Two new constructions of approximately mutually unbiased bases

2018 ◽  
Vol 16 (04) ◽  
pp. 1850038
Author(s):  
Gang Wang ◽  
Min-Yao Niu ◽  
Fang-Wei Fu
Keyword(s):  
Finite Fields ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Character Sum ◽  
Mutually Unbiased Bases ◽  
Orthonormal Bases ◽  
Extremal Sets

Two orthonormal bases [Formula: see text] and [Formula: see text] of a [Formula: see text]-dimensional complex inner-product space [Formula: see text] are called mutually unbiased bases (MUBs) if and only if [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. The size of any set containing pairwise MUBs of [Formula: see text] cannot exceed [Formula: see text]. If [Formula: see text] is a power of a prime, then extremal sets containing [Formula: see text] MUBs are known to exist, which are called the complete MUBs of [Formula: see text]. We have not known whether there exist complete MUBs when [Formula: see text] is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of [Formula: see text] are provided based on the mixed character sum of two special kinds of functions over finite fields.

A Type of Orthonormal Bases on 2-*-Inner Product Spaces

2020 ◽  
Vol 57 (4) ◽  
pp. 541-551
Author(s):  
Behrooz Mohebbi Najmabadi ◽  
Tayebe Lal Shateri ◽  
Ghadir Sadeghi
Keyword(s):  
Product Space ◽  
Orthonormal Basis ◽  
Dense Subset ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Orthonormal Bases ◽  
Inner Product Spaces ◽  
Bessel Inequality

In this paper, we define an orthonormal basis for 2-*-inner product space and obtain some useful results. Moreover, we introduce a 2-norm on a dense subset of a 2-*-inner product space. Finally, we obtain a version of the Selberg, Buzano’s and Bessel inequality and its results in an A-2-inner product space.

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

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