The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ),where s is a bayessian state on an orthomodular lattice L.Having introduced the notion of entropy hs(φ,)of p...The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ),where s is a bayessian state on an orthomodular lattice L.Having introduced the notion of entropy hs(φ,)of partition of a Boolean algebra B with respect to a state s and a state preserving homomorphism φ,we prove afew results on that,define the entropy of a dynamical system h_s(Φ),and show its invarianee.The concept of sufficientfamilies is also given and we establish that hs(Φ) comes out to be equal to the supremum of h_s(φ,),where variesover any sufficient family.The present theory has then been extended to the quantum dynamical system ( L,s,φ),whichas an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system(B,s_o,φ),where B is a Boolean algebra and so is a state on B.展开更多
There are two kinds of isometric isomorphism in probabilistic metric space theory. The first is that a PM space (E, F) is isometrically isomorphic to another PM space (E', F'), and the second is that a PM spac...There are two kinds of isometric isomorphism in probabilistic metric space theory. The first is that a PM space (E, F) is isometrically isomorphic to another PM space (E', F'), and the second is that a PM space (E, F) is isometrically isomorphic to a generating space of quasi-metric family (E', d(r), r is an element of (0, 1)). This paper establishes the connection between the two kinds of isometric isomorphism.展开更多
In two previous papers, we explained the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes and we describe some crystal families. This paper main...In two previous papers, we explained the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes and we describe some crystal families. This paper mainly consists in the study of three crystal families of space E5, the (di-iso hexagons)-al, the hypercube 5 dim and the (hypercube 4 dim)-al crystal families. For each studied family, we explain their name, we describe their cell and we list their point groups which are classified into isomorphism classes. Then we give a WPV symbol to each group. (WPV means Weigel Phan Veysseyre). Our method is based on the description of the cell of the holohedry of each crystal family and of the results given by the Software established by one of us. The advantage to classify the point groups in isomorphism classes is to give their mathematical structures and to compare their WPV symbols. So the study of all crystal families of space E5 is completed. Some crystal families of space E5 can be used to describe di incommensurate structures and quasi crystals.展开更多
文摘The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ),where s is a bayessian state on an orthomodular lattice L.Having introduced the notion of entropy hs(φ,)of partition of a Boolean algebra B with respect to a state s and a state preserving homomorphism φ,we prove afew results on that,define the entropy of a dynamical system h_s(Φ),and show its invarianee.The concept of sufficientfamilies is also given and we establish that hs(Φ) comes out to be equal to the supremum of h_s(φ,),where variesover any sufficient family.The present theory has then been extended to the quantum dynamical system ( L,s,φ),whichas an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system(B,s_o,φ),where B is a Boolean algebra and so is a state on B.
文摘There are two kinds of isometric isomorphism in probabilistic metric space theory. The first is that a PM space (E, F) is isometrically isomorphic to another PM space (E', F'), and the second is that a PM space (E, F) is isometrically isomorphic to a generating space of quasi-metric family (E', d(r), r is an element of (0, 1)). This paper establishes the connection between the two kinds of isometric isomorphism.
文摘In two previous papers, we explained the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes and we describe some crystal families. This paper mainly consists in the study of three crystal families of space E5, the (di-iso hexagons)-al, the hypercube 5 dim and the (hypercube 4 dim)-al crystal families. For each studied family, we explain their name, we describe their cell and we list their point groups which are classified into isomorphism classes. Then we give a WPV symbol to each group. (WPV means Weigel Phan Veysseyre). Our method is based on the description of the cell of the holohedry of each crystal family and of the results given by the Software established by one of us. The advantage to classify the point groups in isomorphism classes is to give their mathematical structures and to compare their WPV symbols. So the study of all crystal families of space E5 is completed. Some crystal families of space E5 can be used to describe di incommensurate structures and quasi crystals.
