Based on bipolar dynamic logic (BDL) and bipolar quantum linear algebra (BQLA) this work introduces bipolar quantum logic gates and quantum cellular combinatorics with a logical interpretation to quantum entanglement....Based on bipolar dynamic logic (BDL) and bipolar quantum linear algebra (BQLA) this work introduces bipolar quantum logic gates and quantum cellular combinatorics with a logical interpretation to quantum entanglement. It is shown that: 1) BDL leads to logically definable causality and generic particle-antiparticle bipolar quantum entanglement;2) BQLA makes composite atom-atom bipolar quantum entanglement reachable. Certain logical equivalence is identified between the new interpretation and established ones. A logical reversibility theorem is presented for ubiquitous quantum computing. Physical reversibility is briefly discussed. It is shown that a bipolar matrix can be either a modular generalization of a quantum logic gate matrix or a cellular connectivity matrix. Based on this observation, a scalable graph theory of quantum cellular combinatorics is proposed. It is contended that this work constitutes an equilibrium-based logical extension to Bohr’s particle-wave complementarity principle, Bohm’s wave function and Bell’s theorem. In the meantime, it is suggested that the result may also serve as a resolution, rather than a falsification, to the EPR paradox and, therefore, a equilibrium-based logical unification of local realism and quantum non-locality.展开更多
Using the coherent state representation we derive some new operator identities and study some mathematical relations in comblnatorics. The technique of integral within an ordered product （IWOP） of operators plays an...Using the coherent state representation we derive some new operator identities and study some mathematical relations in comblnatorics. The technique of integral within an ordered product （IWOP） of operators plays an essential role in realizing our goal.展开更多
Partition-and-Recur (PAR) method is a simple and useful formal method. It can be used to design and testify algo-rithmic programs. In this paper, we propose that PAR method is an effective formal method on solving com...Partition-and-Recur (PAR) method is a simple and useful formal method. It can be used to design and testify algo-rithmic programs. In this paper, we propose that PAR method is an effective formal method on solving combinatorics problems. Furthermore, we formally derive combinatorics problems by PAR method, which cannot only simplify the process of algorithmic program's designing, but also improve its automatization, standardization and correctness. We develop algorithms for two typical combinatorics problems, the number of string scheme and the number of error per-mutation scheme. Lastly, we obtain accurate C++ programs which are transformed by automatic transforming system of PAR platform.展开更多
This paper is contributed to the combinatorial properties of the MSS sequences, which are the periodic kneading words of quadratic maps defined on a interval. An explicit expression of adjacency relations on MSS seque...This paper is contributed to the combinatorial properties of the MSS sequences, which are the periodic kneading words of quadratic maps defined on a interval. An explicit expression of adjacency relations on MSS sequences of given lengths is established.展开更多
This paper is contributed to the combinatorial properties of the periodic kneading words of antisymmetric cubic maps defined on a interval. The least words of given lengths, the adjacency relations on the words of giv...This paper is contributed to the combinatorial properties of the periodic kneading words of antisymmetric cubic maps defined on a interval. The least words of given lengths, the adjacency relations on the words of given lengths and the parity-alternative property in some sets of such words are established.展开更多
A special class of cubic polynomials possessing decay of geometry property is studied.This class of cubic bimodal maps has generalized Fibonacci combinatorics.For maps with bounded combinatorics,we show that they have...A special class of cubic polynomials possessing decay of geometry property is studied.This class of cubic bimodal maps has generalized Fibonacci combinatorics.For maps with bounded combinatorics,we show that they have an absolutely continuous invariant probability measure.展开更多
Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors....Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. The locally twisted cube LTQ_n has many good properties. In this paper, we show that the 1-good-neighbor connectivity κ~1(LTQ_n) = 2n-2 and the 1-good-neighbor diagnosability of LTQ_n is 2n-1 under the PMC model for n ≥ 4 and the MM~*model for n ≥ 5.展开更多
