There are many important concepts in linear algebra, such as linear correlation and linear independence, eigenvalues and eigenvectors, and so on. The article provides a graphical explanation of how to distinguish betw...There are many important concepts in linear algebra, such as linear correlation and linear independence, eigenvalues and eigenvectors, and so on. The article provides a graphical explanation of how to distinguish between the concepts of linear correlation and linear independence. The conclusion points out that linear independence means that there are no two (base) vectors with the same direction in a vector graph;otherwise, it is a linear correlation.展开更多
We study the Hausdorff measure of linear Cantor setE, on the unit interval, under the strong seperated condition. We give a necessary and sufficient condition for ?(E)=∣E∣° by using the contracting ratio and th...We study the Hausdorff measure of linear Cantor setE, on the unit interval, under the strong seperated condition. We give a necessary and sufficient condition for ?(E)=∣E∣° by using the contracting ratio and the first gap. This condition is easy to use. Key words linear Cantor set - Hausdorff measure - strong seperated condition CLC number O 174. 12 Foundation item: Supported by the National Natural Science Foundation of China (10171028)Biography: Ma Chao (1975-), male, Ph. D. candidate, research direction: fractal geometry.展开更多
Let 0<A≤1/3 ,K(λ) be the attractor of an iterated function system {ψ1,ψ2} on the line, where 1(x)= AT, ψ1(x) = 1-λ+λx, x∈[0,1]. We call K(λ) the symmetry Cantor sets. In this paper, we obtained the exact H...Let 0<A≤1/3 ,K(λ) be the attractor of an iterated function system {ψ1,ψ2} on the line, where 1(x)= AT, ψ1(x) = 1-λ+λx, x∈[0,1]. We call K(λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K(λ).展开更多
1.Some definitions and notions 1.1. Hausdorff gauge and Packing gaugeA funciton h:[0.1]→ R is said to be a dimension function if it has the following properties:(1)h(t)is continuous and increasing, h(0)=0, h(t)>0 ...1.Some definitions and notions 1.1. Hausdorff gauge and Packing gaugeA funciton h:[0.1]→ R is said to be a dimension function if it has the following properties:(1)h(t)is continuous and increasing, h(0)=0, h(t)>0 when t>0.(2)there exists a constant M>0, h(2t)<Mh(t) for any t>0.展开更多
It is well known that a totally disconnected compact metric space without isolated points is a Cantor set.In this note me give a simple proof of this theorem.
We study the doubling property of binomial measures on generalized ternary Cantor subsets of [0, 1]. We find some new phenomena. There are three different cases. In the first case, we obtain an equivalent condition fo...We study the doubling property of binomial measures on generalized ternary Cantor subsets of [0, 1]. We find some new phenomena. There are three different cases. In the first case, we obtain an equivalent condition for the measure to be doubling. In the other cases, we show that the condition is not necessary. Then facts and partial results are discussed.展开更多
Let 0<λ_1,λ_2<1 and 1-λ_1-λ_2≥max｛λ_1,λ_2｝.Let ~K(λ_1,λ_2) be the attractor of the iterated function system ｛φ_1,φ_2｝on the line,where φ_1(x)=λ_1x and φ_2(x)=1-λ_2+λ_2x,x∈R.~K(λ_1,λ_2) is ...Let 0<λ_1,λ_2<1 and 1-λ_1-λ_2≥max｛λ_1,λ_2｝.Let ~K(λ_1,λ_2) be the attractor of the iterated function system ｛φ_1,φ_2｝on the line,where φ_1(x)=λ_1x and φ_2(x)=1-λ_2+λ_2x,x∈R.~K(λ_1,λ_2) is called a non-symmetry Cantor set. In this paper,it is proved that the exact Hausdorff centred measure of K(λ_1,λ_2) equals 2s(1-λ)s,where λ=max｛λ_1,λ_2｝ and s is the Hausdorff dimension of K(λ_1,λ_2).展开更多
In this work, we establish relations between DNA sequences with missing subsequences (the forbidden words) and the generalized Cantor sets. Various examples associated with some generalized Cantor sets, including Hao...In this work, we establish relations between DNA sequences with missing subsequences (the forbidden words) and the generalized Cantor sets. Various examples associated with some generalized Cantor sets, including Hao’s frame representation and the generalized Sierpinski Set, along with their fractal graphs, are also presented in this work.展开更多
The Cantor’s conclusion that “there is a one-one correspondence between the points on n-dimensional space and the points on the line” appears in the numerous current documents. By introducing a monotonic and contin...The Cantor’s conclusion that “there is a one-one correspondence between the points on n-dimensional space and the points on the line” appears in the numerous current documents. By introducing a monotonic and continuous function, a one-one correspondence between two intervals is built;and by using parametric equations of the curve, a one-one correspondence from the points on the curve to the points on the line is established. Specially, the meanings of multivariate functions are given. By using a n-variable equation with a parameter, a correspondence from n-dimensional space area to a interval is built, so the wrong conclusion is completely denied. The paper enriches calculus and can reduce the teaching difficulty of real function in some degree. The expression of moving curve (surface) limit is given in the paper. More importantly, after the conclusion is corrected, it will be necessary and possible to re-establish the theory and the approach about multivariable differential calculus.展开更多
We consider the homogeneous Cantor sets which are generalization of symmetric perfect sets, and give a formula of the exact Hausdorff measures for a class of such sets.
