SL$ _n(\mathbb{Z}) $-normalizer of a principal congruence subgroup

Guangren Sun; Zhengjun Zhao; Guangren Sun; Zhengjun Zhao

doi:10.3934/math.2022295

AIMS Mathematics

2022, Volume 7, Issue 4: 5305-5313. doi: 10.3934/math.2022295

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Research article Special Issues

SL $_n(\mathbb{Z})$ -normalizer of a principal congruence subgroup

Guangren Sun ,
Zhengjun Zhao ^,

School of Mathematics and Physics, Anqing Normal University, Anqing, Anhui 246133, China

Received: 29 September 2021 Revised: 10 December 2021 Accepted: 22 December 2021 Published: 06 January 2022
MSC : 20H05

Let SL $_n(\mathbb{Q})$ be the set of matrices of order $n$ over the rational numbers with determinant equal to 1. We study in this paper a subset $\Lambda$ of SL $_n(\mathbb{Q})$ , where a matrix $B$ belongs to $\Lambda$ if and only if the conjugate subgroup $B\Gamma_q(n)B^{-1}$ of principal congruence subgroup $\Gamma_q(n)$ of lever $q$ is contained in modular group SL $_n(\mathbb{Z})$ . The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether B belongs to $\Lambda$ . We show that LCD can be described by the prime decomposition of $q$ . Generally $\Lambda$ is not a group, and not even a subsemigroup of SL $_n(\mathbb{Q})$ . Nevertheless, for the case $n = 2$ , we present two families of subgroups that are maximal in $\Lambda$ in this paper.

Keywords:

Citation: Guangren Sun, Zhengjun Zhao. SL $_n(\mathbb{Z})$ -normalizer of a principal congruence subgroup[J]. AIMS Mathematics, 2022, 7(4): 5305-5313. doi: 10.3934/math.2022295

