Differential Equations in Algebras

Krasnov, Yakov

doi:10.1007/978-3-7643-9893-4_12

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…We emphasize that, to the best of our knowledge, Theorems 1.1 and 1.2 (see also Theorems 3.1 and 4.5) have not appeared before (at least, in the formulations presented). In fact, these theorems are intimately connected to idempotents and nilpotents in non-associative algebras underlying multi-linear vector fields (see [2]) and related Peirce numbers (see [9]). Also (cf.…”

Section: Discussionmentioning

confidence: 99%

“…[9], where the case m = 2 was considered). Since (1.6) is a quadratic system in C, it admits the explicit integration…”

Section: )mentioning

confidence: 99%

“…It turns out that the numbers x i , y i and γ i appearing in Theorem 1.1 cannot take arbitrary values (cf. [9], where the case m = 2 was considered). Since (1.6) is a quadratic system in C, it admits the explicit integration…”

Section: )mentioning

confidence: 99%

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Projective dynamics of homogeneous systems: local invariants, syzygies and the Global Residue Theorem

Balanov

Kononovich

Krasnov

2012

Proceedings of the Edinburgh Mathematical Society

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AbstractWe give an explicit formula for the projective dynamics of planar homogeneous polynomial differential systems in terms of natural local invariants and we establish explicit algebraic connections (syzygies) between these invariants (leading to restrictions on possible global dynamics). We discuss multidimensional generalizations together with applications to the existence of first integrals and bounded solutions.

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Section: Discussionmentioning

confidence: 99%

“…[9], where the case m = 2 was considered). Since (1.6) is a quadratic system in C, it admits the explicit integration…”

Section: )mentioning

confidence: 99%

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Projective dynamics of homogeneous systems: local invariants, syzygies and the Global Residue Theorem

Balanov

Kononovich

Krasnov

2012

Proceedings of the Edinburgh Mathematical Society

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“…• The second one (Cauchy-Riemann approach) is focused on solutions to the Dirac equation in the algebra (cf. [17,18]). • The third one is based on the function-theoretic properties known for complex analytic functions, such as Cauchy's theorem, residue theory, Cauchy integral formula, etc.…”

Section: Algebraic Approach To Function Theoriesmentioning

confidence: 99%

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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“…By using a similar technique, we study the biwave equation in Section 4.3. An interesting solution of the three‐dimensional Laplace equation has been elaborated in by defining a related commutative and associative algebra over the field of complex numbers, and other related developments can be consulted in . In this work, we generalize these ideas.…”

Section: Introductionmentioning

confidence: 99%