For real valued functions defined on Cantor triadic set, a derivative with corresponding formula of Newton Leibniz’s type is given. In particular, for the self similar functions and alternately jumping f...For real valued functions defined on Cantor triadic set, a derivative with corresponding formula of Newton Leibniz’s type is given. In particular, for the self similar functions and alternately jumping functions defined in this paper, their derivative and exceptional sets are studied accurately by using ergodic theory on Σ 2 and Duffin Schaeffer’s theorem concerning metric diophantine approximation. In addition, Haar basis of L 2(Σ 2) is constructed and Haar expansion of standard self similar function is given.展开更多
We pursue the study on homogeneous Cantor sets with their translations. We get the fractal structure of intersection I（t）, and find that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero v...We pursue the study on homogeneous Cantor sets with their translations. We get the fractal structure of intersection I（t）, and find that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from shifting numbers with the coding of t. Concretely, a very brief calculation formula of the measure with the coding of t is given.展开更多
In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and ...In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and the smallest such k is 2~m.Moreover,we generalize this result to the middle-1/αCantor set for1<α<2+√5 and m sufficiently large.For the naturally embedded image W of the Cantor dust C×C into the complex plane C,we prove that for any integer m≥3,every element in the closed unit disk in C can be written as the sum of at most 2^(m+8)m-th powers of elements in W.At last,some similar results on p-adic Cantor sets are also obtained.展开更多
In this paper we have found a general subordinator, X, whose range up to time 1, X([0,1)), has similar structure as random re orderings of the Cantor set K(ω).X([0,1)) and K(ω) have the same exact Hausdorff measure...In this paper we have found a general subordinator, X, whose range up to time 1, X([0,1)), has similar structure as random re orderings of the Cantor set K(ω).X([0,1)) and K(ω) have the same exact Hausdorff measure function and the integal test of packing measure.展开更多
We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph model...We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph models introduced by Barabási, Ravasz and Vicsek ([1] [2]). We generalize the graph models using stars and cliques: both algorithm construct graph sequences such that the next iteration is always based on n replicas of the current iteration, where n is the size of the initial graph structure, being a star or a clique. We analyze these self-similar graph sequences using IMSs in function of the size of the initial star and clique, respectively. Our research uses the Cantor set for the description of the fixed set of these IMSs, which we interpret as the limit object of the analyzed self-similar networks.展开更多
Let E = E({nk},{ck}) be a fat uniform Cantor set. We prove that E is a minimally fat set for doubling measures if and only if (nkck)p = ∞ for all p < 1 and that E is a fairly fat set for doubling measures if and o...Let E = E({nk},{ck}) be a fat uniform Cantor set. We prove that E is a minimally fat set for doubling measures if and only if (nkck)p = ∞ for all p < 1 and that E is a fairly fat set for doubling measures if and only if there are constants 0 < p < q < 1 such that (nkck)q < ∞ and (nkck)p = ∞. The classes of minimally thin uniform Cantor sets and of fairly thin uniform Cantor sets are also characterized.展开更多
文摘There are many important concepts in linear algebra, such as linear correlation and linear independence, eigenvalues and eigenvectors, and so on. The article provides a graphical explanation of how to distinguish between the concepts of linear correlation and linear independence. The conclusion points out that linear independence means that there are no two (base) vectors with the same direction in a vector graph;otherwise, it is a linear correlation.
文摘We study the Hausdorff measure of linear Cantor setE, on the unit interval, under the strong seperated condition. We give a necessary and sufficient condition for ?(E)=∣E∣° by using the contracting ratio and the first gap. This condition is easy to use. Key words linear Cantor set - Hausdorff measure - strong seperated condition CLC number O 174. 12 Foundation item: Supported by the National Natural Science Foundation of China (10171028)Biography: Ma Chao (1975-), male, Ph. D. candidate, research direction: fractal geometry.
基金This work is supported partially by the foundation of the National Education Ministry, National
文摘Let 0<A≤1/3 ,K(λ) be the attractor of an iterated function system {ψ1,ψ2} on the line, where 1(x)= AT, ψ1(x) = 1-λ+λx, x∈[0,1]. We call K(λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K(λ).
文摘1.Some definitions and notions 1.1. Hausdorff gauge and Packing gaugeA funciton h:[0.1]→ R is said to be a dimension function if it has the following properties:(1)h(t)is continuous and increasing, h(0)=0, h(t)>0 when t>0.(2)there exists a constant M>0, h(2t)<Mh(t) for any t>0.