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Abstract

1. Introduction

Non-Newtonian fluids are used in a variety of products, including culinary items, personal protective equipment, printing technologies, damping and braking systems, and drag-reducing agents ^[1]. Fluids that are not Newtonian exhibit complex rheological behavior ^[2]. Unlike Newtonian fluids, these materials do not behave linearly between the shear rate or shear and viscosity. Non-Newtonian fluids ^[3,4] are categorized into many groups based on how they react to changes in shear stress or rate. Based on their behavior, non-Newtonian fluids can be classified as shear thinning ^[5,6], shear thickening ^[7,8], thixotropic, visco-plastic, and rheopectic fluids. Following such characteristics, various studies were given by researchers to explore the type of non-Newtonian fluid models and the corresponding daily life applications. In this direction, the Casson fluid model is one of the non-Newtonian fluid models that best capture yield stress and has noteworthy applications in the biomechanics and polymer sectors. Several different goods, including synthetic lubricants, medicinal compounds, paints, coal, china clay, and many others, can be prepared using Casson fluid. Human blood can also be referred to as Casson fluid since it contains a variety of components including human red blood cells, fibrinogen, protein, and globulin in aqueous base plasma. The operative use of Casson fluid in biological treatments, drilling processes, bio-engineering and food processing has captured the attention of researchers like the analysis of diffusion phenomena in stenosed capillary-tissue exchange systems provided by Siddiqui and Mishra ^[9]. In order to depict the blood flow they utilize a modified Casson's fluid description. Through the alteration of a measure known as the retention parameter, the severity of the condition could be evaluated. Another significant aspect of this analysis was the increase of these variables together with the progression of the stenosis. For the investigation of the normal and sick states, the concentration profiles were identified. In the recent past, according to Abolbashari et al. ^[10], engineering and industrial applications have found usage for nano-fluids because of their fantastic thermal conductivity increase. Due to their significance, analyze the fluid non-Newtonian flow with nanoparticles towards the sheet. In special circumstances, the current analytical solution shows a very strong correlation with those of the earlier research that was published. The impact of various flow physical factors were thoroughly explored. Bhattacharyya et al. ^[11] conducted an examination of Casson fluid flow past a shrinking surface with different effects. Self-similar nonlinear ordinary differential equations (ODEs) were obtained. The translated energy and velocity equations both had two exact answers. According to the depicted data, the temperature drops for greater values of radiation parameter, Prandtl number, and power-law exponent. Additionally, the thickness of the thermal boundary layer increases with direct variation in wall temperature. In some instances, temperature overshoot was seen in the temperature field's graphical depiction. Therefore, under some circumstances, heat transfer from the surface happens instead of heat absorption at the surface. In the presence of fluctuating suction, Abd El-Aziz and Yahya ^[12] investigated the impact of Hall current on the Casson fluid flow over a sheet. To obtain non-dimensional flow fields, the resulting equations were analytically solved. The temperature and velocity profiles are studied for various parameter values that entered the problem and were displayed graphically. Physically relevant Nusselt number was sorted out and displayed as graphics. For an axisymmetric Casson stagnation point, Nawaz et al. ^[13] looked at the effects of Joule heating and viscous dissipation. The ensuing boundary value issue was resolved using the homotopy analysis method. The homotopy solutions and the numerical outcomes were compared. The behavior of various parameters was examined. The Casson fluid parameter's resultant effect seems to accelerate the fluid's velocity. The actions of dissipation and Joule heating cause the system to warm up. The thermal and velocity slip effects on a Magnetohydrodynamics (MHD) Casson fluid were studied by Usman et al. ^[14] over a cylinder with Buongiorno's model. The Casson nanofluid model's equations were constructed and reduced by using similarity transformation. The collocation approach was used to arrive at the numerical solution. The graphs that compare the nanofluid fluid flow to the emerging physical parameters provide an analysis of the physical quantities of interest. The effects of Newtonian heating on the Casson fluid flow were examined by Tassaddiq et al. ^[15]. In this study, porous effects for such fluids and MHD effects were also taken into account. Equations involving partial differentials were used to model the fundamental issue. The analytical tool is known as the Laplace transform was used to achieve the "Velocity" and "Temperature" functions. In order to analyze the modeling parameters that were used, graphical representations were used. To verify the facts, numerical calculations were made. The graphical results show that velocity clearly declines for magnetic parameter is intensified and increases as the porosity parameter is increased (conjugate parameter). For all conceivable Casson parameter values, the fluid flow might be controlled. A comparative examination of multiphase flow over a steep channel was carried out by Hussain et al. ^[16]. The combination of microscopic gold particles with Casson fluid creates an intuitive non-Newtonian particulate suspension. Consideration of a uniformly slanted conduit allowed for the consideration of gravitational and magnetic influences. The shear-thinning phenomena is further influenced by heating effects at the channel's edge. For two-phase fluid flow with heat transfer, an exaction was found. In addition, Newtonian-gold particulate flow and Casson-gold particulate flow, which was a non-Newtonian suspension of particle flow, are contrasted. It was discovered that the shear-thinning effects of magnetized Newtonian particle suspension across the inclined channel are more pronounced and experience reduced skin friction. However, it was discovered that Casson fluid was a very helpful suspension for the fabrication and coatings processes, among other things. Additionally, the magnetic field affects the mobility of both phases as a resistive force. Due to the system's significant viscous dissipation, more energy was added. Akolade and Tijani ^[17] discussed the effect of various transport parameters on Casson- nanofluids through a Riga surface. After imposing sufficient similarity factors, the flow equations were derived. To account for all relevant flow characteristics, the spectral quasi-linearization technique (SQLM) was used. Aside from great agreement with previously published work, the approach performance (numerical estimation tools) was examined in terms of CPU time and mistakes. For the current challenge, engineering dimensionless parameters were also supplied. The outcomes demonstrated that the Casson fluid had a lower flow resistance than the Williamson fluid. Furthermore, Casson fluid diffuses and conducts less than Williamson fluid. Siddiqui et al. ^[18] investigated the Casson nanofluid between two spinning and stretchy disks in this study. The effects of thermal radiation, magnetic field, Darcy Forchheimer flow, and heat source were investigated. The shooting strategy was owned to solve the existing equations with the help of MATLAB packages. The temperature profile when the Prandtl number was varied was also discussed. Furthermore, the thermal profile towards the thermophoresis, radiation, and Brownian parameter was investigated. Graphs were used to evaluate the impact of nanoparticle concentration on microorganisms with rising Peclet and Lewis numbers. References ^{[19,20,21,22]} provide an evaluation of the most recent attempts at Casson fluid flow in a variety of configurations.

