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[en] We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C*-subalgebra to discuss a Shubin trace formula
[en] It is proved that each submodule of a multiplication module over a regular ring is a multiplicative module. If A is a ring with commutative multiplication of right ideals, then each projective right ideal is a multiplicative module, and a finitely generated A-module M is a multiplicative module if and only if all its localizations with respect to maximal right ideals of A are cyclic modules over the corresponding localizations of A. In addition, several known results on multiplication modules over commutative rings are extended to modules over not necessarily commutative rings
[en] We give a definition of quasi-coherent modules for any presheaf of sets on the categories of affine commutative and non-commutative schemes. This definition generalizes the usual one. We study the property of a quasi-coherent module to be a sheaf in various topologies. Using presheaves of groupoids, we construct an embedding of commutative geometry in non-commutative geometry
[en] The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richardson coefficients. We define these bijections in terms of arrays and show that they coincide with analogous bijections defined in terms of discretely concave functions using the octahedron recurrence as well as with bijections defined in terms of Young tableaux. The main ingredient in the proof of their coincidence is a functional version of the Robinson-Schensted-Knuth correspondence
[en] We derive suitable uncertainty relations for characteristics functions of phase and number of a single-mode field obtained from the Weyl form of commutation relations. Some contradictions between the product and sums of characteristic functions are noted. - Highlights: • Phase–number uncertainty relations derived from Weyl commutation relations. • Phase–number uncertainty relations that include correlation terms. • Phase–number uncertainty relations in complete parallel with position–momentum.
[en] We construct a noncommutative (Grassmann) extension of the well-known Adler Yang–Baxter map. It satisfies the Yang–Baxter equation, it is reversible and birational. Our extension preserves all the properties of the original map except the involutivity. (letter)
[en] By considering Landau level projections of the N=1 supersymmetrized Landau problem remaining nontrivial under N=1 supersymmetry transformations, the algebraic structures of the N=1 supersymmetric covariant non(anti)commutative superplane analogue of the ordinary N=0 noncommutative Moyal-Voros plane are identified. In contradistinction to all discussions available in the literature on the supersymmetrized Landau problem, the relevant Landau level projections do not amount to taking a massless limit of the Landau system.
[en] A general deformation of the canonical commutation relations which include the standard deformation as special cases is discussed. The associated coherent states are described. The squeezing properties are evaluated. It is shown that in contradistinction to the conventional case of coherent states squeezing may occur. 16 refs