The theory of <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-rung orthopair fuzzy sets (<inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFSs) proposed by Yager effectively describes fuzzy information in the real world. Because <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFSs contain the parameter <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula> and can adjust the range of expressed fuzzy information, they are superior to both intuitionistic and Pythagorean fuzzy sets. Archimedean T-norm and T-conorm (ATT) is an important tool used to generate operational rules based on the <italic>q</italic>-rung orthopair fuzzy numbers (<inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFNs). In comparison, the Bonferroni mean (BM) operator has an advantage because it considers the interrelationships between the different attributes. Therefore, it is an important and meaningful innovation to extend the BM operator to the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFNs based upon the ATT. In this paper, we first discuss <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-rung orthopair fuzzy operational rules by using ATT. Furthermore, we extend BM operator to the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFNs and propose the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-rung orthopair fuzzy Archimedean BM <inline-formula><tex-math notation="LaTeX">$(q\hbox{-}{ROFABM})$</tex-math></inline-formula> operator and the <italic>q</italic>-rung orthopair fuzzy weighted Archimedean BM <inline-formula><tex-math notation="LaTeX">$(q\hbox{-}{ROFWABM})$</tex-math></inline-formula> operator and study their desirable properties. Then, a new multiple-attribute decision-making (MADM) method is developed based on <inline-formula><tex-math notation="LaTeX">$q\hbox{-}{ROFWABM}$</tex-math></inline-formula> operator. Finally, we use a practical example to verify effectiveness and superiority by comparing to other existing methods.

Extends results of math-ph/0407067

Extends results of math-ph/0407067

Extends results of math-ph/0407067

This paper has been superseded by math-ph/0102032, "Bures geometry of the three-level quantum systems. II".

In this part of the series five-dimensional tangent vectors are introduced first as equivalence classes of parametrized curves and then as differential-algebraic operators that act on scalar functions. I then examine their basic algebraic properties and their parallel transport in the particular case where space-time possesses a special local symmetry. After that I give definition to five-dimensional tangent vectors associated with dimensional curve parameters and show that they can be identified with the five-vectors introduced formally in part I (math-ph/9805004). In conclusion I speak about differential forms associated with five-vectors.

The goal of this note is to give an elementary and very short solution to equations of motion for the Kovalevskaya top. For this we use some results from original papers by Kovalevskay, Kötter and Weber and also the authors Lax representation (see math-ph/0111024)

In a previous paper (math-ph/0202002) an Euler angle parameterization for SU(4) was given. Here we present the derivation of a generalized Euler angle parameterization for SU(N). The formula for the calculation of the Haar measure for SU(N) as well as its relation to Marinov's volume formula for SU(N) will also be derived. As an example of this parameterization's usefulness, the density matrix parameterization and invariant volume element for a qubit/qutrit, three qubit and two three-state systems, also known as two qutrit systems, will also be given.

This paper shows how C-numerical-range related new strucures may arise from practical problems in quantum control--and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function |\tr \{C^\dagger UAU^\dagger\}|. In quantum control of n qubits one may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii) in restricting the dynamics to {\em local} operations on each qubit, i.e. to the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2). Interestingly, the latter then leads to a novel entity, the {\em local} C-numerical range W_{\rm loc}(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying paper (math-ph/0702005). We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. We conclude by relating the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient flow algorithms.

Let H be a self-adjoint operator such that exp(-aH) is of trace class for some a<1. Let V be a symmetric operator, Kato bounded relative to H. We show that log Tr[exp(-H+xV)] is a real analytic function of x in a hood of x=0. We show that the Gibbs states of H+xV form a real analytic Banach manifold. This work has been extended in math-ph/9910031.