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Jean Mawhin - One of the best experts on this subject based on the ideXlab platform.

  • Existence and multiplicity results for nonlinear second order difference equations with Dirichlet boundary conditions
    2020
    Co-Authors: Cristian Bereanu, Jean Mawhin
    Abstract:

    We use Brouwer Degree to prove existence and multiplicity results for the solutions of some nonlinear second order difference equations with Dirichlet boundary conditions. We obtain in particular upper and lower solutions theorems, Ambrosetti-Prodi type results, and sharp existence conditions for nonlinearities which are bounded from below or from above.

  • Bolzano’s theorems for holomorphic mappings
    Chinese Annals of Mathematics Series B, 2017
    Co-Authors: Jean Mawhin
    Abstract:

    The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano’s ones for real functions on an interval is deduced in a very simple way from Cauchy’s theorem for holomorphic functions. A more complicated proof, using Cauchy’s argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from ℂ^ n to ℂ^ n are obtained using Brouwer Degree.

  • reduction and continuation theorems for Brouwer Degree and applications to nonlinear difference equations
    Opuscula Mathematica, 2008
    Co-Authors: Jean Mawhin
    Abstract:

    The aim of this note is to describe the continuation theorem of [J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence Degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636, J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Reg. Conf. in Math., No 40, American Math. Soc., Providence, RI, 1979] directly in the context of Brouwer Degree, providing in this way a simple frame for multiple applications to nonlinear difference equations, and to show how the corresponding reduction property can be seen as an extension of the well-known reduction formula of Leray and Schauder [J. Leray, J. Schauder, Topologie et equations fonctionnelles, Ann. Scient. Ecole Normale Sup. (3) 51 (1934), 45–78], which is fundamental for their construction of Leray-Schauder's Degree in normed vector spaces.

  • Existence and multiplicity results for periodic solutions of nonlinear difference equations
    Journal of Difference Equations and Applications, 2006
    Co-Authors: Cristian Bereanu, Jean Mawhin
    Abstract:

    We use Brouwer Degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second-order and first-order difference equations. We obtain, in particular upper and lower solutions theorems, Ambrosetti-Prodi type results and sharp existence conditions for nonlinearities which are bounded from below or above.

  • a simple approach to Brouwer Degree based on differential forms
    Advanced Nonlinear Studies, 2004
    Co-Authors: Jean Mawhin
    Abstract:

    We present Heinz' approach to Brouwer Degree in a simpler, shorter and better motivated way. We link it to Kronecker index, use the language of differential forms at an elementary level, and prove the homotopy invariance using an unpublished result of Tartar.

Yi-shi Duan - One of the best experts on this subject based on the ideXlab platform.

  • TOPOLOGICAL STRUCTURE OF VORTEX LINES OF QUANTUM ELECTRON PLASMAS IN THREE-DIMENSIONAL SPACE
    International Journal of Modern Physics B, 2009
    Co-Authors: Yi-shi Duan
    Abstract:

    The topological properties of quantum electron plasmas in three-dimensional space are presented. Starting from phi-mapping topological current theory, the vortex lines are just at the core of wave function obtained. It is shown that the vorticity of the vortex can be expressed by the Hopg index and the Brouwer Degree. We find the vortex lines are unstable in some conditions and the evolution of vortex lines at the bifurcation points is given.

  • TOPOLOGICAL SOLUTION OF BOHMIAN QUANTUM MECHANICS
    Modern Physics Letters A, 2009
    Co-Authors: Ming Yu, Yi-shi Duan
    Abstract:

    The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrodinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer Degree. The evolution of the world line at the bifurcation point is given.

  • Knotlike χ disclinations in the cholesteric liquid crystals
    Physica A-statistical Mechanics and Its Applications, 2006
    Co-Authors: Peng-ming Zhang, Yi-shi Duan, Hong Zhang
    Abstract:

    We investigate the topological properties of N(N⩾1) disclination lines in cholesteric liquid crystals. The topological structure of N disclination lines is obtained with the Hopf index and Brouwer Degree. Furthermore, the knotted χ disclination loops is proposed with the Hopf invariant. And we consider the stability of such configuration based on the higher order interaction. At last, the evolution of the disclinations is discussed.

  • Torsion structure in Riemann-Cartan manifold and dislocation
    General Relativity and Gravitation, 2002
    Co-Authors: Marcello Baldo, Yi-shi Duan
    Abstract:

    The U(1) gauge structure of torsion and dislocation in three dimensional Riemann-Cartan manifold have been studied. The local topological structure of dislocation have been presented by so-called topological method in which the quantum number is by Hopf indices and Brouwer Degree. Furthermore, the relationship between the dislocation lines and Wilson lines of the U(1) gauge theory is discussed by using the Chern-Simons theory.

