In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the common properties of the semi-and indefinite-inner-products and define the semi-indefinite-inner-product and the corresponding structure, the semi-indefinite-inn...

Log-Brunn-Minkowski inequality was conjectured by Boröczky, Lutwak, Yang and Zhang [7], and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors [27], that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequal...

The pseudo-additive relation that the Tsallis entropy satisfies has nothing whatsoever to do with the super-and sub-additivity properties of the entropy. The latter properties, like concavity and convexity, are couched in geometric inequalities and cannot be reduced to equalities. Rather, the pseudo-additivity relation is a functional equation that determines the functional forms of the random ...

An effective Minkowski spacetime having no deformation in formal is established by introducing a well-defined proper time from the κ-deformed Minkowski spacetime related to the standard basis. The κ-Minkowski spacetime connects with a special curved spacetime through this effective spacetime. In the framework of such an effective Minkowski spacetime as a mapped coordinate system of the κ-deform...

In this paper the connection between quantum field theories on flat noncommutative space(-times) in Euclidean and Lorentzian signature is studied for the case that time is still commutative. By making use of the algebraic framework of quantum field theory and an analytic continuation of the symmetry groups which are compatible with the structure of Moyal space, a general correspondence between ...

The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-d...

– We present a one-dimensional version of the functional form of the geometric Brunn-Minkowski inequality in free (noncommutative) probability theory. The proof relies on matrix approximation as used recently by P. Biane and F. Hiai, D. Petz and Y. Ueda to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered ...

We prove an Alexander type theorem for the spectral unit ball Ωn showing that there are no non-trivial proper holomorphic mappings in Ωn, n ≥ 2. Let Mn denote the space of n× n complex matrices. In order to avoid some trivialities and ambiguities we assume in the whole paper that n ≥ 2. Let ρ(A) := max{|λ| : λ ∈ Spec(A)} be the spectral radius of A ∈ Mn. Denote also by Spec(A) := {λ ∈ C : det(A...