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## On Unitary Submodules in the Polynomial Representations of Rational Cherednik Algebras

We consider representations of rational Cherednik algebras that are particular ideals in the ring of polynomials. We investigate convergence of the integrals that express the Gaussian inner product on these representations. We derive that the integrals converge for the minimal submodules in types *B* and *D* for the singular values suggested by Cherednik with at most one exception; hence the corresponding modules are unitary. The analogous result on unitarity of the minimal submodules in type *A* was obtained by Etingof and Stoica; we give a different proof of convergence of the Gaussian product in this case. We also obtain partial results on unitarity of the minimal submodule in the case of exceptional Coxeter groups and group *B* with unequal parameters.

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n-1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov–Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q,t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

We obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type A_{n-1} for c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SL_m. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution by Berest-Etingof-Ginzburg and Gordon, and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras H_{m/n}(S_n) and H_{n/m}(S_m). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry has a natural interpretation in terms of invariants of torus knots.

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number *m*/*n*. Every such representation is a representation of the symmetric group *S**n* . We compare certain multiplicity spaces in its decomposition into irreducible representations of *S**n* with the spaces of differential forms on a zero-dimensional moduli space associated with the plane curve singularity *x^**m*= *y*^*n* .

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.