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
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
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
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.
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
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
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
In this paper, we construct a scattered Cantor set having the value 1/2 of log2/log3- dimensional Hausdorff measure. Combining a theorem of Lee and Baek, we can see the value 21 is the minimal Hausdorff measure of the...In this paper, we construct a scattered Cantor set having the value 1/2 of log2/log3- dimensional Hausdorff measure. Combining a theorem of Lee and Baek, we can see the value 21 is the minimal Hausdorff measure of the scattered Cantor sets, and our result solves a conjecture of Lee and Baek.展开更多
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
By viewing spacetime as a transfinite Turing computer, the present work is aimed at a generalization and geometrical-topological reinterpretation of a relatively old conjecture that the wormholes of general relativity...By viewing spacetime as a transfinite Turing computer, the present work is aimed at a generalization and geometrical-topological reinterpretation of a relatively old conjecture that the wormholes of general relativity are behind the physics and mathematics of quantum entanglement theory. To do this we base ourselves on the comprehensive set theoretical and topological machinery of the Cantorian-fractal E-infinity spacetime theory. Going all the way in this direction we even go beyond a quantum gravity theory to a precise set theoretical understanding of what a quantum particle, a quantum wave and quantum spacetime are. As a consequence of all these results and insights we can reason that the local Casimir pressure is the difference between the zero set quantum particle topological pressure and the empty set quantum wave topological pressure which acts as a wormhole “connecting” two different quantum particles with varying degrees of entanglement corresponding to varying degrees of emptiness of the empty set (wormhole). Our final result generalizes the recent conceptual equation of Susskind and Maldacena ER = EPR to become ZMG = ER = EPR where ZMG stands for zero measure Rindler-KAM geometry (of spacetime). These results were only possible because of the ultimate simplicity of our exact model based on Mauldin-Williams random Cantor sets and the corresponding exact Hardy’s quantum entanglement probability P(H) = where is the Hausdorff dimension of the topologically zero dimensional random Cantor thin set, i.e. a zero measure set and . On the other hand the positive measure spatial separation between the zero sets is a fat Cantor empty set possessing a Hausdorff dimension equal while its Menger-Urysohn topological dimension is a negative value equal minus one. This is the mathematical quintessence of a wormhole paralleling multiple connectivity in classical topology. It is both physically there because of the positive measure and not there because of the negative topological dimension.展开更多
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
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
The paper succeeds in the obtaining a class of generalized non-uniform Cantor set based on the iteration （1）： Si（x） = αix ＋ bi, x ∈ [0, 1], i = 1,2,…, m, where 0 〈 αi 〈 1, i = 1,2,…,m; bi ＋ αi 〉 0, i =...The paper succeeds in the obtaining a class of generalized non-uniform Cantor set based on the iteration （1）： Si（x） = αix ＋ bi, x ∈ [0, 1], i = 1,2,…, m, where 0 〈 αi 〈 1, i = 1,2,…,m; bi ＋ αi 〉 0, i = 1,2,…,m- 1, b1 = 0 and αm ＋ bm = 1. Providing the sufficient and necessary conditions of its existence Hausdorff measure.展开更多
Define a linear Cantor set C to be the attractor of a linear iterated function system fj （x） =rjx ＋ bj（j = 1,2,…,N）, on the line satisfying the sures with respect to C,we study the centered upper and the centere...Define a linear Cantor set C to be the attractor of a linear iterated function system fj （x） =rjx ＋ bj（j = 1,2,…,N）, on the line satisfying the sures with respect to C,we study the centered upper and the centered lower density for Ф（t） = t^s withunnatural choices and with natural choices of s.展开更多
Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pytha...Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy E = 1/2mv<sup>2</sup> leads to Einstein’s celebrated maximal energy equation E = mc<sup>2</sup> and how in turn this can be dissected into the ordinary energy density E(O) = mc<sup>2</sup>/22 and the dark energy density E(D) = mc<sup>2</sup>(21/22) of the cosmos where m is the mass;v is the velocity and c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.展开更多
The work gives a natural explanation for the ordinary and dark energy density of the cosmos based on conventional quantum mechanical considerations which dates back as far as the early days of the quantum theory and s...The work gives a natural explanation for the ordinary and dark energy density of the cosmos based on conventional quantum mechanical considerations which dates back as far as the early days of the quantum theory and specifically the work of Max Planck who seems to be the first to propose the possibility of a half quanta corresponding to the ground state, i.e. the energy zero point of the vacuum. Combining these old insights with the relatively new results of Hardy’s quantum entanglement and Witten’s topological quantum field theory as well as the fractal version of M-theory, we find a remarkably simple general theory for dark energy and the Casimir effect.展开更多
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.
