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51.
We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set‐representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non‐Boolean situations. On the other hand, a difference‐closed system can be constructed such that even two‐valued states do not allow for extensions. Finally, we consider these questions in a σ‐complete setup and find a large class of such systems with rather interesting state properties. 相似文献
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Shuang Liang Guodong Zhao Haitao Li Xueying Ding 《Mathematical Methods in the Applied Sciences》2019,42(7):2221-2230
This paper studies the structural monostability and structural cycle‐stability of Boolean networks (BNs). Firstly, the structural‐equivalent Boolean networks are converted to the algebraic forms by using the semitensor product of matrices. Secondly, the concepts of structural monostability and structural cycle‐stability for Boolean networks are proposed. On the basis of the algebraic forms of structural‐equivalent Boolean networks, some necessary and sufficient conditions are presented for the structural monostability and structural cycle‐stability of Boolean networks. Finally, an illustrative example is worked out to show the effectiveness of the obtained results. 相似文献
55.
For every simple graph G,a class of multiple clique cluster-whiskered graphs Geπm is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex IndGeπm is sequentially Cohen-Macaulay.The properties of the graphs Geπm and Gπ constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind Gπ and the calculation of Betti numbers of the cover ideal Ic(Geπ").We also prove that the complex △ =IndH is strongly shellable and pure for either a Boolean graph H =Bn or the full clique-whiskered graph H =Gw of G,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△) and the cover ideal Ic(H) have linear quotients. 相似文献
56.
This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved. 相似文献
57.
称Boole矩阵A是正规的,是指A的行秩与列秩相等.本文主要得到两个结果.第一,推广了J.Konieczny在SemigroupForum,vol.44(1992)发表的论文Oncardina-litiesofrowspaceofBooleanmatrices的主要结果.第二,若n阶Boole矩阵的行空间基数大于2 ̄(n-2)-2,则A必是正规的. 相似文献
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Sugata Gangopadhyay 《Discrete Mathematics》2006,306(14):1540-1556
In 1983, Patterson and Wiedemann constructed Boolean functions on n=15 input variables having nonlinearity strictly greater than 2n-1-2(n-1)/2. Construction of Boolean functions on odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions are known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995 and subsequently these functions can be described as repetitions of a particular binary string. As example we elaborate the cases for n=15,21. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating autocorrelation and generalized nonlinearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all GF(2)-linear transformations of GF(2ab) which acts on PG(2,2a). 相似文献
60.
Victor L. Selivanov 《Mathematical Logic Quarterly》2005,51(1):45-61
We establish some results on the Borel and difference hierarchies in φ‐spaces. Such spaces are the topological counterpart of the algebraic directed‐complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non‐collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω‐ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献