We present the simple and direct proof of the determinantal formula for the scalar product of Bethe eigenstate with an arbitrary dual state. We briefly review the direct calculation of the general scalar product with the help of the factorizing operator and the construction of the factorizing operator itself. We also comment on the previous determination of the scalar product of Bethe eigenstat...

Abstract. This paper will introduce noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product ...

6 Vectors 1 6.1 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 2 6.2 The Parallelogram Law of Vector Addition . . . . . . . . . . . 4 6.3 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 5 6.4 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 6 6.5 Linear Combinations of Vectors . . . . . . . . . . . . . . . . . 6 6.6 Linear Dependence and ...

In the fuzzy arithmetic, the definitions of addition and multiplication of fuzzy numbers are based on Zadeh’s extension principle. From theoretical and practical points of view, this multiplication of fuzzy numbers owns several unnatural properties. Recently, to avoid this shortcoming, a new multiplicative operation of product type is introduced, the so-called cross-product of fuzzy numbers. Th...

The state space of a finite quantum system is identified with the set of positive semidefinite matrices of trace 1. The set of all strictly positive definite matrices of trace 1 becomes naturally a differentiable manifold and the Bogoliubov–Kubo– Mori scalar product defines a Riemannian structure on it. Reference 4 tells about the relation of this metric to the von Neumann entropy functional. I...

Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decompositi...