By Vladimir V. Kisil (auth.), Sergei V. Rogosin, Anna A. Koroleva (eds.)

This e-book includes survey papers according to the lectures awarded on the third overseas wintry weather university “Modern difficulties of arithmetic and Mechanics” held in January 2010 on the Belarusian kingdom collage, Minsk. those lectures are dedicated to diverse difficulties of recent research and its functions. a longer presentation of contemporary difficulties of utilized research will permit the reader to get accustomed to new methods of often interdisciplinary personality. the implications mentioned are program orientated and current new perception into utilized difficulties of becoming significance equivalent to functions to composite fabrics, anomalous diffusion, and fluid dynamics.

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22. A contravariant calculus, also know as functional calculus, is a map from functions to operators, which intertwines two representations of the same group in the respective spaces. The duality between co- and contravariant calculi is the particular case of the duality between covariant transform and the inverse covariant transform deﬁned in the next subsection. In many cases a proper choice of spaces makes covariant and/or contravariant calculus a bijection between functions and operators. Subsequently only one form of calculus, either co- or contravariant, is deﬁned explicitly, although both of them are there in fact.

Let us deﬁne the resolvent function ????(????, ????) : ???? → ????. If ????(????, ????) = (???????? − ????????)−1 then ????(????1 , a)????(????2 , ????1−1 a) = ????(????1 ????2 , a). 2(10)] and is a key ingredient of induced representations. 2), cf. , ???? = ????: ???????? (????1 ) : ???? (???? −1 ⋅ ????) → ????(????1−1 ???? −1 , ????)???? (????1−1 ???? −1 ⋅ ????) ) ( ′ ???? ¯ ⋅ ???? − ????¯′ ???? ′ ′ −1 . V. Kisil For any ???? ∈ ???? we can deﬁne a ????-invariant vacuum vector as ???????? (???? −1 ⋅ ????) = ???? ⊗ ????0 (???? −1 ⋅ ????) ∈ ???? (????, ???? ). It generates the associated with ???????? family of coherent states ???????? (????, ????) = (???????? − ????)−1 ????, where ???? ∈ ????.

3) with such a mother wavelet will have a property: ????ˆ(????ℎ) = ⟨????, ????(????ℎ)????0 ⟩ = ⟨????, ????(????)????(ℎ)????0 ⟩ = ????(ℎ)ˆ ???? (????). Thus the wavelet transform is uniquely deﬁned by cosets on the homogeneous space ????/????. In this case we previously spoke about the reduced wavelet transform [65]. A representation ????0 is called square integrable mod ???? if the induced wavelet transform [????????0 ](????) of the vacuum vector ????0 (????) is square integrable on ????. Erlangen Program at Large: An Overview 31 The image of an induced covariant transform has a similar property: ????ˆ(????ℎ) = ???? (????((????ℎ)−1 )????) = ???? (????(ℎ−1 )????(???? −1 )????) = ???? (????(???? −1 )????)????(ℎ−1 ).