We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$-frequent hypercyclicity and hypercyclicity.

Two boundary value problems are solved for potential steady-state 2D Darcian seepage flows towards a line sink in a homogeneous isotropic soil from a ponded land surface, which is not flat but profiled. The aim of this shaping is ‘uniformisation’ of the velocity and travel time between this surface and a horizontal drain modelled by a line sink. The complex potential domain is a half-strip, which is mapped onto a reference plane. Either the velocity magnitude or a vertical coordinate along the land surface are control variables. Either a complexified velocity or complex physical coordinate is reconstructed by solving mixed boundary-value problems with the help of the Keldysh-Sedov formula via singular integrals, the kernel of which are the control functions. The flow nets, isotachs and breakthrough curves are found by computer algebra routines. A designed soil hump above the drain ameliorates an unwanted ‘preferential flow’ (shortcut) and improves leaching of salinised soil of a cropfield during a pre-cultivation season.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle ${\mathbb T}$ , there is a function f holomorphic in the unit disc ${{\mathbb D}}$ , having $\psi $ as radial limit a.e. on ${\mathbb T}$ . We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point p is replaced by (asymptotically) total approach to p.

In this paper we study the Hilbert transformations over ${{L}^{2}}(\mathbb{R})$ and ${{L}^{2}}(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over ${{L}^{2}}(\mathbb{R})$ commutative with the $ax\,+\,b$ group, we show that the operator is of the form $\lambda I+\eta H$, where $I$ and $H$ are the identity operator and Hilbert transformation, respectively, and $\lambda ,\eta $ are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand–Naimark representation of the $ax\,+\,b$ group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the $ax\,+\,b$ group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle ${{L}^{2}}(\mathbb{T})$.

We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions at t = 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates under L2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

For $p\,>\,0$ and for a given set $E$ of type ${{G}_{\delta }}$ in the boundary of the unit disc $\partial \mathbb{D}$ we construct a holomorphic function $f\,\in \,\mathbb{O}\left( \mathbb{D} \right)$ such that

$${{\int_{\mathbb{D}\backslash \left[ 0,\,1 \right]E}{\left| f \right|}}^{p}}\,d{{\mathfrak{L}}^{2}}\,<\,\infty \,\text{and}\,E\,=\,{{E}^{p}}\left( f \right)\,=\,\{\,z\,\in \,\partial \mathbb{D}\,:\,\int _{0}^{1}\,{{\left| f\left( tz \right) \right|}^{p}}\,dt\,=\,\infty \}.$$

In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb{D}$.

In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.