In this paper, we used the prediction application of artificial intelligence to examine the thermal stagnation point magnetized flow field of non-Newtonian fluid flow due to inclined heated stretched surfaces with thermos-physical effects namely viscous dissipation, mixed convection, heat generation, and variable thermal conductivity. The flow is mathematically formulated for a cylindrical surface and later mathematical reduction is done for thermal flow over a flat surface. The ultimate results are obtained by using the shooting method and are offered as line graphs. The quantity of interest includes the temperature and NN at both flat and cylindrical surfaces. Two different ANN models are constructed to forecast the heat transfer coefficient at surfaces. We are confident that the thermal findings by the use of ANN models will helpful for investigators affiliated with the area of thermal engineering.

2. Flow formulation

Stretching inclined surfaces, such as the cylinder and plate, are used to introduce non-Newtonian fluid flow. By taking into account the magnetic field ^[23,24,25], mixed convection, stagnation point flow ^[26], temperature-dependent conductivity ^[27], viscous dissipation ^[28], and heat generation ^[29] the novelty of the flow field is strengthened. By utilizing the concentration equation, the mass transfer is taken into account. It is assumed that the strength of temperature and concentration is considered higher at surfaces in comparison with strength far away from the surface. The concluding Casson fluid flow equations ^[26] for the given problem are expressed as

$\frac{\partial \left(\widehat{r}\widehat{u}\right)}{\partial \widehat{x}}+\frac{\partial \left(\widehat{r}\widehat{v}\right)}{\partial \widehat{r}} = 0,$

(1)

$\widehat{u}\frac{\partial \widehat{u}}{\partial \widehat{x}}+\widehat{v}\frac{\partial \widehat{u}}{\partial \widehat{r}} = \nu \left(1+\frac{1}{\beta }\right)\left(\frac{{\partial }^{2}\widehat{u}}{\partial {\widehat{r}}^{2}}+\frac{1}{\widehat{r}}\frac{\partial \widehat{u}}{\partial \widehat{r}}\right)+{\widehat{u}}_{e}\frac{{\widehat{u}}_{e}}{\partial \widehat{x}}-\frac{\sigma {{B}_{0}}^{2}}{\rho }(\widehat{u}-{\widehat{u}}_{e}) \\ +{g}_{0}(\widehat{T}-{\widehat{T}}_{\infty }\left){B}_{T}{cos}(\alpha \right)+{g}_{0}(\widehat{C}-{\widehat{C}}_{\infty }\left){B}_{C}{cos}(\alpha \right),$

(2)

$\rho {c}_{\rho }\left(\widehat{u}\frac{\partial \widehat{T}}{\partial \widehat{x}}+\widehat{v}\frac{\partial \widehat{T}}{\partial \widehat{r}}\right) = \frac{1}{\widehat{r}}\frac{\partial }{\partial \widehat{r}}\left(\kappa \frac{\partial \widehat{T}}{\partial \widehat{r}}\right)+\stackrel{̄}{\mu }\left(1+\frac{1}{\beta }\right){\left(\frac{\partial \widehat{u}}{\partial \widehat{r}}\right)}^{2}-\frac{1}{\widehat{r}}\frac{\partial }{\partial \widehat{r}}\left(\widehat{r}\widehat{q}\right)+{Q}_{0}\left(\widehat{T}-{\widehat{T}}_{\infty }\right).$

(3)

$\widehat{u}\frac{\partial \widehat{C}}{\partial \widehat{x}}+\widehat{v}\frac{\partial \widehat{C}}{\partial \widehat{r}} = {D}_{m}\frac{{\partial }^{2}\widehat{C}}{\partial {\widehat{r}}^{2}}.$

