Single input single output system was studied. With proportion, differential, integral results of deviation between given input and output as controller input, the logic rules in control process was analyzed, these lo...Single input single output system was studied. With proportion, differential, integral results of deviation between given input and output as controller input, the logic rules in control process was analyzed, these logic rule with Pan-Boolean algebra was described, therefore a PID Pan-Boolean algebra control algorithm was obtained. The simulation results indicates that the new control algorithm is more effective compared to the traditional PID algorithm, having advantages such as more than 3 adjustable parameters of controllers, better result, and so on.展开更多
It is well known that Zadeh's fuzzy logic is not a Boolean algebra because it does not satisfy the Law of Excluded Middle or the Law of Contradiction, but the two-valued propositional logic does. On the other hand...It is well known that Zadeh's fuzzy logic is not a Boolean algebra because it does not satisfy the Law of Excluded Middle or the Law of Contradiction, but the two-valued propositional logic does. On the other hand, a recently proposed measure-based fuzzy logic(MBFL) satisfies all the axioms of Boolean algebra. In this paper, a complete and thorough proof is given for this.展开更多
In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we o...In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we obtain a new version of the Orlicz Pettis theorem,for Banach spaces.We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.展开更多
In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations an...In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].展开更多
To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation o...To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation of Boolean functions is proposed. The proposed method is used to construct Boolean functions with an odd number of variables and with maximum algebraic immunity. We also discuss the nonlinearity of the constructed functions. Moreover, a lower bound is determined for the number of Boolean functions with maximum algebraic immunity.展开更多
The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approach...The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.展开更多
Several nonmonotonic logic systems together with their algebraic semantics are discussed.NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to ...Several nonmonotonic logic systems together with their algebraic semantics are discussed.NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.展开更多
Boolean homomorphisms of a hypercube, which correspond to the morphisms in the category of finite Boolean algebras, coincide with the linear isometries of the category of finite binary metric vector spaces.
In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physica...In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physical entities and more associated with mental events, thinking shifts from analytical to conceptual-abstract. Fundamental topics taken from the abstract algebra (aka: modern algebra) are unquestionably abstract. Historically, fundamental concepts taught from the abstract algebra are detached from physical reality with one exception: Boolean operations. Even so, many abstract algebra texts present Boolean operations from a purely mathematical operator perspective that is detached from physical entities. Some texts on the abstract algebra introduce logic gate circuits, but treat them as perceptual symbols. For majors of pure or applied mathematics, detachments from physical entities is not relevant. For students of Computer and Electrical Engineering (CpE/EE), mental associations of Boolean operations are essential, and one might argue that studying pure Boolean axioms are unnecessary mental abstractions. But by its nature, the CpE/EE field tends to be more mentally abstract than the other engineering disciplines. The depth of the mathematical abstractions that we teach to upper-division CpE/EE majors is certainly up for questioning.展开更多
基金Project (J51801) supported by Shanghai Education Commission Key DisciplineProject(08ZY79)supported by Shanghai Education Commission Research FundProject(DZ207004)supported by Shanghai Second Polytechnic University Fund
文摘Single input single output system was studied. With proportion, differential, integral results of deviation between given input and output as controller input, the logic rules in control process was analyzed, these logic rule with Pan-Boolean algebra was described, therefore a PID Pan-Boolean algebra control algorithm was obtained. The simulation results indicates that the new control algorithm is more effective compared to the traditional PID algorithm, having advantages such as more than 3 adjustable parameters of controllers, better result, and so on.
文摘It is well known that Zadeh's fuzzy logic is not a Boolean algebra because it does not satisfy the Law of Excluded Middle or the Law of Contradiction, but the two-valued propositional logic does. On the other hand, a recently proposed measure-based fuzzy logic(MBFL) satisfies all the axioms of Boolean algebra. In this paper, a complete and thorough proof is given for this.
文摘In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we obtain a new version of the Orlicz Pettis theorem,for Banach spaces.We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.
基金Supported by a grant of National Natural Science Foundation of China(12001243,61976244,12171294,11961016)the Natural Science Basic Research Plan in Shaanxi Province of China(2020JQ-762,2021JQ-580)。
文摘In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].
基金This work was supported by the National Natural Science Foundation of China under Grants No. 61103191, No. 61070215 the Funds of Key Lab of Fujian Province University Network Security and Cryptology under Crant No. 2011003 and the Open Research Fund of State Key Laboratory of Inforrmtion Security.
文摘To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation of Boolean functions is proposed. The proposed method is used to construct Boolean functions with an odd number of variables and with maximum algebraic immunity. We also discuss the nonlinearity of the constructed functions. Moreover, a lower bound is determined for the number of Boolean functions with maximum algebraic immunity.
文摘The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.
文摘Several nonmonotonic logic systems together with their algebraic semantics are discussed.NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.
文摘Boolean homomorphisms of a hypercube, which correspond to the morphisms in the category of finite Boolean algebras, coincide with the linear isometries of the category of finite binary metric vector spaces.
文摘In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physical entities and more associated with mental events, thinking shifts from analytical to conceptual-abstract. Fundamental topics taken from the abstract algebra (aka: modern algebra) are unquestionably abstract. Historically, fundamental concepts taught from the abstract algebra are detached from physical reality with one exception: Boolean operations. Even so, many abstract algebra texts present Boolean operations from a purely mathematical operator perspective that is detached from physical entities. Some texts on the abstract algebra introduce logic gate circuits, but treat them as perceptual symbols. For majors of pure or applied mathematics, detachments from physical entities is not relevant. For students of Computer and Electrical Engineering (CpE/EE), mental associations of Boolean operations are essential, and one might argue that studying pure Boolean axioms are unnecessary mental abstractions. But by its nature, the CpE/EE field tends to be more mentally abstract than the other engineering disciplines. The depth of the mathematical abstractions that we teach to upper-division CpE/EE majors is certainly up for questioning.
基金Supported by the National Basic Research Program of China （No. 2007CB311201）, the National Natural Science Foundation of China （No.60833008 No.60803149）, and the Foundation of Guangxi Key Laboratory of Information and Communication （No.20902）.
