Bipolar Quantum Logic Gates and Quantum Cellular Combinatorics—A Logical Extension 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.

Formal Derivation of the Combinatorics Problems with PAR Method

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.

Combinatorics on Words in Symbolic Dynamics:The Quadratic Map

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.

Combinatorics on Words in Symbolic Dynamics:the Antisymmetric Cubic Map

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.

The 1-Good-neighbor Connectivity and Diagnosability of Locally Twisted Cubes

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 nodeto contain at least g fault-free neighbors. The locally twisted cube LTQn has many goodproperties. In this paper, we show that the 1-good-neighbor connectivity al （LTQn） =2n-2and the 1-good-neighbor diagnosability of LTQn is 2n - 1 under the PMC model for n ≥ 4and the MM model for n ≥ 5.

Introducing “Arithmetic Calculus” with Some Applications: New Terms, Definitions, Notations and Operators

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.

The Tightly Super 3-Extra Connectivity and Diagnosability of Locally Twisted Cubes

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.

Some Properties of the g-Good-Neighbor (g-Extra) Diagnosability of a Multiprocessor System

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* model.

Combinator演算族的π演算语义

以SKI演算作为Combinator演算族的代表,通过形式化的手段给出了SKI演算的π演算语义;通过一个实例验证了所论方法的正确性.所给出的转换方法证明了π演算的表达能力:π演算为图灵完备的.由于高阶函数式语言与Combinator演算族之间存在着自然的转换,所给的转换思想不仅为在π演算的理论框架下研究Combinator演算族提供了基础,也为探讨高阶函数式语言的表示和实现问题提供了新途径.

DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE

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.

Root-Patterns to Algebrising Partitions

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.

k割宽临界树的构造

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.

Enumeration of Strength Three Orthogonal Arrays and Their Implementation in Parameter Design

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.

Evaluation and Design of Alphabetic Communication Boards

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.

THE STRUCTURE OF INVERTIBLE SUBSTITUTIONS ON A THREE-LETTER ALPHABET

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.

Generalized Krein Parameters of a Strongly Regular Graph

We consider the real three-dimensional Euclidean Jordan algebra associated to a strongly regular graph. Then, the Krein parameters of a strongly regular graph are generalized and some generalized Krein admissibility conditions are deduced. Furthermore, we establish some relations between the classical Krein parameters and the generalized Krein parameters.

趣谈(上)Functional Programming

本文并非是一个系统性的 FP 教程,仅仅是揭示了 FP 背后的一些数学道理,介绍与之相关的一些有趣话题,我相信您一定能从文章中有所收获。本文的例子主要使用 Python 解说。不过即使您不懂 Python,稍微耐心一些也可以看懂。

Decay of Correlations for Fibonacci Unimodal Interval Maps

We consider a class of generalized Fibonacci unimodal maps for which the central return times {Sn} satisfy that sn= sn-1 ＋ ksh-2 for some k≥ 1. We show that such a unimodal map admits a unique absolutely continuous invariant probability with exactly stretched exponential decay of correlations if its critical order lies in （1, k ＋ 1）.

Quantitative Stability of the Brunn-Minkowski Inequality for Sets of Equal Volume(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

The authors prove a quantitative stability result for the Brunn-Minkowski inequality on sets of equal volume: If |A| = |B| > 0 and |A + B|^(1/n) =(2 + δ)|A|^(1/n) for some small δ, then, up to a translation, both A and B are close(in terms of δ) to a convex set K.Although this result was already proved by the authors in a previous paper, the present paper provides a more elementary proof that the authors believe has its own interest. Also,the result here provides a stronger estimate for the stability exponent than the previous result of the authors.

Transitivity of varietal hypercube networks

The varietal hypercube VQn is a variant of the hypercube Qn and has better properties than Qn with the same number of edges and vertices. This paper proves that VQn is vertex-transitive. This property shows that when VQn is used to model an interconnection network, it is high symmetrical and obviously superior to other variants of the hypercube such as the crossed cube.