(4)

The radioactive flux and thermal conductivity relations are given as:

$\widehat{q} = -\frac{16{\sigma }^{*}{T}_{\infty }^{3}}{3{k}^{*}}\frac{\partial \widehat{T}}{\partial \widehat{r}},$

$\kappa \left(\widehat{T}\right) = {\kappa }_{\infty }\left(1+\varepsilon \frac{\widehat{T}-{\widehat{T}}_{\infty }}{{\widehat{T}}_{w}-{\widehat{T}}_{\infty }}\right),$

(5)

The flow conditions are:

$\widehat{u} = {\widehat{U}}_{w} = a\widehat{x}, , \widehat{C} = {\widehat{C}}_{w}, \widehat{v} = 0\widehat{T} = {\widehat{T}}_{w}, \text{at}~~ \widehat{r} = {R}_{c}, \\ \widehat{u} = {\widehat{u}}_{e} = d\widehat{x}, \widehat{C}\to {\widehat{C}}_{\infty }, \widehat{T}\to {\widehat{T}}_{\infty }, \text{as} ~~ \stackrel{~}{r}\to \infty.$

(6)

Here, $\nu$ is kinematic viscosity, ${R}_{c}$ stands for radius of cylinder, ${B}_{T}$ is the thermal expansion coefficient, ${B}_{C}$ denote the coefficient of solutal expansion, $\rho, \widehat{T}, \widehat{C}, {Q}_{0}, \varepsilon , {\kappa }_{\infty }~~ \text{and}~~ {D}_{m}$ are fluid density, temperature, concentration, heat generation coefficient, small parameter, heat conductivity far from the surface and mass diffusivity respectively. By using NN-ANN model, we are interested in comparing how flow parameters affect the Nusselt number and temperature of Casson fluid flow toward both surfaces. Consequently, we must convert these partial differential equations (PDEs) into corresponding ODEs in order to solve Eq (1) through (4). For this reason, we have own the following set of variables ^[26,27]:

$\eta = \frac{{{{\hat r}^2} - {R_c}^2}}{{2{R_c}}}\sqrt {\frac{{{U_0}}}{{\nu L}}} \, , \widehat{u} = \widehat{x}\frac{{U}_{0}}{L}F'\left(\eta \right) , \widehat{v} = -\frac{{R}_{c}}{\widehat{r}}\sqrt{\frac{\nu {U}_{0}}{L}}F\left(\eta \right) ,$

${\phi }_{n}\left(\eta \right) = \frac{\widehat{C}-{\widehat{C}}_{\infty }}{{\widehat{C}}_{w}-{\widehat{C}}_{\infty }}, {\theta }_{n}\left(\eta \right) = \frac{\widehat{T}-{\widehat{T}}_{\infty }}{{\widehat{T}}_{w}-{\widehat{T}}_{\infty }}.$

(7)

Owning Eq (7), Eqs (1)-(3) turns into

$\left(F'''(1+2{\gamma }_{p}\eta )+2{\gamma }_{p}F''\right)\left(1+1/{\beta }_{p}\right)+FF''-F{'}^{2}+{G}_{T}{\theta }_{n}{cos}\left(\alpha \right)+{G}_{C}{\phi }_{n}{cos}\left(\alpha \right) \\ -{{M}_{p}}^{2}(F'-{A}_{p})+{{A}_{p}}^{2} = 0,$

(8)

$\left({\theta }_{n}''(1+2{\gamma }_{p}\eta )+2{\gamma }_{p}{\theta }_{n}'\right)\left(1+\frac{4}{3}{R}_{p}\right)+{\varepsilon }_{T}\left(({\theta }_{n}{'}^{2}+{\theta }_{n}{\theta }_{n}'')(1+2\eta {\gamma }_{p})+2{\theta }_{n}{\gamma }_{p}{\theta }_{n}'\right)\\ +{Pr}{E}_{n}\left(1+2\eta {\gamma }_{p}\right)F'{'}^{2}\left(1+\frac{1}{{\beta }_{p}}\right)+{Pr}{H}_{p}\theta +{Pr}{\theta }_{n}'F = 0,$