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).
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.
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.
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the National Natural Science Foundation of China (No.10771164)the Education Committee of Fujian Province (No.JA08155)
文摘In this paper, we construct a scattered Cantor set having the value 1/2 of log2/log3- dimensional Hausdorff measure. Combining a theorem of Lee and Baek, we can see the value 21 is the minimal Hausdorff measure of the scattered Cantor sets, and our result solves a conjecture of Lee and Baek.
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.
文摘By viewing spacetime as a transfinite Turing computer, the present work is aimed at a generalization and geometrical-topological reinterpretation of a relatively old conjecture that the wormholes of general relativity are behind the physics and mathematics of quantum entanglement theory. To do this we base ourselves on the comprehensive set theoretical and topological machinery of the Cantorian-fractal E-infinity spacetime theory. Going all the way in this direction we even go beyond a quantum gravity theory to a precise set theoretical understanding of what a quantum particle, a quantum wave and quantum spacetime are. As a consequence of all these results and insights we can reason that the local Casimir pressure is the difference between the zero set quantum particle topological pressure and the empty set quantum wave topological pressure which acts as a wormhole “connecting” two different quantum particles with varying degrees of entanglement corresponding to varying degrees of emptiness of the empty set (wormhole). Our final result generalizes the recent conceptual equation of Susskind and Maldacena ER = EPR to become ZMG = ER = EPR where ZMG stands for zero measure Rindler-KAM geometry (of spacetime). These results were only possible because of the ultimate simplicity of our exact model based on Mauldin-Williams random Cantor sets and the corresponding exact Hardy’s quantum entanglement probability P(H) = where is the Hausdorff dimension of the topologically zero dimensional random Cantor thin set, i.e. a zero measure set and . On the other hand the positive measure spatial separation between the zero sets is a fat Cantor empty set possessing a Hausdorff dimension equal while its Menger-Urysohn topological dimension is a negative value equal minus one. This is the mathematical quintessence of a wormhole paralleling multiple connectivity in classical topology. It is both physically there because of the positive measure and not there because of the negative topological dimension.
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.
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.
基金Supported by the Scientific Research of Hanshan Teacher's College(2004)
文摘The paper succeeds in the obtaining a class of generalized non-uniform Cantor set based on the iteration （1）： Si（x） = αix ＋ bi, x ∈ [0, 1], i = 1,2,…, m, where 0 〈 αi 〈 1, i = 1,2,…,m; bi ＋ αi 〉 0, i = 1,2,…,m- 1, b1 = 0 and αm ＋ bm = 1. Providing the sufficient and necessary conditions of its existence Hausdorff measure.
文摘Define a linear Cantor set C to be the attractor of a linear iterated function system fj （x） =rjx ＋ bj（j = 1,2,…,N）, on the line satisfying the sures with respect to C,we study the centered upper and the centered lower density for Ф（t） = t^s withunnatural choices and with natural choices of s.
文摘Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy E = 1/2mv<sup>2</sup> leads to Einstein’s celebrated maximal energy equation E = mc<sup>2</sup> and how in turn this can be dissected into the ordinary energy density E(O) = mc<sup>2</sup>/22 and the dark energy density E(D) = mc<sup>2</sup>(21/22) of the cosmos where m is the mass;v is the velocity and c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.
文摘The work gives a natural explanation for the ordinary and dark energy density of the cosmos based on conventional quantum mechanical considerations which dates back as far as the early days of the quantum theory and specifically the work of Max Planck who seems to be the first to propose the possibility of a half quanta corresponding to the ground state, i.e. the energy zero point of the vacuum. Combining these old insights with the relatively new results of Hardy’s quantum entanglement and Witten’s topological quantum field theory as well as the fractal version of M-theory, we find a remarkably simple general theory for dark energy and the Casimir effect.
