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The present paper proposes new integral representations of -Whittaker functions corresponding to an arbitrary semisimple Lie algebra with the integrand expressed in terms of matrix elements of the fundamental representations of . For the classical Lie algebras , , and a modification of this construction is proposed, providing a direct generalization of the integral representation of -Whittaker functions first introduced by Givental. The Givental representation has a recursive structure with respect to the rank of the Lie algebra , and the proposed generalization to all classical Lie algebras retains this property. It was observed elsewhere that an integral recursion operator for the -Whittaker function in the Givental representation coincides with a degeneration of the Baxter -operator for -Toda chains. In this paper -operators for the affine Lie algebras , and a twisted form of are constructed. It is then demonstrated that the relation between integral recursion operators for the generalized Givental representations and degenerate -operators remains valid for all classical Lie algebras.

Bibliography: 33 titles.

Boolean functions are among the fundamental objects of discrete mathematics, especially in those of its subdisciplines which fall under mathematical logic and mathematical cybernetics. The language of Boolean functions is convenient for describing the operation of many discrete systems such as contact networks, Boolean circuits, branching programs, and some others. An important parameter of discrete systems of this kind is their complexity. This characteristic has been actively investigated starting from Shannon's works. There is a large body of scientific literature presenting many fundamental results. The purpose of this survey is to give an account of the main results over the last sixty years related to the complexity of computation (realization) of Boolean functions by contact networks, Boolean circuits, and Boolean circuits without branching.

Bibliography: 165 titles.