We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$, where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm {GL}(n)\times \mathrm {GL}(n)$ Rankin–Selberg L-functions $L(1/2,\pi \otimes \pi _0)$ for a fixed cuspidal automorphic representation $\pi _0$ over the family of $\pi $ with analytic conductors bounded by a quantity that is tending to infinity. Our proof uses the integral representations of the L-functions, period with regularised Eisenstein series and the invariance properties of the analytic newvectors.

This study presents a new exact-analytical solution for convective heat transfer of thermally fully-developed laminar nanofluid flows in a circular tube for the first time. In this problem, the pipe wall is exposed to a constant temperature. The solution is based on the Whittaker function and perturbation technique. In the nanofluid model, it is assumed that nanoparticles and base-fluid behave as a single-phase with average properties. In this study, the effects of Reynolds number, volume fraction of the particles, Peclet number, and particle diameter are investigated on the average heat transfer coefficient, surface mass transfer, and Nusselt number.

This paper presents the feasibility of torsional surface wave propagation in an anisotropic layer sandwiched between two anisotropic inhomogeneous media. The anisotropy considered in the upper layer and the lower half-space is of transversely isotropic kind while the sandwiched anisotropic layer is a porous layer. The directional rigidities and density have been considered linearly and exponentially varying in the half-space and in the upper layer respectively, while it is taken as a variable in the sandwiched layer. The compact form of dispersion equation governing the propagation of the torsional surface wave has been derived by using the Whittaker function under appropriate boundary conditions. The dispersion of the torsional wave and the effects of inhomogeneity parameters, initial stress and poroelastic constant have been calculated numerically and demonstrated through graphs.