This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and ... more This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction.
A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using ... more A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models).
ABSTRACT We study complex structures in real two-dimensional commutative algebras and show their ... more ABSTRACT We study complex structures in real two-dimensional commutative algebras and show their connection with homotopy properties of the multiplications. An application to the Riccati equation in rank three algebras is also discussed.
We construct polynomial dynamical systems x ′ = P ( x ) with symmetries present in the local phas... more We construct polynomial dynamical systems x ′ = P ( x ) with symmetries present in the local phase portrait. This point of view on symmetry yields the approaches to ODEs construction being amenable to classical methods of the Spectral Analysis.
Necessary and sufficient conditions are given for a quadratic polynomial to be a divisor of a non... more Necessary and sufficient conditions are given for a quadratic polynomial to be a divisor of a nonzero harmonic polynomial inR n.
Modern Methods in Operator Theory and Harmonic Analysis, 2019
In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively ... more In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively capture its topological and algebraic properties. This suggests to extend the linear spectral theory to non-linear operators by considering spectra of linearizations in small neighborhoods of the fixed points. In the present paper, we develop this approach for quadratic maps. Several standard concepts such as asymptotic laws for splitting/gluing zeros of polynomial maps) are considered from new (and, possibly, unexpected) angles.
Using the syzygy method, established in our earlier paper, we characterize the combinatorial stra... more Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point... more In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n-1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $\lambda=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.
In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point... more In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least n− 1 nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension n. We also discuss the exceptionality of the eigenvalue λ = 1 2 which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.
Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang... more Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprßigTrends Math, Birkhäuser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
In this paper, we present our recent results on the concepts of differential and integral equatio... more In this paper, we present our recent results on the concepts of differential and integral equations occurred in the non-associative algebras. We study also how topological and dynamical properties of a differential and integral equations occurred in the associated algebras can be described in the purely algebraic language.
Modern Methods in Operator Theory and Harmonic Analysis pp 199-216, 2019
In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively ... more In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively capture its topological and algebraic properties. This suggests to extend the linear spectral theory to non-linear operators by considering spectra of linearizations in small neighborhoods of the fixed points. In the present paper, we develop this approach for quadratic maps. Several standard concepts such as asymptotic laws for splitting/gluing zeros of polynomial maps) are considered from new (and, possibly, unexpected) angles.
We construct polynomial dynamical systems x = P(x) with symmetries present in the local phase por... more We construct polynomial dynamical systems x = P(x) with symmetries present in the local phase portrait. This point of view on symmetry yields the approaches to ODEs construction being amenable to classical methods of the Spectral Analysis.
This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and ... more This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction.
A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using ... more A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models).
ABSTRACT We study complex structures in real two-dimensional commutative algebras and show their ... more ABSTRACT We study complex structures in real two-dimensional commutative algebras and show their connection with homotopy properties of the multiplications. An application to the Riccati equation in rank three algebras is also discussed.
We construct polynomial dynamical systems x ′ = P ( x ) with symmetries present in the local phas... more We construct polynomial dynamical systems x ′ = P ( x ) with symmetries present in the local phase portrait. This point of view on symmetry yields the approaches to ODEs construction being amenable to classical methods of the Spectral Analysis.
Necessary and sufficient conditions are given for a quadratic polynomial to be a divisor of a non... more Necessary and sufficient conditions are given for a quadratic polynomial to be a divisor of a nonzero harmonic polynomial inR n.
Modern Methods in Operator Theory and Harmonic Analysis, 2019
In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively ... more In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively capture its topological and algebraic properties. This suggests to extend the linear spectral theory to non-linear operators by considering spectra of linearizations in small neighborhoods of the fixed points. In the present paper, we develop this approach for quadratic maps. Several standard concepts such as asymptotic laws for splitting/gluing zeros of polynomial maps) are considered from new (and, possibly, unexpected) angles.
Using the syzygy method, established in our earlier paper, we characterize the combinatorial stra... more Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point... more In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n-1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $\lambda=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.
In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point... more In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least n− 1 nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension n. We also discuss the exceptionality of the eigenvalue λ = 1 2 which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.
Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang... more Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprßigTrends Math, Birkhäuser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
In this paper, we present our recent results on the concepts of differential and integral equatio... more In this paper, we present our recent results on the concepts of differential and integral equations occurred in the non-associative algebras. We study also how topological and dynamical properties of a differential and integral equations occurred in the associated algebras can be described in the purely algebraic language.
Modern Methods in Operator Theory and Harmonic Analysis pp 199-216, 2019
In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively ... more In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively capture its topological and algebraic properties. This suggests to extend the linear spectral theory to non-linear operators by considering spectra of linearizations in small neighborhoods of the fixed points. In the present paper, we develop this approach for quadratic maps. Several standard concepts such as asymptotic laws for splitting/gluing zeros of polynomial maps) are considered from new (and, possibly, unexpected) angles.
We construct polynomial dynamical systems x = P(x) with symmetries present in the local phase por... more We construct polynomial dynamical systems x = P(x) with symmetries present in the local phase portrait. This point of view on symmetry yields the approaches to ODEs construction being amenable to classical methods of the Spectral Analysis.
Modern Methods in Operator Theory and Harmonic Analysis pp 199-216, 2019
In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively ... more In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively capture its topological and algebraic properties. This suggests to extend the linear spectral theory to non-linear operators by considering spectra of linearizations in small neighborhoods of the fixed points. In the present paper, we develop this approach for quadratic maps. Several standard concepts such as asymptotic laws for splitting/gluing zeros of polynomial maps) are considered from new (and, possibly, unexpected) angles.
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Papers by Yakov Krasnov