We construct the positive principal series representations for Uq(gR) where g is of type Bn, Cn, F4 or G2, parametrized by Rn where n is the rank of g. We show that under the representations, the generators of the Langlands dual group Uq(LgR) are related to the generators of Uq(gR) by the transcendental relations. This gives a new and very simple analytic relation between the Langlands dual pair. We define the modified quantum group Uqq(gR)=Uq(gR) ⊗ Uq(LgR) of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the q-tori polynomials.

We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in (R+)d. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of P-pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge–Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

We look for bounded periodic solutions for a parametric fractional problem involving a continuous nonlinearity with subcritical growth. By using a variant of Caffarelli and Silvestre extension method adapted to the periodic case and variational tools we prove the existence of at least three bounded periodic solutions when the parameter varies in an appropriate range.

We derive an upper bound on the free energy of a Bose gas at density $ \varrho $ and temperature T. In combination with the lower bound derived previously by Seiringer (Commun. Math. Phys. 279(3): 595–636, 2008), our result proves that in the low density limit, i.e., when $ a^3 \varrho \ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\Delta f$, the free energy difference per volume between interacting and ideal Bose gases, is equal to $ 4\pi a(2\varrho^{2}-[\varrho-\varrho_{c}]^{2}_{+}) $. Here, $\varrho_c (T)$ denotes the critical density for Bose–Einstein condensation (for the ideal Bose gas), and $ [⋅]_{+}= max \{⋅ , 0\}$ denotes the positive part.

A hyperbolic 3-manifold M which fibers over the circle admits a flow called the suspension flow. We show that such a flow can be isotoped to be uniformly quasigeodesic in the hyperbolic metric on M; i.e., the flow lines lifted to hyperbolic space are K-bilipschitz embeddings of R (K > 1 fixed).