Modeling of solute transport is a key issue in the area of soil physics and hydrogeology. The most common approach (the convection-dispersion equation) considers an average convection flow rate and Fickian-like disper...Modeling of solute transport is a key issue in the area of soil physics and hydrogeology. The most common approach (the convection-dispersion equation) considers an average convection flow rate and Fickian-like dispersion. Here,we propose a solute transport model in porous media of continuously expanding scale, according to the combinatorics principle. The model supposed actual porous media as a combinative body of many basic segments. First, we studied the solute transport process in each basic segment body, and then deduced the distribution of pore velocity in each basic segment body by difference approximation, finally assembled the solute transport process of each basic segment body into one of the combinative body. The simulation result coincided with the solute transport process observed in test. The model provides useful insight into the solute transport process of the non-Fickian dispersion in continuously expanding scale.展开更多
Diagnosability of a multiprocessor system G is one important measure of the reliability of interconnection networks. In 2016, Zhang et al. proposed the g-extra diagnosability of G, which restrains that every component...Diagnosability of a multiprocessor system G is one important measure of the reliability of interconnection networks. In 2016, Zhang et al. proposed the g-extra diagnosability of G, which restrains that every component of G – S has at least (g + 1) vertices. The locally twisted cube LTQn is applied widely. In this paper, we show that LTQn is tightly (4n – 9) super 3-extra connected for n ≥ 6 and the 3-extra diagnosability of LTQn under the PMC model and MM* model is 4n - 6 for n ≥ 5 and n ≥ 7, respectively.展开更多
New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equ...New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements;and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned;and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.展开更多
Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed a measure for fault tolerance of the system, which is called the g-good-neighbor diagnosability that restrains ever...Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed a measure for fault tolerance of the system, which is called the g-good-neighbor diagnosability that restrains every fault-free node containing at least g fault-free neighbors. In 2015, Zhang et al. proposed a measure for fault diagnosis of the system, namely, g-extra diagnosability, which restrains that every fault-free component has at least g+1 fault-free nodes. In this paper, we obtain some properties of the g-good-neighbor (g-extra) diagnosability of the system and give the g-good-neighbor (g-extra) diagnosability of some graphs under the PMC model and MM<sup>*</sup> model.展开更多
Factor properties and their structures are important themes in combinatorics on words.Let D be the infinite one-sided sequence over the alphabet{a,b}generated by the period-doubling substitutionσ(a)=ab andσ(b)=aa.In...Factor properties and their structures are important themes in combinatorics on words.Let D be the infinite one-sided sequence over the alphabet{a,b}generated by the period-doubling substitutionσ(a)=ab andσ(b)=aa.In this paper,we determine the derived sequence D_(ω)(D)for any factorω■D,and study some factor spectra using the structures of derived sequences.We also prove the reflexivity property of derived sequences.展开更多
We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as I wσ 1σ ...We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as I wσ 1σ 2…σ k,where I w is the inner automorphism associated with w, and σ j∈ S for 1≤j≤k. As a consequence,M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.展开更多
This paper is on a distribution question of Ⅰ type in the combinatorics (see: 'Ke Zhao, Wei WanDi, combinatorial theory, science press, 1981. '). Suppose n distrinct balls, m distinct boxes. The capacity set...This paper is on a distribution question of Ⅰ type in the combinatorics (see: 'Ke Zhao, Wei WanDi, combinatorial theory, science press, 1981. '). Suppose n distrinct balls, m distinct boxes. The capacity set of every box is Now,place the n balls into the m boxes. Thenumber of distinct placed ways is expressed by , and simply by And it is called the distribution number.Main result is:Theorem The exponential generating function of sequence isThus .there is the recurrence relationhere (integer set).展开更多
The study of the confluences of the roots of a given set of polynomials—root-pattern problem— does not appear to have been considered. We examine the situation, which leads us on to Young tableaux and tableaux repre...The study of the confluences of the roots of a given set of polynomials—root-pattern problem— does not appear to have been considered. We examine the situation, which leads us on to Young tableaux and tableaux representations. This in turn is found to be an aspect of multipartite partitions. We discover, and show, that partitions can be expressed algebraically and can be “differentiated” and “integrated”. We show a complete set of bipartite and tripartite partitions, indicating equivalences for the root-pattern problem, for select pairs and triples. Tables enumerating the number of bipartite and tripartite partitions, for small pairs and triples are given in an appendix.展开更多