  • The second Chern class in Spinning System
    arXiv: High Energy Physics - Theory, 1999
    Co-Authors: Yi-shi Duan, Li-bin Fu
    Abstract:

    Topological property in a spinning system should be directly associated with its wavefunction. A complete decomposition formula of SU(2) gauge potential in terms of spinning wavefunction is established rigorously. Based on the $\phi $-mapping theory and this formula, one proves that the second Chern class is inherent in the spinning system. It is showed that this topological invariant is only determined by the Hopf index and Brouwer Degree of the spinning wavefunction.

Anna Siffert - One of the best experts on this subject based on the ideXlab platform.

  • Harmonic Self-Maps of $$\mathrm {SU}(3)$$ SU ( 3 )
    Journal of Geometric Analysis, 2017
    Co-Authors: Anna Siffert
    Abstract:

    By constructing solutions of a singular boundary value problem, we prove the existence of a countably infinite family of harmonic self-maps of $$\mathrm {SU}(3)$$ with non-trivial, i.e., $$\ne 0,\pm 1$$ , Brouwer Degree.

  • Harmonic Self-Maps of \mathrm {SU}(3)
    Journal of Geometric Analysis, 2017
    Co-Authors: Anna Siffert
    Abstract:

    By constructing solutions of a singular boundary value problem, we prove the existence of a countably infinite family of harmonic self-maps of \(\mathrm {SU}(3)\) with non-trivial, i.e., \(\ne 0,\pm 1\), Brouwer Degree.

  • Infinite families of harmonic self-maps of spheres
    Journal of Differential Equations, 2016
    Co-Authors: Anna Siffert
    Abstract:

    Abstract For each of the spheres S n , n ≥ 5 , we construct a new infinite family of harmonic self-maps, and prove that their members have Brouwer Degree ±1 or ±3. These self-maps are obtained by solving a singular boundary value problem. As an application we show that for each of the special orthogonal groups SO ( 4 ) , SO ( 5 ) , SO ( 6 ) and SO ( 7 ) there exist two infinite families of harmonic self-maps.

  • Harmonic Self-maps of $\mbox{SU}(3)$
    arXiv: Classical Analysis and ODEs, 2015
    Co-Authors: Anna Siffert
    Abstract:

    By constructing solutions of a singular boundary value problem we prove the existence of a countably infinite family of harmonic self-maps of $\mbox{SU}(3)$ with non-trivial, i.e. $\neq 0,\pm 1$, Brouwer Degree.

  • Infinite families of harmonic self-maps of spheres
    arXiv: Classical Analysis and ODEs, 2015
    Co-Authors: Anna Siffert
    Abstract:

    For each of the spheres $\mathbb{S}^{n}$, $n\geq 5$, we construct a new infinite family of harmonic self-maps, and prove that their members have Brouwer Degree $\pm1$ or $\pm3$. These self-maps are obtained by solving a singular boundary value problem.

Robert F Brown - One of the best experts on this subject based on the ideXlab platform.

  • Additivity and Product Properties
    A Topological Introduction to Nonlinear Analysis, 2020
    Co-Authors: Robert F Brown
    Abstract:

    We have some unfinished business from Chapter 9: the proof of two of the properties of the Brouwer Degree. We will need two more tools from algebraic topology for these proofs.

  • Properties of the Leray–Schauder Degree
    A Topological Introduction to Nonlinear Analysis, 2014
    Co-Authors: Robert F Brown
    Abstract:

    You may have noticed that once the Brouwer Degree was defined and its properties established, we used it in the chapter that followed only in a sort of formal way. In defining the Leray-Schauder Degree we needed to know that there was a well-defined integer, called the Brouwer Degree, represented by the symbol d(I∈ - f∈, U∈), but we did not have to specify how that integer was defined. Furthermore, and this is the point I want to emphasize, in the proof that the Leray-Schauder Degree is well-defined, which is really a theorem about the Brouwer Degree, all we needed to know about that Degree was two of its properties: homotopy and reduction of dimension. In this chapter, I will list and demonstrate properties of the Leray-Schauder Degree; properties which, basically, are consequences of the corresponding properties of the Brouwer Degree. Again all we will need to know about the Brouwer Degree is its existence and properties. As in the previous chapter, no homology groups will ever appear. Then, once we have established these properties of the Leray-Schauder Degree, that’s all we’ll need to know about that Degree for the rest of the book. In other words, after this chapter we can forget how the Degree was defined.

  • Compact Linear Operators
    A Topological Introduction to Nonlinear Analysis, 2014
    Co-Authors: Robert F Brown
    Abstract:

    In a sense, nonlinear analysis doesn’t require a very long attention span. A few chapters ago, we were concerned with algebraic topology in the theory of the Brouwer Degree; the previous chapter gave us a brief but bracing dip into the sea of point-set topology; and in this chapter we will discuss some topics in classical “linear” functional analysis.

