(Γ(b, d))δ s = lim sup δ + Ñδ(A)δ s = m s (A), i.e. m s (Γ(b, d)) = max{m s A, m s B}. For s > D, then s > dim B A > dim B B. With Proposition., lim sup δ + N δ (A)δ s = lim sup δ + N δ (B)δ s =. Noticing Eq.(), we have m s (Γ(b, a)) = lim sup δ + N δ (Γ(a, b))δ s =. Corollary. Let A : z = f(x), x [a, b] and B : z = g(y), y [c, d] be continuous curves. Then And additionally, if dim B A dim B B, then dim B (Γ(b, d)) max{dim B A, dim B B}. () dim B (Γ(b, d)) = max{dim B A, dim B B}. (3) Proof. () b a = d c. By Proposition. and Proposition., for any s > max{dim B A, dim B B}, m s (Γ(b, d)) =, then dim B (Γ(b, d)) < s, hence dim B (Γ(b, d)) max{dim B A, dim B B}. Additionally, if dim B A dim B B, without loss of generality, we suppose dim B A > dim B B. According to Proposition., it can be proved that m s (Γ(b, d)) = + if s < dim B A, and m s (Γ(b, d)) = if s > dim B A. Hence dim B (Γ(b, d)) = dim B A. () b a d c. Let ϕ : R 3 R 3, (x, y, z) ( x a b a, y, z); ϕ : R 3 R 3, (x, y, z) (x, y c d c, z); ϕ 3 : R 3 R 3, (x, y, z) ( x a b a, y c d c, z). It is easy to prove that ϕ, ϕ, ϕ 3, are three Lipschitz homeomorphic transformations. And ϕ (A) : z = f(a + (b a)x), x [, ]; ϕ (B) : z = g(c + (d c)y), y [, ]; ϕ 3 (A B) = ϕ (A) ϕ (B); and ϕ 3 (Γ(b, d)) = Γ(, ). Because of (), dim B (Γ(, )) = max{dim B ϕ (A), dim B ϕ (B)}. Because Lipschitz homeomorphic transformation conserves box-counting dimensions[], dim B (Γ(b, d)) = dim B (Γ(, )) = max{dim B ϕ (A), dim B ϕ (B)} = max{dim B A, dim B B}. IJNS homepage:

8 International Journal of Nonlinear Science,Vol.(),No.3,pp. - Proposition. Let A : z = f(x), x [a, b] and B : z = g(y), y [c, d] be continuous curves. If b a = d c, and max{dim B A, dim B B} > min{dim B A, dim B B}, then m s (Γ(b, d)) = max{m s (A), m s (B)}. () Proof. Without loss of generality, suppose dim B A = max{dim B A, dim B B} = d, and similar to the proof of Proposition., a = c = and b = d = l. Then d > dim B B, and according to the proof of Proposition., we have Ñ δ (A) Ñδ(B) l/δ Ñδ(Γ(b, d)) Ñδ(A) + Ñδ(B). When s > dim B B, with Proposition., m s (Γ(a, b)) lim inf δ +(Ñδ(A)δ s Ñδ(B)δ s l/δ δ s ) lim inf δ + Ñδ(A)δ s lim sup δ + Ñδ(B)δ s lim sup δ + l/δ δ s = m s (A) m s (B) = m s (A), and m s (Γ(a, b)) lim inf δ +(Ñδ(A)δ s +Ñδ(B)δ s ) lim inf δ + Ñδ(A)δ s +lim sup δ + Ñδ(B)δ s = m s (A). When s dim B B, According to Proposition., m s (Γ(a, b)) is a non-increasing function of s, so m s (Γ(a, b)) m s (Γ(a, b)) = m s (A) = + = m s (A), where dim B B < s < d = dim B A. Then m s (Γ(a, b)) = m s (A). Similar to the proof of Corollary., we can prove Corollary.. Corollary. Let A : z = f(x), x [a, b] and B : z = g(y), y [c, d] be continuous curves. If max{dim B A, dim B B} > min{dim B A, dim B B}, then dim B (Γ(b, d)) = max{dim B A, dim B B}. (5) Remark. In Propositions and Corollaries, the conditions dim B A dim B B or max{dim B A, dim B B} > min{dim B A, dim B B} are indispensable. For example, let A : z = f(x), x [a, b] and z = g(y) = C f(y), y [a, b]. For x = y, z = f(x) + g(y) = C, so dim B (Γ(b, d)) =, despite the values of dim B A and dim B B. References [] Mistakidis E S, Panagiotopoulos P D, Panagouli O K: Fractal surfaces and interfaces in structures. Methods and algorithms. Chaos, Solitons & Fractals. (5):55-57(99) [] Geronimo J S, Hardin D.: Fractal interpolation surfaces and a related -D multiresolution analysis. Journal of Mathematical Analysis and Applications. 7:5-5(993) [3] Malysz R.: The Minkowski dimension of the bivariate fractal interpolation surfaces. Chaos Solitons & Fractals. 7: 7-5() [] Xie H, Feng Z, Chen Z.: On star product fractal surfaces and their dimensions. Applied Mathematics and Mechanics. (): 3-9(999) [5] Dai G, Liu Y, Feng Z.: On box dimensions of profile curves of SPS. International Journal of Nonlinear Science.(3): -9() [] Falconer K.: Fractal geometry: mathematical foundations and Application. John Wiley & Sons, Chichester.( 997) [7] Borodich FM.: Non-classical scaling of microcrack patterns and crack propagation in multiple scale analyses and coupled physical systems. Presses Point Et Chaussees, Paris. 93-5(997) [] Kolmogorov A N, Fomin S V.: Elements of the theory of functions and functional analysis-volume : measure. The Lebesgue Integral. Hilbert Space. Doven Publications, New York. (999) IJNS for contribution: editor@nonlinearscience.org.uk