Complex Structures in Real Algebras. I. Two-Dimensional Commutative Case

Balanov, Zalman; Krasnov, Yakov

doi:10.1081/agb-120022810

Cited by 11 publications

(20 citation statements)

References 12 publications

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“…Here we will restrict ourselves with simple formulations that do not require any technical preparations. Theorem 2.6 (see [4]). The following statements hold.…”

Section: Two-dimensional Resultsmentioning

confidence: 95%

“…Observe that in [4], the reader can nd a detailed study of complex structures, squares roots and isotopies in algebras with zero divisors and/or 2-nilpotents. Here we will restrict ourselves with simple formulations that do not require any technical preparations.…”

Section: Two-dimensional Resultsmentioning

confidence: 99%

“…In this section, we will discuss the following two problems: For the background related to the interplay between ODEs and non-associative algebras as well as for many interesting results on bounded and periodic solutions to the Riccati equations, we refer to [25] (see also [4,6,15,20,26]). …”

Section: Admits a (Non-trivial) Periodic Solution If And Only If The mentioning

confidence: 99%

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Complex structures in algebra, topology and differential equations

Balanov

Krasnov

2014

Georgian Mathematical Journal

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In this paper, we present our recent results on the so-called complex structures in algebras in the context relevant to the following four problems which, at rst glance, have nothing in common: (i) solubility of polynomial equations in non-associative algebras, (ii) topological/geometric properties of quadratic maps, (iii) existence of bounded solutions to quadratic di erential systems, and (iv) ellipticity of the Dirac equation. The unsolved problems related to (i)-(iv) are formulated as well.

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“…Here we will restrict ourselves with simple formulations that do not require any technical preparations. Theorem 2.6 (see [4]). The following statements hold.…”

Section: Two-dimensional Resultsmentioning

confidence: 95%

Section: Two-dimensional Resultsmentioning

confidence: 99%

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Complex structures in algebra, topology and differential equations

Balanov

Krasnov

2014

Georgian Mathematical Journal

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“…Therefore, the topological classification of cubic systems in A (together with the one of systems of the formẋ = x 2 (t) * x 2 (t)) should give (i) a better understanding the hierarchy of the quadratic system equivalences mentioned above as well as (ii) a reasonable classification of real commutative two-dimensional algebras (cf. [31], [6]). This stream of ideas and applications goes beyond the scope of the present paper.…”

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confidence: 99%

On orbital topological equivalence of cubic ODEs in two-dimensional algebras

Balanov¹,

Krawcewicz²,

Zur³

2005

TMNA

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Cubic differential systems in real commutative two-dimensional algebras are classified up to orbital topological equivalence via the solubility of polynomial equations in algebras. As a by-product, existence of bounded solutions in such systems is studied via complex structures in the algebras. Application to the existence of periodic solutions to n-dimensional differential systems "cubic at infinity" is given.

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“…Since the original paper of Albert, a wide amount of papers have appeared in the literature that deal with isotopisms of distinct types of algebras as division [2,3,4,5], Jordan [6,7,8], alternative [9,10], absolute valued [11,12], structural [13] and real two-dimensional commutative [14] algebras. Isotopism of Lie algebras were already considered by Albert himself [1] and, shortly after, in 1944, by Bruck [15].…”

Section: Introductionmentioning

confidence: 99%

An approach to the isotheory by means of extended pseudoisotopisms

Ganfornina

Núñez

2016

AIP Conference Proceedings

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A perturbed-mean-field approach to the decay rates of excited vibrational states in extended systems: An application to I 2(Ne) n J. Chem. Phys. 100, 4355 (1994) Abstract. Based on the traditional concept of isotopism, extended isotopisms were introduced by the authors in 2006 in order to provide a fundamental basis to the isotheory of Santilli. Since that first attempt, distinct studies on extended isotopisms have focused on the construction of partial Latin squares having a Santilli autotopism in their autotopism group. In order to deal with new structures, we introduce in this paper the concept of extended pseudoisotopism. This is based on the use of onto linear transformations that are not necessarily injective. Some examples are exposed throughout the paper.

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Complex Structures in Real Algebras. I. Two-Dimensional Commutative Case

Cited by 11 publications

References 12 publications

Complex structures in algebra, topology and differential equations

Complex structures in algebra, topology and differential equations

On orbital topological equivalence of cubic ODEs in two-dimensional algebras

An approach to the isotheory by means of extended pseudoisotopisms

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