Research on the Properties of Lattice-Valued Finite State Machines in Computer Technology

In this paper, we discuss some algebraic properties of Lattice valued finite state machine and prove that if there are homomorphic mapping satisfying certain conditions between two Lattice valued finite state machines, the first one is strongly connected (cycle), then then the second one is the same. And if the homomorphism is strongly homomorphic, one of the Lattice valued finite state machines is complete if and only if another Lattice valued finite state machine is complete. Discuss the completeness, strong connectivity, circulation and exchange capacity between the product of a Lattice valued finite state machine and the original Lattice valued finite state machine and get some results.

A New Globally Exponential Stability Criterion for Neural Networks with Discrete and Distributed Delays

This paper concerns the problem of the globally exponential stability of neural networks with discrete and distributed delays. A novel criterion for the globally exponential stability of neural networks is derived by employing the Lyapunov stability theory, homomorphic mapping theory, and matrix theory. The proposed result improves the previously reported global stability results. Finally, two illustrative numerical examples are given to show the effectiveness of our results.

FROM DESCRIPTIVISM TO CONSTRUCTIVISM ‐ A CHALLENGE TO SYMBOLIC MODELING. II

Classical modeling is not isomorphic, on the contrary the “objects of reality” or the like is the source of a homomorphic mapping performed to produce the model – just the way a piece of landscape is portrayed by its map with some bewildering details left out. We are thus taught that this process of modeling (or abstraction) is a plain mapping procedure –we call this descriptivism or representationalism. The prevailing object‐oriented modeling approach – or realistic approach – has some serious shortcomings due to the negligence of some aspects of the observer function, which for instance has resulted in a “world definition” made from “outside” the living consciousness (realism or materialism). By reversing this picture instead taking off from the impressions arisen within the subject's (the observer/knower's) conscious experience – the subject‐oriented approach –and ask how a living consciousness organizes itself to handle the task of living, we gain new insights in the process of conceptualization and learning. We learn that the dualistic world view is superfluous and should better be replaced by a neutral monistic approach, where the hypothetical existence of an independent outside reality (realism) can be substituted by the idea of a reality constructed from inside a living consciousness –nothing else but a model whose main purpose is to guide human anticipation and facilitate communication. Taking that stances the main tasks of human consciousness just become modeling – creating the outside model reality and the inside domain of feelings. In such a framework also the classical truth – in the sense of a God‐given modeling truth become meaningless – and must be substituted by the Pierce‐ain pragmatical or consensual truth. In the subject‐oriented approach states, properties etc. are not given any observer independent existence. On the contrary they emerge at the moment of their measurement as advocated by the Copenhagen interpretation. Bell's theorem also states: Given the quantum mechanics, either the idea of Einstein locality or the idea of an observer independent reality must be abandoned. The subject‐oriented approach clearly abandons the idea of a pre‐given observer independent reality – in favor of a cognitive agent created private reality, which then become the base for defining an “objective reality” in the form of a consensual scientific agreement.