(9)

${\phi }_{n}''(1+2\eta {\gamma }_{p})+2{\gamma }_{p}{\phi }_{n}'+ScF{\phi }_{n}' = 0.$

(10)

and flow conditions reduced as:

$F' = 1, F = 0, {\theta }_{n} = 1, {\phi }_{n} = 1, \text{at}~~ \eta = 0,$

$F' = {A}_{p}, {\theta }_{n} = 0, {\phi }_{n} = 0, \text{at}~~ \eta \to \mathrm{\infty }.$

(11)

The parameters are given as:

${\beta }_{p} = \frac{\stackrel{̄}{\mu }\sqrt{2{\pi }_{c}}}{{\tau }_{r}} , {R}_{p} = \frac{4{\sigma }^{*}{\stackrel{~}{T}}_{\infty }^{3}}{\kappa {k}^{*}} , {\gamma }_{p} = \sqrt{\frac{\upsilon L}{{c}^{2}{U}_{0}}} ,$

${A_p} = \frac{d}{a}, {M}_{p} = \sqrt{\frac{\sigma {B}_{0}^{2}L}{\rho {U}_{0}}} , Pr = \frac{\stackrel{̄}{\mu }{c}_{p}}{\kappa } , {E}_{n} = \frac{{U}_{0}^{2}(\widehat{x}/L{)}^{2}}{{c}_{p}({\widehat{T}}_{w}-{\widehat{T}}_{\infty })} , {G}_{T} = \frac{{g}_{0}{\beta }_{T}({\widehat{T}}_{w}-{\widehat{T}}_{\infty }){L}^{2}}{{U}_{0}\widehat{x}}, {G}_{C} =$

$\frac{{g}_{0}{\beta }_{C}({\widehat{C}}_{w}-{\widehat{C}}_{\infty }){L}^{2}}{{U}_{0}\widehat{x}}, Sc = \frac{\nu }{{D}_{m}}, {H}_{p} = \frac{L{Q}_{0}}{{U}_{0}\rho {c}_{p}}.$

(12)

The Casson fluid parameter, Prandtl number, Eckert number, thermal radiation parameter, Solutal Grashof number, thermal Grashof number, velocities ratio parameter, and Schmidt number are symbolized by ${\beta }_{p}, {Pr}, {E}_{n}, {R}_{p}, {G}_{C}, {G}_{T}, {A}_{p}\text{and}Sc$ . The heat transfer at inclined surfaces is considered a key physical effect and to evaluate heat transfer we considered the Nusselt number. The mathematical relationship ^[26] of NN is:

$\left.\begin{array}{l}N{u}_{\widehat{x}} = \frac{\widehat{x}{q}_{w}}{\kappa ({\widehat{T}}_{w}-{\widehat{T}}_{\infty })}, {q}_{w} = -\kappa \left(1+\frac{16{\sigma }^{*}{\widehat{T}}_{\infty }^{3}}{3{k}^{*}\kappa }\right){\left(\frac{\partial \widehat{T}}{\partial \widehat{r}}\right)}_{\widehat{r} = {R}_{c}}, \\ \frac{N{u}_{\widehat{x}}}{\sqrt{{\boldsymbol{Re}}_{\widehat{x}}}} = -\left(1+\frac{4}{3}{R}_{p}\right){\theta }_{n}'\left(0\right)\end{array}\right\}.$

(13)

3. Numerical scheme

The lower order equations, Eqs (8)-(10), can be used to examine the non-Newtonian fluid flow regime. Due to their coupling and nonlinearity, they are challenging to solve analytically. The flow differential equations are reported using a number of various solution approaches ^{[30,31,32,33,34,35]}, however, the shooting method will be employed to get numerical solution for the present research problem. The name of the shooting technique is taken from an analogy with target shooting: we shoot at the target and watch where it lands; based on the misses, we can modify our aim and shoot again in the hopes that it would hit close to the target. To convert boundary value problems into similar initial value problems, the shooting method was developed. The supportive material in this regard can be accessed in Refs. ^[25,26,27]. Having an initial value system is a vital first step in this regard. We can achieve it by presuming:

