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Reduction of semialgebraic constructible functions

Ludwig Bröcker (2005)

Annales Polonici Mathematici

Let R be a real closed field with a real valuation v. A ℤ-valued semialgebraic function on Rⁿ is called algebraic if it can be written as the sign of a symmetric bilinear form over R[X₁,. .., Xₙ]. We show that the reduction of such a function with respect to v is again algebraic on the residue field. This implies a corresponding result for limits of algebraic functions in definable families.

Refinement type equations: sources and results

Rafał Kapica, Janusz Morawiec (2013)

Banach Center Publications

It has been proved recently that the two-direction refinement equation of the form f ( x ) = ∑ n ∈ c n , 1 f ( k x - n ) + ∑ n ∈ ℤ c n , - 1 f ( - k x - n ) can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation f ( x ) = ∑ n ∈ ℤ c ₙ f ( k x - n ) , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation f ( x ) = ∫ ℝ c ( y ) f ( k x - y ) d y has also various interesting applications....

Regular mappings between dimensions

Guy David, Stephen Semmes (2000)

Publicacions Matemàtiques

The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat...

Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type ☆ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type ☆ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step...

Relations between Shy Sets and Sets of ν p -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

An example of a non-zero non-atomic translation-invariant Borel measure ν p on the Banach space ℓ p ( 1 ≤ p ≤ ∞ ) is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " ν p -almost every element of ℓ p has a property P" implies that “almost every” element of ℓ p (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

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