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

This e-book comprises survey papers in accordance with the lectures awarded on the third overseas iciness institution “Modern difficulties of arithmetic and Mechanics” held in January 2010 on the Belarusian country collage, Minsk. those lectures are dedicated to assorted difficulties of recent research and its purposes. a longer presentation of contemporary difficulties of utilized research will allow the reader to get accustomed to new ways of usually interdisciplinary personality. the consequences mentioned are program orientated and current new perception into utilized difficulties of turning out to be significance corresponding to functions to composite fabrics, anomalous diffusion, and fluid dynamics.

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**Example text**

1) generated by ???? is called the induced covariant transform. The following is the main motivating example. 2. 7. Choose a vacuum vector ????0 to be a joint eigenvector for all operators ????(ℎ), ℎ ∈ ????, that is ????(ℎ)????0 = ????(ℎ)????0 , where ????(ℎ) is a complex number depending of ℎ. Then ???? is obviously a character of ????. 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 ????/????.

Let ???? be a square integrable representation of ???? modulo a subgroup ???? ⊂ ???? and let ???? = ????/???? be the corresponding homogeneous space with a quasi-invariant measure ????????. Then integration over ???????? with an appropriate weight produces an invariant pairing. 52)] of the reconstruction formula mentioned in the previous example. Let ???? be not a square integrable representation (even modulo a subgroup) or let ????0 be an inadmissible vector of a square integrable representation ????. An invariant pairing in this case is not associated with an integration over any non-singular invariant measure on ????.

However it emerged later that the new deﬁnition is a useful replacement for the classical one across all ranges of problems. In the following subsections we will support the last claim by consideration of the simple known problem: characterisation of an ???? × ???? matrix up to similarity. Even that “freshman” question can be only sorted out by the classical spectral theory for a small set of diagonalisable matrices. Our solution in terms of new spectrum will be full and thus unavoidably coincides with one given by the Jordan normal form of matrices.