${Y}_{1} = F\left(\eta \right), {Y}_{2} = F'\left(\eta \right), {Y}_{3} = F''\left(\eta \right),$

${Y}_{4} = {\theta }_{n}\left(\eta \right), {Y}_{5} = {\theta }_{n}'\left(\eta \right),$

${Y}_{6} = {\phi }_{n}\left(\eta \right), {Y}_{7} = {\phi }_{n}'\left(\eta \right),$

(14)

under Eq (14), Eqs (8)-(10) reduces to

${Y}_{1}' = {Y}_{2},$

${Y}_{2}' = {Y}_{3},$

${Y}_{3}' = \frac{1}{\left(1+\frac{1}{{\beta }_{p}}\right)\left(1+2\eta {\gamma }_{p}\right)}\left[\begin{array}{l}-2{\gamma }_{p}{Y}_{3}\left(1+\frac{1}{{\beta }_{p}}\right)+{Y}_{2}^{2}-{Y}_{1}{Y}_{3}-{G}_{T}{Y}_{4}{cos}\alpha -{G}_{C}{Y}_{6}{cos}\alpha \\ -{{M}_{p}}^{2}({Y}_{2}-{A}_{p})-{{A}_{p}}^{2}\end{array}\right],$

${Y}_{4}' = {Y}_{5},$

${Y}_{5}' = -\frac{1}{\left(1+\frac{4}{3}{R}_{p}\right)\left(1+2\eta {\gamma }_{p}\right)+{\varepsilon }_{T}(1+2\eta {\gamma }_{p}{Y}_{4}}\left[\begin{array}{l}(1+\frac{4}{3}{R}_{p})\left(2{\gamma }_{p}{Y}_{5}\right)+{\varepsilon }_{T}\left(\right(1+2\eta {\gamma }_{p}){Y}_{5}^{2}\\ +2{\gamma }_{p}{Y}_{4}{Y}_{5})+{Pr}{Y}_{1}{Y}_{5}+{Pr}{H}_{p}{Y}_{4}\\ +{Pr}{E}_{n}(1+2\eta {\gamma }_{p})(1+\frac{1}{{\beta }_{p}}){Y}_{3}^{2}\end{array}\right]$

${Y}_{6}' = {Y}_{7},$

${Y}_{7}' = \frac{-Sc{Y}_{1}{Y}_{6}-2{\gamma }_{p}{Y}_{7}}{(1+2\eta {\gamma }_{p})}.$

(15)

The flow equations turns down to:

${Y}_{1} = 0, {Y}_{2} = 1, {Y}_{6} = 1, {Y}_{4} = 1, \text{at}~~ \eta = 0,$

${Y}_{2}\to {A}_{p}, {Y}_{6}\to 0, {Y}_{4}\to 0, \text{as}~~ \eta = \mathrm{\infty }.$

(16)

The self-coding scheme of shooting is carried out in Matlab for the present problem and outcomes are shared in terms of graphs and numerical data.

4. Artificial neural networking model

Mathematical modeling is done for the Casson flow over heated stretched surfaces. The equations for non-Newtonian flow are numerically solved. Both a flat plate and a cylindrical surface are used to evaluate the Nusselt number. The Nusselt number at surfaces is predicted using ANN models in both situations. Researchers firmly think that the multilayer perceptron (MLP) and ANN model ^{[36,37,38,39,40]} have excellent learning capabilities and can be utilized to anticipate a variety of physical occurrences. In MLP networks, three separate layers are utilized. The first layer uses the inputs, while the centrally important layer is referred to as the hidden layer. The output layer, which is the final layer, is where the prediction data is stored. The NN-ANN-I and II are two ANN models that we have created. Whereas NN-ANN-II is designed to provide predictions of Nusselt number values at cylindrical surfaces, NN-ANN-I is built to forecast the Nusselt number at flat surfaces. The five parameters are employed as inputs and the NN is taken into account as an output for both NN-ANN models. Variable thermal conductivity, heat generation, Prandtl number, the Casson fluid, and the thermal radiation parameters are the flow parameters.

