Normal Subgroups of Hecke Groups <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">λ</Emphasis>)

Let λ ≥ 2 and let H(λ) be the Hecke group associated to λ. Also let H(λ)\U be the Riemann surface associated to the Hecke group H(λ). In this article, we study the even subgroup H _e (λ) and the power subgroups H ^m (λ) of the Hecke groups H(λ). We also study some genus 0 normal subgroups of finite index of H(λ). Finally, we discuss free normal subgroups of H(λ).     

The concrete columns as a sensible thermal energy storage medium and a heater

This study investigated storage possibility of sensible thermal energy in the concrete columns of multi-storey buildings and the heating performance of the indoors with the stored energy. In the suggested system, the dry air heated in an energy center will be circulated in stainless steel pipes through columns. The sensible thermal energy would firstly be stored by means of forced convection in column medium. Then, the stored thermal energy will transfer by natural convection and radiation from the column surfaces to indoor spaces. The transient thermal calculations are realized for a flat of the 11-storey building in Kayseri city of Turkey. The thermal energy requirement of the flat is nearby 5.3 kW as an average of a winter season. The simplified transient calculations were carried out over a concrete hollow cylindrical column having outer radius of 0.31 m and inner radius of 0.05 m corresponding an averaged column section in the sample flat. The flow temperature was selected between T = 350 and 500 K, which are considerably lower than the temperature of 573 K assumed as a limit for thermal strength of the concrete in the literature. The flow velocity ranges were selected between V = 1.0 and 5.0 m/s. The initial temperature was assumed as 293 K. After the first energy charging process of 23 h, for T = 350 K and V = 1.0 m/s, the total heat flux from the column surfaces into indoors are nearby 5.5 kW. The first charging time required to reach the energy requirement of 5.3 kW is decreased by increasing the flow velocity and temperature. Also for 5.0 m/s–350 K and 5.0 m/s–450 K, this time can decrease to 10 and 4.5 h, respectively. In addition, with 4.0 m/s–360 K or 2.0 m/s–400 K, after the energy charging of 8 h, the energy requirement of 5.3 kW can be provided by the energy discharging of 16 h and the energy charging of 8 h during 7 days. The results are very attractive in terms of the building heating systems of the future.     

The normal subgroup structure of the extended Hecke groups

We consider the extended Hecke groups generated by T(z) = −1/z, S(z) = −1/(z + λ) and R(z) = 1/z with λ ≥ 2. In this paper, firstly, we study the fundamental region of the extended Hecke groups . Then, we determine the abstract group structure of the commutator subgroups , the even subgroup , and the power subgroups of the extended Hecke groups . Also, finally, we give some relations between them.     

Principal congruence subgroups of the Hecke groups and related results

In this paper, first, we determine the quotient groups of the Hecke groups H(λ _q ), where q ≥ 7 is prime, by their principal congruence subgroups H _p (λ _q ) oflevel p, where p is also prime. We deal with the case of q = 7 separately, because of its close relation with the Hurwitz groups. Then, using the obtained results, we find the principal congruence subgroups of the extended Hecke groups $ \overline H $ \overline H (λ _q ) for q ≥ 5 prime. Finally, we show that some of the quotient groups of the Hecke group H(λ _q ) and the extended Hecke group $ \overline H $ \overline H (λ _q ), q ≥ 5 prime, by their principal congruence subgroups H _p (λ _q ) are M*-groups.     

The effect of B2O3 addition to the microstructure and magnetic properties of Ni0.4Zn0.6Fe2O4 ferrite

The effects of 0.01 and 0.1 mol B₂O₃ addition to the microstructure and magnetic properties of a Ni–Zn ferrite composition expressed by a molecular formula of Ni_0.4Zn_0.6Fe₂O₄ were investigated. The toroid-shaped samples prepared by pressing the milled raw materials used in the preparation of the composition were sintered in the range of 1000–1300 °C. The addition of 0.01 mol B₂O₃ increased the grain growth and densification giving rise to reduced intergranular and intragranular porosity due to liquid-phase sintering. The sintered toroid sample at 1300 °C gave the optimum magnetic properties of B_r=170 mT, H_c=0.025 kA/m and a high initial permeability value of μ_i=4000. The increment of the B₂O₃ content to 0.1 mol resulted in a pronounced grain growth and also gave rise to large porosity due to the evaporation of B₂O₃ at higher sintering temperatures. Hence, it resulted in an air-gap effect in the hysteresis curves of these samples.