The mappings of degree 1

• Format: Article
• Language: English
• Published: Hindawi Limited 2006-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://www.hindawi.com/GetArticle.aspx?doi=10.1155/JBB/2006/90837

<p>The maps of the form <mml:math> <mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle='true'> <mml:msubsup> <mml:mo>&#x2211;</mml:mo> <mml:mrow> <mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>&#x22C5;</mml:mo><mml:mi>x</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:mstyle> </mml:math>, called 1-degree maps, are introduced and investigated. For noncommutative algebras and modules over them 1-degree maps give an analogy of linear maps and differentials. Under some conditions on the algebra <mml:math> <mml:mi>&#x1d49c;</mml:mi> </mml:math>, contractibility of the group of 1-degree isomorphisms is proved for the module <mml:math> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>&#x1d49c;</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math>. It is shown that these conditions are fulfilled for the algebra of linear maps of a finite-dimensional linear space. The notion of 1-degree map gives a possibility to define a nonlinear Fredholm map of <mml:math> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>&#x1d49c;</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> and a Fredholm manifold modelled by <mml:math> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>&#x1d49c;</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math>. 1-degree maps are also applied to some problems of Markov chains.</p>