Table 1 provides the symbolic information for models. Network architecture is given in Figure 1. We have gathered 80 samples for the five inputs, leading to 80 sample values for the Nusselt number. 15% of the data is used for validation while the remaining 70% is used to train the network. Testing also makes advantage of the 15%. Table 2 provides all available details in this regard. Ten (10) neurons are taken into account in the hidden layer, and the network is trained using the Levenberg-Marquardt method. The transfer functions used in hidden and output have the following expressions:

Table 1. Neural networking detail.

Model	Surface	Input					Output
NN-ANN Model-I	$\left({\gamma }_{p=}0.0\right)$	$\left({\varepsilon }_{T}\right)$	$\left({H}_{p}\right)$	$({Pr})$	$\left({\beta }_{p}\right)$	$\left({R}_{p}\right)$	NN
NN-ANN Model-II	$\left({\gamma }_{p=}0.5\right)$	$\left({\varepsilon }_{T}\right)$	$\left({H}_{p}\right)$	$({Pr})$	$\left({\beta }_{p}\right)$	$\left({R}_{p}\right)$	NN

| Show Table

DownLoad: CSV

Figure 1. Architecture of ANN.

Samples	NN-ANN Model-I	NN-ANN Model-II
Total samples	80	80
Training	56	56
Validation	12	12
Testing	12	12

${\boldsymbol{\varepsilon }}_{\boldsymbol{T}}$	${\boldsymbol{\theta }}_{\boldsymbol{n}}\bf{'}\left(0\right)$		$-(\mathbf{1}+\mathbf{4}/\mathbf{3}{{\boldsymbol{R}}_{\boldsymbol{p}}})$
${\boldsymbol{\varepsilon }}_{\boldsymbol{T}}$	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5(Cylinder)	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5 (Cylinder)
0.1	−0.3868	−0.5200	0.5415	0.7280
0.2	−0.3782	−0.5152	0.5295	0.7213
0.3	−0.3702	−0.5107	0.5183	0.7150
0.4	−0.3625	−0.5064	0.5075	0.7090
0.5	−0.3553	−0.5023	0.4974	0.7032
0.6	−0.3484	−0.4985	0.4878	0.6979
0.7	−0.3419	−0.4947	0.4787	0.6926
0.8	−0.3357	−0.4912	0.4700	0.6877
0.9	−0.3298	−0.4877	0.4617	0.6828
1.0	−0.3187	−0.4813	0.4462	0.6738

$\boldsymbol{P}\boldsymbol{r}$	${\boldsymbol{\theta }}_{\boldsymbol{n}}\boldsymbol{'}\left(\mathbf{0}\right)$		$-(\mathbf{1}+\mathbf{4}/\mathbf{3}{{\boldsymbol{R}}_{\boldsymbol{p}}})$
$\boldsymbol{P}\boldsymbol{r}$	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5(Cylinder)	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5 (Cylinder)
0.7	−0.4278	−0.5235	0.5989	0.7329
0.8	−0.4588	−0.5283	0.6423	0.7396
0.9	−0.4879	−0.5333	0.6831	0.7466
1.0	−0.5152	−0.5385	0.7213	0.7539
2.0	−0.5652	−0.5492	0.7913	0.7689
3.0	−0.6114	−0.5601	0.8560	0.7841
4.0	−0.6537	−0.5708	0.9152	0.7991
5.0	−0.6931	−0.5815	0.9703	0.8141
6.0	−0.7301	−0.5914	1.0221	0.8280
7.0	−0.8893	−0.6333	1.245	0.8866