The cutwidth of a graph G is the minimum number of overlap edges when G is embedded into a path Pn.The cutwidth problem for a graph G is to determine the cutwidth of G.A graph G with cutwidth k is k-cutwidth critical ...The cutwidth of a graph G is the minimum number of overlap edges when G is embedded into a path Pn.The cutwidth problem for a graph G is to determine the cutwidth of G.A graph G with cutwidth k is k-cutwidth critical if every proper subgraph of G has cutwidth less than k and G is homeomorphically minimal.In this paper,we completely investigated methods of forming a k-cutwidth(k>1)critical tree T.展开更多
This paper describes the construction and enumeration of mixed orthogonal arrays (MOA) to produce optimal experimental designs. A MOA is a multiset whose rows are the different combinations of factor levels, discrete ...This paper describes the construction and enumeration of mixed orthogonal arrays (MOA) to produce optimal experimental designs. A MOA is a multiset whose rows are the different combinations of factor levels, discrete values of the variable under study, having very well defined features such as symmetry and strength three (all main interactions are taken in consideration). The applied methodology blends the fields of combinatorics and group theory by applying the ideas of orbits, stabilizers and isomorphisms to array generation and enumeration. Integer linear programming was used in order to exploit the symmetry property of the arrays under study. The backtrack search algorithm was used to find suitable arrays in the underlying space of possible solutions. To test the performance of the MOAs, an engineered system was used as a case study within the stage of parameter design. The analysis showed how the MOAs were capable of meeting the fundamental engineering design axioms and principles, creating optimal experimental designs within the desired context.展开更多
Communication boards provide a low-cost means of facilitating communication with patients who are unable to speak;however the process is slow and frustrating. A computer model was used to calculate the cumulative freq...Communication boards provide a low-cost means of facilitating communication with patients who are unable to speak;however the process is slow and frustrating. A computer model was used to calculate the cumulative frequency-weighted path length for letter selection (“chart index”) for three conventional communication boards using different search strategies;and exhaustively generate and evaluate designs for a novel communication board based on a frequency-ordered arrangement of letters. For all arrangements, a 46% to 53% reduction in chart indices was achieved when “2 Dimensional” (2D) rather than “1 Dimensional” (1D) search strategies were employed. A further 23% to 30% reduction in chart indices was achieved through use of frequency-ordered sequences with optimal row groupings. Conventional communication boards can be used more efficiently by employing a 2D search strategy. Novel communication boards based on optimised arrangements of frequency-ordered letter sequences potentially provide a faster means of communication than conventional communication boards.展开更多
文摘Based on bipolar dynamic logic (BDL) and bipolar quantum linear algebra (BQLA) this work introduces bipolar quantum logic gates and quantum cellular combinatorics with a logical interpretation to quantum entanglement. It is shown that: 1) BDL leads to logically definable causality and generic particle-antiparticle bipolar quantum entanglement;2) BQLA makes composite atom-atom bipolar quantum entanglement reachable. Certain logical equivalence is identified between the new interpretation and established ones. A logical reversibility theorem is presented for ubiquitous quantum computing. Physical reversibility is briefly discussed. It is shown that a bipolar matrix can be either a modular generalization of a quantum logic gate matrix or a cellular connectivity matrix. Based on this observation, a scalable graph theory of quantum cellular combinatorics is proposed. It is contended that this work constitutes an equilibrium-based logical extension to Bohr’s particle-wave complementarity principle, Bohm’s wave function and Bell’s theorem. In the meantime, it is suggested that the result may also serve as a resolution, rather than a falsification, to the EPR paradox and, therefore, a equilibrium-based logical unification of local realism and quantum non-locality.
