We consider a parametric nonlinear Robin problem driven by a nonhomogeneous differential operator. Using variational tools together with suitable truncation and perturbation techniques, we prove a bifurcation-type theorem describing the dependence of the set of positive solutions on the parameter.

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We study the structures of Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1, k+n+2) by using twisted cohomology groups and the intersection forms on them. We apply our results to algebraic statistics; we give an algorithm for numerical evaluation of the normalizing constants of two way contingency tables with fixed marginal sums.

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In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q-deformed bosonic operators, which are constructed from the L-operator of higher rank found in the recent work by Garbali, de Gier and Wheeler. The L-operator is obtained from the universal R-matrix of the quantum affine algebra of type A_r⁽¹⁾ by the use of the q-oscillator representation. Thus our formula may be regarded as a bridge between two approaches to studying integrable stochastic systems by means of the quantum affine algebra and the affine Hecke algebra.

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We investigate a singular perturbation for Hamilton-Jacobi equations in an open subset of two dimensional Euclidean space, where the set is determined through a Hamiltonian and the Hamilton-Jacobi equations are the dynamic programming equations for optimal control of the Hamiltonian flow of the Hamiltonian. We establish the convergence of solutions of the Hamilton-Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary differential equations on a graph. The perturbation is singular in the sense that the domain degenerates to the graph in the limiting process. Our result can be seen as a perturbation analysis, in the viewpoint of optimal control, of the Hamiltonian flow.

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