Brouwer degree, equivariant maps and tensor powers

Balanov, Zalman; Krawcewicz, Wiesław; Kushkuley, Alexander

doi:10.1155/s1085337598000621

Cited by 7 publications

(15 citation statements)

References 9 publications

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“…Moreover, a phase portrait to (1.2) is unique (up to OTE) in the case i = 1, 3, 4. This observation together with Propositions 5.3 and 5.4 justify the algebraic characterization of Figures (4), (8) and (15) in Table 5.1 (as well as the fact that Figures (11) and (13) can be realized for system (1.1)). To differ algebraically Figure (11) from Figure (13) observe, first, that according to [4], [9], F A 4 has two (homogeneous) roots both of multiplicity two only in the case of Figure (13).…”

Section: Type E(a) Esupporting

confidence: 72%

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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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Section: Type E(a) Esupporting

confidence: 72%

“…The problem of the existence of bounded solutions to (1.2) has attracted much attention for a long time (cf. [19], [27], [23], [8]). Definition 1.1.…”

mentioning

confidence: 99%

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

Balanov¹,

Krawcewicz²,

Zur³

2005

TMNA

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“…Earlier we investigated the classification of otopy classes in cases of gradient local maps in R n ( [8,9]), gradient local fields on manifolds ( [11]) and equivariant local maps on a representation of a compact Lie group ( [5,6]). It is worth pointing out that the ideas presented here were inspired by [1,2,3,4,18,21,22]. Moreover, our paper develops and clarifies the material contained in [7,17].…”

Section: Introductionsupporting

confidence: 52%

The Hopf type theorem for equivariant gradient local maps

Bartłomiejczyk

Nowak-Przygodzki

2017

J. Fixed Point Theory Appl.

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Abstract. We construct a degree-type otopy invariant for equivariant gradient local maps in the case of a real finite-dimensional orthogonal representation of a compact Lie group. We prove that the invariant establishes a bijection between the set of equivariant gradient otopy classes and the direct sum of countably many copies of Z.Mathematics Subject Classification. Primary 55P91; Secondary 54C35.

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“…In this section we generalize the construction of the degree for S 1 -equivariant potential operators due to Rybicki [38,40]; see also [4,22] for a homotopy-theoretic approach. We use the following notation, and refer to [1] for basic representation theory.…”

Section: Degree Theory For Equivariant Potential Operatorsmentioning

confidence: 99%

Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Bartsch¹,

Gebhard²

2016

Math. Ann.

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The paper deals with singular first order Hamiltonian systems of the formwhere J ∈ R 2×2 defines the standard symplectic structure in R 2 , and the Hamiltonian H is of N -vortex type:This is defined on the configuration space {(z 1 , . . . , z N ) ∈ Ω 2N : z j = z k for j = k} of N different points in the domain Ω ⊂ R 2 . The function F : Ω N → R may have additional singularities near the boundary of Ω N . We prove the existence of a global continuum of periodic solutions z(t) = (z 1 (t), . . . , z N (t)) ∈ Ω N that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium Z(t) ∈ R 2N of the N -vortex problem in the whole plane (where F = 0). Examples for Z include Thomson's vortex configurations, or equilateral triangle solutions. The domain Ω need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of 2π-periodic H 1/2 functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for S 1 -equivariant gradient maps that we adapt to this class of potential operators.MSC 2010: Primary: 37J45; Secondary: 37N10, 76B47

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Brouwer degree, equivariant maps and tensor powers

Abstract: Abstract. A construction of equivariant maps based on factorization through symmetric powers of a faithful representation is presented together with several examples of related equivariant maps. Applications to differential equations are also discussed.

Cited by 7 publications

References 9 publications

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

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

The Hopf type theorem for equivariant gradient local maps

Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

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