FROM DESCRIPTIVISM TO CONSTRUCTIVISM ‐ A CHALLENGE TO SYMBOLIC MODELING. I

Classical modeling is not isomorphic, on the contrary the “objects of reality” or the like is the source of a homomorphic mapping performed to produce the model ‐ just the way a piece of landscape is portrayed by its map with some bewildering details left out. We are thus taught that this process of modeling (or abstraction) is a plain mapping procedure ‐ we call this descriptivism or representationalism. The prevailing object‐oriented modeling approach ‐ or realistic approach ‐ has some serious shortcomings due to the negligence of some aspects of the observer function, which for instance has resulted in a “world definition” made from “outside” the living consciousness (realism or materialism). By reversing this picture instead taking off from the impressions arisen within the subject's (the observer/knower's) conscious experience ‐ the subject‐oriented approach ‐ and ask how a living consciousness organize s itself to handle the task of living, we gain new insights in the process of conceptualization and learning. We learn that the dualistic worldview is superfluous and should better be replaced by a neutral monistic approach, where the hypothetical existence of an independent outside reality (realism) can be substituted by the idea of a reality constructed from inside a living consciousness ‐nothing else but a model whose main purpose is to guide human anticipation and facilitate communication. Taking that stances the main tasks of human consciousness just become modeling ‐ creating the outside model reality and th e inside domain of feelings. In such a framework also the classical truth ‐ in the sense of a God‐given modeling truth become meaningless ‐ and must be substituted by the Pierce‐ain pragmatical or consensual truth. In the subject‐oriented approach states, properties etc. are not given any observer independent existence. On the contrary they emerge at the moment of their measurement as advocated by the Copenhagen interpretation. Bell's theorem also states: Given the quantum mechanics, either the idea of Einstein locality or the idea of an observer independent reality must be abandoned. The subject‐oriented approach clearly abandons the idea of a pre‐given observer independent reality ‐ in favor of a cognitive agent created private reality, which then become the base for defining an “objective reality” in the form of a consensual scientific agreement.

On Commutative Continuation of Partial Endomorphisms of Groups

Given a homomorphic mapping θ of a subgroup A of a group G onto another subgroup B of G, necessary and sufficient conditions for the existence of a supergroup G* of G and an endomorphism θ* of G* such that θ* coincides with θ on A were derived by B. H. Neumann and Hanna Neumann (3). The homomorphism θ is called a partial endomorphism of G and θ* is said to continue, or extend, θ. Necessary and sufficient conditions for the simultaneous continuation of two partial endomorphisms of a group G to total endomorphisms of one supergroup G* ⊇ G were derived by the author (2).

Embedding Theorems for Groups

By a partial endomorphism of a group G we mean a homomorphic mapping μ of a subgroup A of G onto a subgroup B of G. If μ is denned on the whole of G then it is called a total endomorphism. We call a partial endomorphism totally extendable (or extendable) if there exists a supergroup G*⊇G with a total endomorphism μ* which extends μ in the sense that gμ* = gμ, whenever the right-hand side is defined (3).

An embedding theorem for groups

Let G by any given group. A homomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, where A and B need not be distinct, is called a partial endomorphism of G. When μ is defined on the whole of G, that is when A = G, we call μ a total endomorphism of G; or simply an endomorphism of G.A partial (or total) endomorphism μ* of a supergroup G* of G is said to extend (or continue) μ if μ* is defined on a supergroup A* of A, that is, μ* is defined for at least the elements for which μ. is defined, and moreover μ* coincides with μ on A.

Simultaneous Extension of Partial Endomorphisms of Groups

Let μ be a homomorphic mapping of some subgroup A of the group G onto a subgroup Ḃ (not necessarily distinct from A) of G; then we call μ a partial endomorphism of G. If A coincides with G, that is, if the homomorphism is defined on the whole of G, we speak of a total endomorphism; this is what is usually called an endomorphism of G. A partial (or total) endomorphism μ*extends or continues a partial endomorphism μ if the domain of μ* contains the domain of μ, that is, μ* is defined for (at least) all those elements for which μ. is defined, and moreover μ* coincides with μ where μ is defined.

The Homomorphic Mapping of Certain Matric Algebras onto Rings of Diagonal Matrices

The problem of determining the conditions under which a finite set of matrices A1A2, … , Ak has the property that their characteristic roots λ1j, λ2j, … , λki (j = 1, 2, …, n) may be so ordered that every polynomial f(A1A2 … , Ak) in these matrices has characteristic roots f(λ1j, λ2j …,λki) (j = 1, 2, … , n) was first considered by Frobenius [4]. He showed that a sufficient condition for the (Ai〉 to have this property is that they be commutative. It may be shown by an example that this condition is not necessary.J. Williamson [9] considered this problem for two matrices under the restriction that one of them be non-derogatory. He then showed that a necessary and sufficient condition that these two matrices have the above property is that they satisfy a certain finite set of matric equations.