Let M({nk}k≥1,{ck}k≥1) be the collection of homogeneous Moran sets determined by {nk}k≥1and {ck}k≥1, where {nk}k≥1 is a sequence of positive integers and {ck}k≥1 a sequence of positive numbers. Then the maximal ...Let M({nk}k≥1,{ck}k≥1) be the collection of homogeneous Moran sets determined by {nk}k≥1and {ck}k≥1, where {nk}k≥1 is a sequence of positive integers and {ck}k≥1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in M are determined. The result is proved that for any value s between the maximal and minimal values, there exists an element in M{nk}k≥1, {ck}k≥1) such that its Hausdorff dimension is equal to s. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.展开更多
At the end of 1982, Palmiter, Brinster et al. reported that mice of superb growth had been created through microinjection of a hybrid gene coding for rat growth hormone into the male pronuclei of mouse fertilized eggs...At the end of 1982, Palmiter, Brinster et al. reported that mice of superb growth had been created through microinjection of a hybrid gene coding for rat growth hormone into the male pronuclei of mouse fertilized eggs. This work ushered in a new approach to develop new breeds of domestic animals with a method of foreign gene trans-formation. We infer that fish, on a lower level of evolution in vertebrata,展开更多
Mauremys mutica(Cantor,1842)is an endangered species in China.Main phenotypic variations inbody color,body weight,body shape,clutch size,egg size,and hatchling size were revealed betweenthe southern and northern popul...Mauremys mutica(Cantor,1842)is an endangered species in China.Main phenotypic variations inbody color,body weight,body shape,clutch size,egg size,and hatchling size were revealed betweenthe southern and northern populations.Both populations have the phenomenon of'larger male'sexualsize dimorphism(SSD),especially in the southern population.Furthermore,genetic variations betweenthe two populations were analyzed by RAPD band patterns of 30 random individuals in each population.The average genetic distance was 0.299±0.108 among the samples of two populations.The average ge-netic distance between southern and northern populations was 0.305±0.046.Cluster analysis indicatedthat all the individuals from the southern and northern populations were clustered among themselves toform two distinct clades.A total of 20 population-specific RAPD fragments were scored from 16 primers,and could be used as RAPD markers for distinguishing the southern and the northern population.Basedon the nucleotide sequences of two RAPD markers,two pairs of SCAR primers(SC1-S and SC2-S)weredesigned,which could be used as SCAR markers for the southern population.According to the significantphenotypic and genetic variations,we suggested that the northern population and southern populationmight be considered as two separate taxa,the'northern taxon'and the'southern taxon',and the con-servation should be respectively conducted on the two taxa.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
In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical...In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of count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
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
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
Let C be the Cantor triadic set and let Ca= C+a = The authors give the dimensions of and Hp. In addition the characteristic of Hp is described by means of some measure supported on C.
Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space (Rn), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of rea...Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space (Rn), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of real numbers is of measure zero. The proposition just mentioned is basically consistent with Poincare's famous remark which renders a deep insight into the paradoxical structural nature of Cantor's continuum consisting precisely of all distinct real numbers.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
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 construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, s... In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.展开更多
In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in ...In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in the cosmos according to Einstein’s prediction must be dark energy or not there at all. This percentage is in almost complete agreement with actual measurements.展开更多
Using powerful concepts and tools borrowed from the seminal arsenal connecting physics fundamentals with esoteric set theoretical operations developed in recent years by Alexandria E-infinity theoretician M. S. El Nas...Using powerful concepts and tools borrowed from the seminal arsenal connecting physics fundamentals with esoteric set theoretical operations developed in recent years by Alexandria E-infinity theoretician M. S. El Naschie, this paper explores the deep implications of some of the dualities Dr El Naschie has identified and analyzed in his exposes, connecting them with our own Xonic Quantum Physics (XQP) which places dynamical action rather than spacetime and energy at the core of the System of the World.展开更多
基金Project supported by the National Climbing Project"Nonlinear Science"and the Scientific Foundation of the State Education Commission of China.
文摘Let M({nk}k≥1,{ck}k≥1) be the collection of homogeneous Moran sets determined by {nk}k≥1and {ck}k≥1, where {nk}k≥1 is a sequence of positive integers and {ck}k≥1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in M are determined. The result is proved that for any value s between the maximal and minimal values, there exists an element in M{nk}k≥1, {ck}k≥1) such that its Hausdorff dimension is equal to s. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.
基金Project supported by the Science Fund of Academia Sinica
文摘At the end of 1982, Palmiter, Brinster et al. reported that mice of superb growth had been created through microinjection of a hybrid gene coding for rat growth hormone into the male pronuclei of mouse fertilized eggs. This work ushered in a new approach to develop new breeds of domestic animals with a method of foreign gene trans-formation. We infer that fish, on a lower level of evolution in vertebrata,
基金the National Basic Research Program of China(No.2004CB117405)the Key Research Program of Ocean and Fishery Bureau of Guangdong province(No.2001A09)
文摘Mauremys mutica(Cantor,1842)is an endangered species in China.Main phenotypic variations inbody color,body weight,body shape,clutch size,egg size,and hatchling size were revealed betweenthe southern and northern populations.Both populations have the phenomenon of'larger male'sexualsize dimorphism(SSD),especially in the southern population.Furthermore,genetic variations betweenthe two populations were analyzed by RAPD band patterns of 30 random individuals in each population.The average genetic distance was 0.299±0.108 among the samples of two populations.The average ge-netic distance between southern and northern populations was 0.305±0.046.Cluster analysis indicatedthat all the individuals from the southern and northern populations were clustered among themselves toform two distinct clades.A total of 20 population-specific RAPD fragments were scored from 16 primers,and could be used as RAPD markers for distinguishing the southern and the northern population.Basedon the nucleotide sequences of two RAPD markers,two pairs of SCAR primers(SC1-S and SC2-S)weredesigned,which could be used as SCAR markers for the southern population.According to the significantphenotypic and genetic variations,we suggested that the northern population and southern populationmight be considered as two separate taxa,the'northern taxon'and the'southern taxon',and the con-servation should be respectively conducted on the two taxa.
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In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of count
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.
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a grant from the National Science Foundation of China.
文摘Let C be the Cantor triadic set and let Ca= C+a = The authors give the dimensions of and Hp. In addition the characteristic of Hp is described by means of some measure supported on C.
基金Supperted by Special Foundation of Dalian Univ. of Technology.
文摘Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space (Rn), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of real numbers is of measure zero. The proposition just mentioned is basically consistent with Poincare's famous remark which renders a deep insight into the paradoxical structural nature of Cantor's continuum consisting precisely of all distinct real numbers.
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 Natural Science Foundation of Education Committee of Anhui Province (No.2001kj198zc).
文摘 In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.
文摘In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in the cosmos according to Einstein’s prediction must be dark energy or not there at all. This percentage is in almost complete agreement with actual measurements.
文摘Using powerful concepts and tools borrowed from the seminal arsenal connecting physics fundamentals with esoteric set theoretical operations developed in recent years by Alexandria E-infinity theoretician M. S. El Naschie, this paper explores the deep implications of some of the dualities Dr El Naschie has identified and analyzed in his exposes, connecting them with our own Xonic Quantum Physics (XQP) which places dynamical action rather than spacetime and energy at the core of the System of the World.
