Tiling a Circular Disc with Congruent Pieces

In this note, we prove that any monohedral tiling of the closed circular unit disc with $$k \le 3$$ k ≤ 3 topological discs as tiles has a k-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer, and Guy in 1994.

On the q-deformed circular imaginary unit and hyperbolic imaginary unit: q-deformed rotation in two dimension and q-deformed special relativity in 1 + 1 dimension

In this paper, the [Formula: see text]-deformed circular unit and hyperbolic imaginary unit are studied. With a help of these units, the invariant [Formula: see text]-deformed length is defined. As applications, the [Formula: see text]-deformed rotation in two dimension and [Formula: see text]-deformed special relativity in [Formula: see text] dimension are also investigated.

Plastic Zone Analysis of Deep-Buried, Noncircular Tunnel and Application on the High-Speed Railway in the Karst Area

With the conformal mapping function provided by Verruijt, the outland of a noncircular tunnel can be mapped to a circular unit in the complex plane and then spread the analytic function into a Laurent series. The stress unified solution of oval and horseshoe cross section can be determined using Muskhelishvili’s complex variables function method. Subsequently, the solution can be taken into the Griffith strength failure criterion and determine the scale and shape of plastic zone in the tunnel surrounding rock. Aiming at the critical safety thickness between a concealed cave and tunnel in the karst area and determining whether the plastic zone of tunnel surrounding rock is connected with the plastic zone of cave as a judgment standard, the model of critical safety thickness among the concealed caves and tunnels is established. The numerical model is established in comparison with the computing method of rock plate critical safety thickness in actual engineering based on the Doumo tunnel engineering of Shanghai-Kunming (Guizhou segment) high-speed railway. The following conclusions can be drawn: the analytical approximation method has less indexes, and the output of this method is approximately close to actual engineering and numerical analysis, in which it is reliable and rational.

ON THE CLASS NUMBER AND THE FUNDAMENTAL UNIT OF THE REAL QUADRATIC FIELD

For a real quadratic field $k=\mathbb {Q}(\sqrt {pq})$, let tk be the exact power of 2 dividing the class number hk of k and ηk the fundamental unit of k. The aim of this paper is to study tk and the value of Nk/ℚ(ηk). Various methods have been successfully applied to obtain results related to this topic. The idea of our work is to select a special circular unit ℰk of k and investigate C(k)=〈±ℰk 〉. We examine the indices [E(k):C(k)] and [C(k):CS (k)] , where E(k) is the group of units of k, and CS (k) is that of circular units of k defined by Sinnott. Then by using the Sinnott’s index formula [E(k):CS (k)]=hk, we obtain as much information about tk and Nk/ℚ (ηk) as possible.