${\boldsymbol{\beta }}_{\boldsymbol{p}}$	${\boldsymbol{\theta }}_{\boldsymbol{n}}\boldsymbol{'}\left(\mathbf{0}\right)$		$-(\mathbf{1}+\mathbf{4}/\mathbf{3}{{\boldsymbol{R}}_{\boldsymbol{p}}})$
${\boldsymbol{\beta }}_{\boldsymbol{p}}$	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5(Cylinder)	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5 (Cylinder)
0.1	−0.4052	−0.6093	0.5673	0.8530
0.2	−0.3953	−0.6095	0.5534	0.8533
0.3	−0.3911	−0.6089	0.5475	0.8525
0.4	−0.3888	−0.6085	0.5443	0.8519
0.5	−0.3863	−0.6082	0.5408	0.8515
0.6	−0.3855	−0.6080	0.5397	0.8512
0.7	−0.3850	−0.6078	0.5390	0.8509
0.8	−0.3845	−0.6077	0.5383	0.8508
0.9	−0.3840	−0.6076	0.5376	0.8506
1.0	−0.3838	−0.6075	0.5373	0.8505

${\boldsymbol{R}}_{\boldsymbol{p}}$	${\boldsymbol{\theta }}_{\boldsymbol{n}}\boldsymbol{'}\left(\mathbf{0}\right)$		$-(\mathbf{1}+\mathbf{4}/\mathbf{3}{{\boldsymbol{R}}_{\boldsymbol{p}}})$
${\boldsymbol{R}}_{\boldsymbol{p}}$	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5(Cylinder)	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5 (Cylinder)
0.1	−0.5235	−0.4278	0.5933	0.4848
0.2	−0.5215	−0.4058	0.6605	0.5140
0.3	−0.5200	−0.3868	0.7280	0.5415
0.4	−0.5188	−0.3703	0.7954	0.5678
0.5	−0.5179	−0.3557	0.8631	0.5929
0.6	−0.5171	−0.3428	0.9308	0.6170
0.7	−0.5166	−0.3314	0.9988	0.6407
0.8	−0.5161	−0.3212	1.0667	0.6638
0.9	−0.5157	−0.3121	1.1345	0.6867
1.0	−0.5154	−0.3038	1.2026	0.7089

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[4]	J. E. Humphreys, Arithmetic groups, Springer-Verlag, Berlin, Lecture Notes in Mathematics.789, 1980. https://dx.doi.org/10.1007/BFb0094567
[5]	J. Lehner, M. Newman, Weierstrass points on $\Gamma_0(N)$ , Ann. Math., 79 (1964), 360–368. https://dx.doi.org/10.2307/1970550 doi: 10.2307/1970550
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[9]	H. J. S. Smith, On systems of linear indeterminate equations and congruences, Philos. Trans., 151 (1861), 293–326. https://dx.doi.org/10.1080/03081081003709819 doi: 10.1080/03081081003709819

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${\boldsymbol{\varepsilon }}_{\boldsymbol{T}}$	${\boldsymbol{\theta }}_{\boldsymbol{n}}\boldsymbol{'}\left(\mathbf{0}\right)$		$-(\mathbf{1}+\mathbf{4}/\mathbf{3}{{\boldsymbol{R}}_{\boldsymbol{p}}})$
${\boldsymbol{\varepsilon }}_{\boldsymbol{T}}$	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5(Cylinder)	${\gamma }_{p}=0$ (Plate)	${\gamma }_{p}=0$ .5 (Cylinder)
0.1	−1.3699	−0.9213	1.9179	1.2898
0.2	−1.3414	−0.9107	1.8780	1.2750
0.3	−1.3145	−0.9005	1.8403	1.2607
0.4	−1.2890	−0.8908	1.8046	1.2471
0.5	−1.2649	−0.8815	1.7709	1.2341
0.6	−1.2420	−0.8726	1.7388	1.2216
0.7	−1.2203	−0.8641	1.7084	1.2097
0.8	−1.1995	−0.8559	1.6793	1.1983
0.9	−1.1798	−0.8480	1.6517	1.1872
1.0	−1.1609	−0.8404	1.6253	1.1766

AIMS Mathematics

SLn(Z) _n(\mathbb{Z}) -normalizer of a principal congruence subgroup

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SL $_n(\mathbb{Z})$ -normalizer of a principal congruence subgroup