The first widely accepted proof of the Fundamental Theorem of Algebra was published by Gauß in 1799 in his Ph.D. thesis, although to current standards this proof has gaps. Argand gave a proof (with only small gaps) in 1814 which was based on a flawed proof of d’Alembert of 1746. Many more proofs followed, including three more proofs by Gauß. For a more about the history of the Fundamental Theor...

We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional vector space admits eigenvectors. The argument avoids the use of the fundamental theorem of algebra, which can then be deduced from it. Our presentation avoids any appeal to the theory of determinants. 2010 Mathematics Subject Classification. 15-01, 15A18.

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

Peterson’s Intermediate Syllogisms, generalizing Aristotelian syllogisms by intermediate quantifiers ‘Many’, ‘Most’ and ‘Almost all’, are studied. It is demonstrated that, by associating certain values V, W and U on standard Lukasiewicz MV–algebra with the first and second premise and the conclusion, respectively, the validity of a corresponding intermediate syllogism is determined by a simple ...

In this paper we generalize the max plus algebra system of real matrices to the class of real tensors and derive its fundamental properties. Also we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverses of tensors is characterized.

Quantum computational logics provide a fertile common ground for a uni ed treatment of vagueness and uncertainty. In this paper we describe an approach to the logic of quantum computation that has been recently taken up and developed by the present authors. Special attention will be devoted to a generalisation of Chang's MV algebras (called quasi-MV algebra) which abstracts over the algebra who...

In this paper we study an outer measure on MV -algebras. In Section 1 the definition of MV -algebra and Mundici theorem are remind. In Section 2 there is defined an outer measure, measurable elements and there is proved Choquet lema for this structure. In conclusion some properties of measurable elements are resumed.

The focus of this paper is the class of perfect GMV-algebras, which includes all non-commutative analogues of perfect MV-algebras. As in the commutative case, we show that each perfect GMV-algebra possesses a single negation, it is generated by its infinitesimal elements, and can be realized as an interval in a lexicographical product of the lattice-ordered group of integers and an arbitrary la...

We present a preliminary study of applying the concepts of algebraic geometry over fields to the theory of MV-algebras. We proceed along lines similar to B. Plotkin and others where an algebraic geometry over groups is developed.

Here, we report a spontaneously formed nanolamellar BaTiO3–CoFe2O4 bicrystal. 11̄0 interfaces join the BaTiO3 and CoFe2O4 single crystalline periodically arranged lamellae that have a common 111 direction. The superlattice of approximately 2 nm wavelength is magnetoelectric with a frequency dependent coupling coefficient of 20 mV/Oe cm at 100 Hz. The BaTiO3 component is a ferroelectric relaxor w...