基金The project supported by National Natural Science Foundation of China under Grant No. 10475056
文摘Using the coherent state representation we derive some new operator identities and study some mathematical relations in comblnatorics. The technique of integral within an ordered product （IWOP） of operators plays an essential role in realizing our goal.
文摘Partition-and-Recur (PAR) method is a simple and useful formal method. It can be used to design and testify algo-rithmic programs. In this paper, we propose that PAR method is an effective formal method on solving combinatorics problems. Furthermore, we formally derive combinatorics problems by PAR method, which cannot only simplify the process of algorithmic program's designing, but also improve its automatization, standardization and correctness. We develop algorithms for two typical combinatorics problems, the number of string scheme and the number of error per-mutation scheme. Lastly, we obtain accurate C++ programs which are transformed by automatic transforming system of PAR platform.
基金the National Natural Science Foundation of China (Grant No.10731040)
文摘This paper is contributed to the combinatorial properties of the MSS sequences, which are the periodic kneading words of quadratic maps defined on a interval. An explicit expression of adjacency relations on MSS sequences of given lengths is established.
基金the National Natural Science Foundation of China (Grant No.10731040)
文摘This paper is contributed to the combinatorial properties of the periodic kneading words of antisymmetric cubic maps defined on a interval. The least words of given lengths, the adjacency relations on the words of given lengths and the parity-alternative property in some sets of such words are established.
文摘A special class of cubic polynomials possessing decay of geometry property is studied.This class of cubic bimodal maps has generalized Fibonacci combinatorics.For maps with bounded combinatorics,we show that they have an absolutely continuous invariant probability measure.
基金supported by the National Natural Science Foundation of China(61772010)
文摘Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. The locally twisted cube LTQ_n has many good properties. In this paper, we show that the 1-good-neighbor connectivity κ~1(LTQ_n) = 2n-2 and the 1-good-neighbor diagnosability of LTQ_n is 2n-1 under the PMC model for n ≥ 4 and the MM~*model for n ≥ 5.
文摘Modeling of solute transport is a key issue in the area of soil physics and hydrogeology. The most common approach (the convection-dispersion equation) considers an average convection flow rate and Fickian-like dispersion. Here,we propose a solute transport model in porous media of continuously expanding scale, according to the combinatorics principle. The model supposed actual porous media as a combinative body of many basic segments. First, we studied the solute transport process in each basic segment body, and then deduced the distribution of pore velocity in each basic segment body by difference approximation, finally assembled the solute transport process of each basic segment body into one of the combinative body. The simulation result coincided with the solute transport process observed in test. The model provides useful insight into the solute transport process of the non-Fickian dispersion in continuously expanding scale.
文摘Diagnosability of a multiprocessor system G is one important measure of the reliability of interconnection networks. In 2016, Zhang et al. proposed the g-extra diagnosability of G, which restrains that every component of G – S has at least (g + 1) vertices. The locally twisted cube LTQn is applied widely. In this paper, we show that LTQn is tightly (4n – 9) super 3-extra connected for n ≥ 6 and the 3-extra diagnosability of LTQn under the PMC model and MM* model is 4n - 6 for n ≥ 5 and n ≥ 7, respectively.
文摘New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements;and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned;and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.
文摘Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed a measure for fault tolerance of the system, which is called the g-good-neighbor diagnosability that restrains every fault-free node containing at least g fault-free neighbors. In 2015, Zhang et al. proposed a measure for fault diagnosis of the system, namely, g-extra diagnosability, which restrains that every fault-free component has at least g+1 fault-free nodes. In this paper, we obtain some properties of the g-good-neighbor (g-extra) diagnosability of the system and give the g-good-neighbor (g-extra) diagnosability of some graphs under the PMC model and MM<sup>*</sup> model.