  • properties of the Brouwer Degree
    2014
    Co-Authors: Robert F Brown
    Abstract:

    This chapter is devoted to the properties of the Brouwer Degree that we will need in order to extend it to the Leray–Schauder Degree. In all that follows, we assume that U is an open subset of R n and that we have a map \(f: \overline{U} \rightarrow \mathbf{R}^{n}\) such that \(F = f^{-1}(\mathbf{0})\) is admissible in U, that is, compact and disjoint from ∂ U, so the Brouwer Degree deg(f, U) is well defined. The properties of the Degree are given names for easy identification; the terminology I’m using for this purpose is pretty much standard. Some of the properties will carry over to the infinite-dimensional case and others are needed in order to make the transition to that more general setting.

  • A Topological Introduction To Nonlinear Analysis
    1993
    Co-Authors: Robert F Brown
    Abstract:

    I Fixed Point Existence Theory.- 1 The Topological Point of View.- 2 Ascoli-Arzela Theory.- 3 Brouwer Fixed Point Theory.- 4 Schauder Fixed Point Theory.- 5 The Forced Pendulum.- 6 Equilibrium Heat Distribution.- 7 Generalized Bernstein Theory.- II Degree Theory.- 8 Brouwer Degree.- 9 Properties of the Brouwer Degree.- 10 Leray-Schauder Degree.- 11 Properties of the Leray-Schauder Degree.- 12 The Mawhin Operator.- 13 The Pendulum Swings Back.- III Bifurcation Theory.- 14 A Separation Theorem.- 15 Compact Linear Operators.- 16 The Degree Calculation.- 17 The Krasnoselskii-Rabinowitz Bifurcation Theorem.- 18 Nonlinear Sturm-Liouville Theory.- 19 More Sturm-Liouville Theory.- 20 Euler Buckling.- IV Appendices.- A Singular Homology.- B Additivity and Product Properties.- C Bounded Linear Transformations.- C Bounded Linear Transformations.- References.

Zalman Balanov - One of the best experts on this subject based on the ideXlab platform.

  • on algebraic problems behind the Brouwer Degree of equivariant maps
    Journal of Algebra, 2020
    Co-Authors: Zalman Balanov, Mikhail Muzychuk, Haopin Wu
    Abstract:

    Abstract Given a finite group G and two unitary G-representations V and W, possible restrictions on topological Degrees of equivariant maps between representation spheres S ( V ) and S ( W ) are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in S ( V ) (denoted by α ( V ) ). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is α ( V ) > 1 ? and (ii) does there exist an equivariant map with the Degree easy to calculate? In the present paper, we address both questions. We show that α ( V ) > 1 for each irreducible non-trivial C [ G ] -module if and only if G is solvable. This provides a new solvability criterion for finite groups. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial α-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the Degree.

  • On Some Applications of Group Representation Theory to Algebraic Problems Related to the Congruence Principle for Equivariant Maps
    arXiv: Representation Theory, 2017
    Co-Authors: Zalman Balanov, Mikhail Muzychuk, Haopin Wu
    Abstract:

    Given a finite group $G$ and two unitary $G$-representations $V$ and $W$, possible restrictions on Brouwer Degrees of equivariant maps between representation spheres $S(V)$ and $S(W)$ are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in $S(V)$ (denoted $\alpha(V)$). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is ${\alpha}(V)>1$? and (ii) does there exist an equivariant map with the Degree easy to calculate? In the present paper, we address both questions. We show that ${\alpha}(V)>1$ for each irreducible non-trivial $C[G]$-module if and only if $G$ is solvable. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial ${\alpha}$-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the Brouwer Degree.

  • A short treatise on the equivariant Degree theory and its applications
    Journal of Fixed Point Theory and Applications, 2010
    Co-Authors: Zalman Balanov, Wieslaw Krawcewicz, Sławomir Rybicki, Heinrich Steinlein
    Abstract:

    The aim of this survey is to give a profound introduction to equivariant Degree theory, avoiding as far as possible technical details and highly theoretical background. We describe the equivariant Degree and its relation to the Brouwer Degree for several classes of symmetry groups, including also the equivariant gradient Degree, and particularly emphasizing the algebraic, analytical, and topological tools for its effective calculation, the latter being illustrated by six concrete examples. The paper concludes with a brief sketch of the construction and interpretation of the equivariant Degree.

  • Brouwer Degree equivariant maps and tensor powers
    Abstract and Applied Analysis, 1998
    Co-Authors: Zalman Balanov, Wieslaw Krawcewicz, A Kushkuley
    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 differen- tial equations are also discussed.