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The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object's symmetries. In this paper, we introduce new concepts in an algebraic structure called BCI-algebra, where we present the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. The relationship between bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy ideals is investigated, and various conditions are provided for a bipolar fuzzy ideal to be a bipolar fuzzy BCI-positive implicative ideal. Furthermore, conditions are presented for a bipolar fuzzy (closed) ideal to be a bipolar fuzzy BCI-commutative ideal. [ABSTRACT FROM AUTHOR]

Using the algebraic state space representation, this paper studies the robust consensus of networked evolutionary games (NEGs) with attackers and forbidden profiles. Firstly, an algebraic form is established for NEGs with attackers and forbidden profiles. Secondly, based on the algebraic form, a necessary and sufficient condition is presented for the robust constrained reachability of NEGs. Thirdly, a series of robust reachable sets is constructed by using the robust constrained reachability, based on which a constructive procedure is proposed to design state feedback controls for the robust consensus of NEGs with attackers and forbidden profiles. Finally, an illustrative example is given to show that the main results are effective. [ABSTRACT FROM AUTHOR]

It has been shown in previous papers that classes of (minimal asymmetric) informationally-complete positive operator valued measures (IC-POVMs) in dimension d can be built using the multiparticle Pauli group acting on appropriate fiducial states. The latter states may also be derived starting from the Poincaré upper half-plane model H. To do this, one translates the congruence (or non-congruence) subgroups of index d of the modular group into groups of permutation gates, some of the eigenstates of which are the sought fiducials. The structure of some IC-POVMs is found to be intimately related to the Kochen-Specker theorem. [ABSTRACT FROM AUTHOR]

In this paper, we bring together two concepts related to uncertainty and vagueness: fuzzy numbers and intervals. With them, we build a new structure whose elements we call interval fuzzy segments. We have undertaken this based on the conviction that the fuzzy numbers are a correct representation of the real numbers under situations of indeterminacy. We also believe that if it makes sense to consider the set of real numbers between two real bounds, then it also makes sense to consider the set of all the fuzzy numbers between two fuzzy number bounds. In this way, we extend the concept of real interval to the concept of interval fuzzy segment defined by two fuzzy bounds and a transition mapping that leads from the lower fuzzy bound to the upper fuzzy bound and this transition mapping generates the set of all the fuzzy numbers comprised between those fuzzy bounds. At the same time, this transition mapping brings the concept of interval fuzzy segment closer to the concept of line segment. [ABSTRACT FROM AUTHOR]