基金supported by National Natural Science Foundation of China(11701024)the Fundamental Research Funds for the Central Universities(2019RC17)。
文摘Factor properties and their structures are important themes in combinatorics on words.Let D be the infinite one-sided sequence over the alphabet{a,b}generated by the period-doubling substitutionσ(a)=ab andσ(b)=aa.In this paper,we determine the derived sequence D_(ω)(D)for any factorω■D,and study some factor spectra using the structures of derived sequences.We also prove the reflexivity property of derived sequences.
文摘We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as I wσ 1σ 2…σ k,where I w is the inner automorphism associated with w, and σ j∈ S for 1≤j≤k. As a consequence,M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.
文摘This paper is on a distribution question of Ⅰ type in the combinatorics (see: 'Ke Zhao, Wei WanDi, combinatorial theory, science press, 1981. '). Suppose n distrinct balls, m distinct boxes. The capacity set of every box is Now,place the n balls into the m boxes. Thenumber of distinct placed ways is expressed by , and simply by And it is called the distribution number.Main result is:Theorem The exponential generating function of sequence isThus .there is the recurrence relationhere (integer set).
文摘The study of the confluences of the roots of a given set of polynomials—root-pattern problem— does not appear to have been considered. We examine the situation, which leads us on to Young tableaux and tableaux representations. This in turn is found to be an aspect of multipartite partitions. We discover, and show, that partitions can be expressed algebraically and can be “differentiated” and “integrated”. We show a complete set of bipartite and tripartite partitions, indicating equivalences for the root-pattern problem, for select pairs and triples. Tables enumerating the number of bipartite and tripartite partitions, for small pairs and triples are given in an appendix.
基金Supported by Soft Science Foundation of Henan Province(Grant No.192400410212)the Science and Technology Key Project of Henan Province of China(Grant No.22210211008)。
文摘The cutwidth of a graph G is the minimum number of overlap edges when G is embedded into a path Pn.The cutwidth problem for a graph G is to determine the cutwidth of G.A graph G with cutwidth k is k-cutwidth critical if every proper subgraph of G has cutwidth less than k and G is homeomorphically minimal.In this paper,we completely investigated methods of forming a k-cutwidth(k>1)critical tree T.
文摘This paper describes the construction and enumeration of mixed orthogonal arrays (MOA) to produce optimal experimental designs. A MOA is a multiset whose rows are the different combinations of factor levels, discrete values of the variable under study, having very well defined features such as symmetry and strength three (all main interactions are taken in consideration). The applied methodology blends the fields of combinatorics and group theory by applying the ideas of orbits, stabilizers and isomorphisms to array generation and enumeration. Integer linear programming was used in order to exploit the symmetry property of the arrays under study. The backtrack search algorithm was used to find suitable arrays in the underlying space of possible solutions. To test the performance of the MOAs, an engineered system was used as a case study within the stage of parameter design. The analysis showed how the MOAs were capable of meeting the fundamental engineering design axioms and principles, creating optimal experimental designs within the desired context.
文摘Communication boards provide a low-cost means of facilitating communication with patients who are unable to speak;however the process is slow and frustrating. A computer model was used to calculate the cumulative frequency-weighted path length for letter selection (“chart index”) for three conventional communication boards using different search strategies;and exhaustively generate and evaluate designs for a novel communication board based on a frequency-ordered arrangement of letters. For all arrangements, a 46% to 53% reduction in chart indices was achieved when “2 Dimensional” (2D) rather than “1 Dimensional” (1D) search strategies were employed. A further 23% to 30% reduction in chart indices was achieved through use of frequency-ordered sequences with optimal row groupings. Conventional communication boards can be used more efficiently by employing a 2D search strategy. Novel communication boards based on optimised arrangements of frequency-ordered letter sequences potentially provide a faster means of communication than conventional communication boards.
