We describe a generalization of Drinfeld’s description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald’s difference operators. 1. The center of Uq(g), where g is a simple Lie algebra. 1.1. Let g be a simple Lie algebra over C of rank r. Let h be a Cartan subalgebra in g. Let W be the Weyl...

Thie paper presents a representation echeme for polyhedral objects in arbitrary dimensions. Each object ie represented as the algebraic sum of convex polyhedra (cells). Each cell in turn ie represented ae the intereection of halfspacee and encoded in a vector. The notion of vertices is abandoned completely aa it ie not needed for the eet and eearch operators we intend to support. We ehow how th...

Problem 7.8. (a) Let ω ∈ Ω(V ). Show that there is a basis φ1, . . . , φn of V ∗ such that ω = (φ1 ∧ φ2) + · · ·+ (φ2r−1 ∧ φ2r). (b) Show that the r-fold wedge product ω∧· · ·∧ω is non-zero and decomposable, and that the (r + 1)-fold wedge product is 0. Thus r is well-determined; it is called the rank of ω. (c) If ω = ∑ i<j aijψi ∧ ψj, and A is the upper triangular matrix with Aij = aij for i <...

For any C-algebra A let A denote the set of all positive elements in A. A state on a unital C-algebra A is a linear functional ω : A → C such that ω(a) ≥ 0 for every a ∈ A and ω(I) = 1 where I is the unit of A. By S(A) we will denote the set of all states on A. For any Hilbert space H we denote by B(H) the set of all bounded linear operators on H . A linear map φ : A → B between C-algebras is c...

A zero sum game is a simultaneous move game between 2 players. Such a game is represented by a matrix A ∈ Rm×n. The strategies of the “row” (resp. “column”) player are the rows (resp. columns) of A. If the row player plays strategy i ∈ [m] and the column player plays j ∈ [n] then the outcome is Aij . 1 Interpret this as that the row player pays Aij amount of money to the column player, therefor...

The question of controllability of natural and man-made network systems has recently received considerable attention. In the context of the human brain, the study of controllability may not only shed light into the organization and function of different neural circuits, but also inform the design and implementation of minimally invasive yet effective intervention protocols to treat neurological...

we investigate a class of fourth-order differential operators with eigenparameter dependent boundary conditions and transmission conditions. a self-adjoint linear operator a is defined in a suitable hilbert space h such that the eigenvalues of such a problem coincide with those of a . we discuss asymptotic behavior of its eigenvalues and completeness of its eigenfunctions. finally, we obtain th...

‎in this study‎, ‎properties of spectral characteristic are investigated for‎ ‎singular sturm-liouville operators in the case where an eigen‎ ‎parameter not only appears in the differential equation but is‎ ‎also linearly contained in the jump conditions‎. ‎also weyl function‎ ‎for considering operator has been defined and the theorems which‎ ‎related to uniqueness of solution of inverse proble...