By Dmitry S. Kaliuzhnyi-verbovetskyi, Victor Vinnikov

During this booklet the authors increase a conception of unfastened noncommutative features, in either algebraic and analytic settings. Such services are outlined as mappings from sq. matrices of all sizes over a module (in specific, a vector area) to sq. matrices over one other module, which recognize the dimensions, direct sums, and similarities of matrices. Examples contain, yet aren't restricted to, noncommutative polynomials, strength sequence, and rational expressions. Motivation and proposal for utilizing the idea of loose noncommutative capabilities frequently comes from loose likelihood. a major program quarter is "dimensionless" matrix inequalities; those come up, e.g., in quite a few optimization difficulties of procedure engineering. between different similar components are these of polynomial identities in earrings, formal languages and finite automata, quasideterminants, noncommutative symmetric services, operator areas and operator algebras, and quantum keep watch over

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**Foundations of free noncommutative function theory**

During this booklet the authors improve a thought of loose noncommutative capabilities, in either algebraic and analytic settings. Such services are outlined as mappings from sq. matrices of all sizes over a module (in specific, a vector house) to sq. matrices over one other module, which admire the dimensions, direct sums, and similarities of matrices.

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**Example text**

0 .. ⎥ . ⎥ ⎥ ⎥ ∈ Ωn0 +···+n , 0⎥ ⎥ ⎥ Z ⎦ ··· .. .. 0 .. 29) ⎡ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣ ⎛⎡ 0 X ⎜⎢ ⎜⎢ 0 ⎜⎢ ⎜⎢ . ⎢ f⎜ ⎜⎢ .. ⎜⎢ ⎜⎢ .. ⎝⎣ . 0 0 Z1 0 X1 .. .. .. .. ··· ··· 0 1 ··· .. .. −1 X 0 1 f (X ) ΔR f (X , X )(Z ) 0 .. .. 0 f (X 1 ) .. ··· ⎤⎞ 0 ⎟ .. ⎥ ⎟ . ⎥ ⎥⎟ ⎥⎟ ⎥⎟ 0 ⎥⎟ ⎥⎟ ⎥⎟ Z ⎦⎠ X ··· .. .. .. ··· ··· .. f (X −1 ) 0 ΔR f (X 0 , . . , X )(Z 1 , . . , Z ) ΔR−1 f (X 1 , . . , X )(Z 2 , . . , Z .. ΔR f (X −1 , X )(Z ) f (X ) ⎤ ⎥ )⎥ ⎥ ⎥ ⎥. 12. Let f ∈ T k (Ω(0) , .

Z k+1 ) −→ ΔR f (X 0 , . . , X k+1 )(Z 1 , . . , Z k+1 ) from N1 n0 ×n1 × · · · × Nk nk−1 ×nk × Mk nk ×nk+1 to N0 n0 ×nk+1 is (k + 1)-linear over R. Furthermore, ΔR f ∈ T k+1 (Ω(0) , . . , Ω(k) , Ω(k) ; N0,nc , . . , Nk,nc , Mk,nc ). Proof. The fact that ΔR f X 0 , . . , X k+1 Z 1 , . . , Z k+1 is linear as a function of Z j , for every j = 1, . . 21). 9. 21) implies that the function ΔR f X 0 , . . , X k+1 Z 1 , . . , Z k+1 respects direct sums and similarities in variables X 0 , . .

Nk,nc ) → T k+1 (Ω(0) , . . , Ω(k) , Ω(k) ; N0,nc , . . , Nk,nc , Mk,nc ). A nc function of order k is a function of k + 1 arguments, in nc sets Ω(0) ⊆ M0,nc , . . , Ω(k) ⊆ Mk,nc , with values certain k-linear mappings, which respects (in an appropriate way) direct sums, similarities, and intertwinings. Higher order nc diﬀerence-diﬀerential operators will be obtained by iterating ΔR . (From now on, we will concentrate on right nc diﬀerence-diﬀerential operators. 1. Higher order nc functions Let M0 , .