文摘It is well known that a totally disconnected compact metric space without isolated points is a Cantor set.In this note me give a simple proof of this theorem.
基金supported by NSFC(11101447),supported by NSFC(11201500)
文摘We study the doubling property of binomial measures on generalized ternary Cantor subsets of [0, 1]. We find some new phenomena. There are three different cases. In the first case, we obtain an equivalent condition for the measure to be doubling. In the other cases, we show that the condition is not necessary. Then facts and partial results are discussed.
文摘Let 0<λ_1,λ_2<1 and 1-λ_1-λ_2≥max｛λ_1,λ_2｝.Let ~K(λ_1,λ_2) be the attractor of the iterated function system ｛φ_1,φ_2｝on the line,where φ_1(x)=λ_1x and φ_2(x)=1-λ_2+λ_2x,x∈R.~K(λ_1,λ_2) is called a non-symmetry Cantor set. In this paper,it is proved that the exact Hausdorff centred measure of K(λ_1,λ_2) equals 2s(1-λ)s,where λ=max｛λ_1,λ_2｝ and s is the Hausdorff dimension of K(λ_1,λ_2).
文摘In this work, we establish relations between DNA sequences with missing subsequences (the forbidden words) and the generalized Cantor sets. Various examples associated with some generalized Cantor sets, including Hao’s frame representation and the generalized Sierpinski Set, along with their fractal graphs, are also presented in this work.
文摘The Cantor’s conclusion that “there is a one-one correspondence between the points on n-dimensional space and the points on the line” appears in the numerous current documents. By introducing a monotonic and continuous function, a one-one correspondence between two intervals is built;and by using parametric equations of the curve, a one-one correspondence from the points on the curve to the points on the line is established. Specially, the meanings of multivariate functions are given. By using a n-variable equation with a parameter, a correspondence from n-dimensional space area to a interval is built, so the wrong conclusion is completely denied. The paper enriches calculus and can reduce the teaching difficulty of real function in some degree. The expression of moving curve (surface) limit is given in the paper. More importantly, after the conclusion is corrected, it will be necessary and possible to re-establish the theory and the approach about multivariable differential calculus.
基金Supported by the National Natural Science Foundation of China (No. 10771075)
文摘We consider the homogeneous Cantor sets which are generalization of symmetric perfect sets, and give a formula of the exact Hausdorff measures for a class of such sets.
文摘For real valued functions defined on Cantor triadic set, a derivative with corresponding formula of Newton Leibniz’s type is given. In particular, for the self similar functions and alternately jumping functions defined in this paper, their derivative and exceptional sets are studied accurately by using ergodic theory on Σ 2 and Duffin Schaeffer’s theorem concerning metric diophantine approximation. In addition, Haar basis of L 2(Σ 2) is constructed and Haar expansion of standard self similar function is given.
基金the National Science Foundation of China (10671180)Jiangsu University 05JDG041
文摘We pursue the study on homogeneous Cantor sets with their translations. We get the fractal structure of intersection I（t）, and find that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from shifting numbers with the coding of t. Concretely, a very brief calculation formula of the measure with the coding of t is given.
文摘In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and the smallest such k is 2~m.Moreover,we generalize this result to the middle-1/αCantor set for1<α<2+√5 and m sufficiently large.For the naturally embedded image W of the Cantor dust C×C into the complex plane C,we prove that for any integer m≥3,every element in the closed unit disk in C can be written as the sum of at most 2^(m+8)m-th powers of elements in W.At last,some similar results on p-adic Cantor sets are also obtained.
文摘In this paper we have found a general subordinator, X, whose range up to time 1, X([0,1)), has similar structure as random re orderings of the Cantor set K(ω).X([0,1)) and K(ω) have the same exact Hausdorff measure function and the integal test of packing measure.
文摘We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph models introduced by Barabási, Ravasz and Vicsek ([1] [2]). We generalize the graph models using stars and cliques: both algorithm construct graph sequences such that the next iteration is always based on n replicas of the current iteration, where n is the size of the initial graph structure, being a star or a clique. We analyze these self-similar graph sequences using IMSs in function of the size of the initial star and clique, respectively. Our research uses the Cantor set for the description of the fixed set of these IMSs, which we interpret as the limit object of the analyzed self-similar networks.
基金supported by National Natural Science Foundation of China (Grant Nos.10971056,10771164)
文摘Let E = E({nk},{ck}) be a fat uniform Cantor set. We prove that E is a minimally fat set for doubling measures if and only if (nkck)p = ∞ for all p < 1 and that E is a fairly fat set for doubling measures if and only if there are constants 0 < p < q < 1 such that (nkck)q < ∞ and (nkck)p = ∞. The classes of minimally thin uniform Cantor sets and of fairly thin uniform Cantor sets are also characterized.
