We look at periodic Jacobi matrices on trees. provide upper and lower bounds the gap of such operators analogous to well known in spectrum Laplacian half-plane with hyperbolic metric. make some conjectures about antibound states an interesting observation for what [3] calls rg-model.

In this note we shall show that a lattice ℤω 1 +ℤω 2 in ℂ has ℚ-linearly dependent quasi-periods if and only ω /ω is equivalent to zero of the Eisenstein series E under action SL (ℤ) on upper half plane ℂ.

In this paper, we discuss free interpolation in the spaces of entire and analytic finite-order functions upper half-plane. A review problems basic results related to such is given. Solvability criteria are formulated terms canonical products nodes measure determined by these nodes.

We describe how problems of geometric construction using straightedge and compass can be introduced to students through project-based learning. discuss these extended the upper half-plane model. Furthermore, we use assess advanced levels in van Hiele model for geometry thinking.

where L is the number of links of the lattice, NC the number of clusters in the configuration C and ‖C‖ the number of links inside the NC clusters, usually called FK-clusters. Criticality is then encoded in the fractal nature of these clusters. The stochastic Loewner evolutions (SLE) [2] are mathematically well-defined processes describing the growth of random sets, called the SLE hulls, and of...

We prove a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on the upper half-plane H. Moreover, we give an accompanying result construction modular forms in form series. As application these results, construct and study classical elliptic

The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element, g2 HzL = -1 ê z, and an element of infinite order, g• HzL = z + 1. Viewing the action of the group elements on a...

Under identical viewing conditions, observers made two types of judgement about the shape of stereoscopically defined surfaces: one required an estimate of viewing distance for correct performance (e.g. setting the depth of a hemi-cylinder to equal its half-height or a dihedral angle to 90 deg), the other did not (matching the depth of, for example, sinusoidal corrugations or hemi-cylinders pre...