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• W2023707737 abstract "We construct a minimal pair of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degrees. We do this by showing the existence of an unbounded nondecreasing function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which forces <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-triviality in the sense that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma element-of 2 Superscript omega> <mml:semantics> <mml:mrow> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>ω<!-- ω --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma in 2^omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-trivial if and only if for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K left-parenthesis gamma up harpoon with barb right n right-parenthesis less-than-or-equal-to upper K left-parenthesis n right-parenthesis plus f left-parenthesis n right-parenthesis plus script upper O left-parenthesis 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo stretchy=false>↾<!-- ↾ --></mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>K(gamma upharpoonright n) leq K(n) + f(n)+ mathcal {O}(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2023707737 date "2005-10-04" @default.
• W2023707737 modified "2023-10-18" @default.
• W2023707737 title "A minimal pair of 𝐾-degrees" @default.
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