Narrow Results

Framed oriented $n$-tangle diagrams in the annulus, subject to the Homfly skein relations, are used to produce an algebra isomorphic to the affine Hecke algebras ${Ḣ}_n$ of type $A$. The use of closed curves and braids gives neat pictures for central elements in the algebras.

Using an inner complex blockage within a square cavity is spreading massively for the cooling process. This study adopts the time-fractional derivative of the incompressible smoothed particle hydrodynamics (ISPH) method for studying the magnetic field, diffusion-thermo, and thermo-diffusion impacts on the double diffusion of a nanofluid in a porous annulus between a square cavity and an astroid shape. The alterations of the pertinent parameters, fractional derivative order $\alpha$ between 0.9 and 1, dimensionless time parameter $\tau$ between 0 and 0.6, the radius of an astroid $R_a$ between 0.1 and 0.45, solid volume fraction $\varphi$ between 0 and 0.06, Hartman parameter Ha between 0 and 100, Darcy parameter Da between $10^{-2}$ and $10^{-5}$, and Soret number Sr between 0.1 and 2 supplemented by Dufour number Du between 0.6 and 0.03 on the velocity field, temperature, concentration, and mean of Nusselt and Sherwood numbers are discussed. The main findings of the ISPH numerical simulations showed that a decrease in a fractional derivative order $\alpha$ delivers the sooner steady-state of the double diffusion which suppresses the performed calculations. The velocity field’s maximum powers by 19.23% as $R_a$ increases from 0.1 to 0.45 and it decreases by 16.67%, 28.89%, and 97.99% as $\varphi$ powers from 0 to 0.06, Ha powers from 0 to 100, and Da decreases from 10${}^{-2}$ to 10${}^{-5}$, respectively. The outlines of $\overline{\mathrm{\text{Nu}}}$ and $\overline{\mathrm{\text{Sh}}}$ are increasing as $R_a$ and $\varphi$ are increased. A growth in Sr supplemented by a reduction in Du is diminishing the distributed concentration and nanofluid velocity within an annulus.

The unsteady flow in an annulus due to a velocity applied to one of the boundaries is addressed. The fluid considered is non-Newtonian, incompressible and electrically conducting. The strength of the applied magnetic field is time-dependent. Both analytical and numerical approaches are presented and compared. The nonlinear effects on the velocity profile are shown.

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H_n. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, .

Symmetric functions of the Murphy operators are also central in H_n. I define geometrically a homomorphism from to the centre of each algebra H_n, and find an element in , independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how the 2-variable Homfly invariant <λ, μ> of the Hopf link arising from decorations Q_λ and Q_μ can be found from the Schur symmetric function s_μ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_λ and V_μ, which is a 1-variable specialisation of <λ, μ>, can be expressed in terms of an N × N minor of the Vandermonde matrix (q^ij).

We consider the buckling eigenvalue problem for a clamped plate in the annulus. We identify the first eigenvalue in dependence of the inner radius, and study the number of nodal domains of the corresponding eigenfunctions. Moreover, in order to investigate the asymptotic behavior of eigenvalues and eigenfunctions as the inner radius approaches the outer one, we provide an analytical study of the buckling problem in rectangles with mixed boundary conditions.

It is known that any infinite periodic frieze comes from a triangulation of an annulus by Theorem 4.6 of [K. Baur, M. J. Parsons and M. Tschabold, Infinite friezes, European J. Combin.54 (2016) 220–237]. In this paper, we show that each infinite periodic frieze determines a triangulation of an annulus in essentially a unique way. Since each triangulation of an annulus determines a pair of friezes, we study such pairs and show how they determine each other. We study associated module categories and determine the growth coefficient of the pair of friezes in terms of modules as well as their quiddity sequences.

In this paper, we will prove a uniqueness theorem in the case of meromorphic functions on the annuli share $q(q\ge 5)$ distinct elements with different multiple values.