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

This booklet includes survey papers in accordance with the lectures offered on the third overseas iciness tuition “Modern difficulties of arithmetic and Mechanics” held in January 2010 on the Belarusian nation collage, Minsk. those lectures are dedicated to diversified difficulties of contemporary research and its functions. a longer presentation of recent difficulties of utilized research will permit the reader to get acquainted with new methods of as a rule interdisciplinary personality. the implications mentioned are program orientated and current new perception into utilized difficulties of growing to be value corresponding to purposes to composite fabrics, anomalous diffusion, and fluid dynamics.

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O ????+ ℎ ????− j ????+ j ????+ ℎ ????− j ????+ j / ???? / ???? (????−2)+j(????−2) o ????+j(????−2) o − O O ????− ???? ℎ ℎ ????+ ℎ ????− ℎ ????+ ℎ ????− j ????+ j / ???? (????−2)+j???? o O ????− j ????+ j ????+ ℎ ????− ℎ ????+ ℎ ????− j ????+ j / ???? ????+j???? o O ????− j ????+ j / ???? / ???? (????−2)+j(????+2) o ????+j(????+2) o − O O ????− ???? ℎ ℎ ????− j ????+ j ... ????− j ????+ j ... 21 ... O ????+ ℎ ????− j ????+ j ????+ ℎ / ???? / ... (????+2)+j(????−2) o − − O ????ℎ ????ℎ ????+ ℎ ????− ℎ ????+ ℎ ????− j ????+ j / ???? (????+2)+j???? o O ????− j ????+ j ????+ ℎ ????− ℎ / ... ????+ ℎ / ???? / ... (????+2)+j(????+2) o − − O ????ℎ ????ℎ ????− j ????+ j ...

A modiﬁcation of the previous construction is obtained if we have two groups ????1 and ????2 represented by ????1 and ????2 on ???? and ???? ∗ respectively. Then we have a covariant transform ????(????, ???? ) → ????(????1 × ????2 , ℂ) deﬁned by the formula ˆ 1 , ????2 ) = ⟨????????1 (????1 )????, ????2 (????2 )????⟩ . ???? : ???? → ????(???? This generalises the above Berezin covariant calculi [66]. 18. Let us restrict the previous example to the case when ???? = ???? is a Hilbert space, ????1 = ????2 = ???? and ???? = ???? with ∥????∥ = 1. Then the range of the covariant transform, ˆ ???? : ???? → ????(????) = ⟨????????(????)????, ????(????)????⟩ , is a subset of the numerical range of the operator ????.

7) Erlangen Program at Large: An Overview (????) The graph Γ???? 43 of j???? ???? is a submanifold of ???????? which is a section of the jet bundle over (????) ???? with a ﬁbre ℂ????+1 . 7). One can prolong any map of functions ???? : ???? (????) → [???????? ](????) to a map ???? (????) of ????-jets by the formula ???? (????) (???????? ???? ) = ???????? (???????? ). 8) (????) ????1 For example such a prolongation of the representation ????1 of the group SL2 (ℝ) in ????2 (????) (as any other representation of a Lie group [108]) will be again a repre(????) sentation of SL2 (ℝ).