Let and be two free semigroups. We define external direct product of two free semigroups as an ordered pair of words such that and .We investigate the presentations of external direct product of free semigroups, state and prove under some conditions that the external direct product of two finitely generated free semigroups is finitely generated, also the external direct product of two finitely presented free semigroups is finitely presented.
{"title":"FREE SEMIGROUP PRESENTATIONS","authors":"Nwawuru Francis","doi":"10.18063/ijmp.v1i1.671","DOIUrl":"https://doi.org/10.18063/ijmp.v1i1.671","url":null,"abstract":"Let and be two free semigroups. We define external direct product of two free semigroups as an ordered pair of words such that and .We investigate the presentations of external direct product of free semigroups, state and prove under some conditions that the external direct product of two finitely generated free semigroups is finitely generated, also the external direct product of two finitely presented free semigroups is finitely presented. ","PeriodicalId":330194,"journal":{"name":"International Journal of Mathematical Physics","volume":"39 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116343973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to exhaust the possibilities offered by the scaled particle theory as far as possible and to confirm the reliability of the virial coefficients found in the literature, especially the estimated ones: B i for i > 11. In a previous article (J.Math.Phys.36,201,1995) a theoretical equation of state for the hard sphere fluid was derived making use of the ideas of the so called scaled particle theory which has been developed by Reiss et al.(J.Chem.Phys.31,369,1959). It contains two parameters which could be calculated. The equation of state agrees with the simulation data up to high densities, where the fluid is metastable. The derivation was besed on a generalized series expansion. The virial coefficients B 2 , B 3 and B 4 are exactly reproduced and B 5 , B 6 and B 7 to within small deviations, but the higher ones up to B 18 are systematically and significantly smaller than the values found in the literature. The scaled particle theory yields a number of equations of which only four were used. In this paper we make use of seven equations to calculate the compressibility factors of the fluid. They agree with the simulation data slightly better than those yielded by the old equation. Moreover, the differences between the calculated virial coefficients B i and those found in the literature up to B 18 are very small (less than 4 percent).
本文的目的是尽可能地穷尽尺度粒子理论提供的可能性，并证实文献中发现的维里系数的可靠性，特别是估计的维里系数:i bbb11。在之前的一篇文章(j.m math . phys.36,201,1995)中，利用Reiss等人(j.m im . phys.31,369,1959)提出的所谓尺度粒子理论，推导出了硬球流体的理论状态方程。它包含两个可以计算的参数。在高密度流体为亚稳态时，状态方程与模拟数据一致。推导是基于一个广义级数展开。维氏系数b2、b3和b4得到了精确的再现，b5、b6和b7在很小的偏差范围内，但b18之前的维氏系数比文献中发现的值系统地和显著地小。缩放粒子理论产生了许多方程，其中只有四个被使用。本文用七个方程来计算流体的压缩系数。它们与模拟数据的一致性略好于旧方程的结果。此外，计算出的病毒系数bi与文献中发现的直到b18之间的差异非常小(小于4%)。
{"title":"Statistical mechanics of hard spheres: the scaled particle theory of the hard sphere fluid revisited","authors":"B. Baeyens","doi":"10.18063/ijmp.v1i1.757","DOIUrl":"https://doi.org/10.18063/ijmp.v1i1.757","url":null,"abstract":"The aim of this paper is to exhaust the possibilities offered by the scaled particle theory as far as possible and to confirm the reliability of the virial coefficients found in the literature, especially the estimated ones: B i for i > 11. In a previous article (J.Math.Phys.36,201,1995) a theoretical equation of state for the hard sphere fluid was derived making use of the ideas of the so called scaled particle theory which has been developed by Reiss et al.(J.Chem.Phys.31,369,1959). It contains two parameters which could be calculated. The equation of state agrees with the simulation data up to high densities, where the fluid is metastable. The derivation was besed on a generalized series expansion. The virial coefficients B 2 , B 3 and B 4 are exactly reproduced and B 5 , B 6 and B 7 to within small deviations, but the higher ones up to B 18 are systematically and significantly smaller than the values found in the literature. The scaled particle theory yields a number of equations of which only four were used. In this paper we make use of seven equations to calculate the compressibility factors of the fluid. They agree with the simulation data slightly better than those yielded by the old equation. Moreover, the differences between the calculated virial coefficients B i and those found in the literature up to B 18 are very small (less than 4 percent).","PeriodicalId":330194,"journal":{"name":"International Journal of Mathematical Physics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115623526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper deals with spectral dynamic analysis of free torsional vibration of doubly symmetric thin-walled beams of open section. Spectral frequency equation is derived in this paper for the case of rotationally restrained doubly-symmetric thin-walled beam with one end rotationally restrained and transversely restrained at the other end. The resulting transcendental frequency equation with appropriate boundary conditions is derived and is solved for varying values of warping parameter and the rotational and transverse restraint parameter. The influence of rotational restraint parameter, transverse restraint parameter and warping parameter on the free torsional vibration frequencies is investigated in detail. A MATLAB computer program is developed to solve the spectral frequency equation derived in this paper. Numerical results for natural frequencies for various values of rotational and transverse restraint parameters for various values of warping parameter are obtained and presented in both tabular as well as graphical form showing the influence of these parameters on the first fundamental torsional frequency parameter.
{"title":"Spectral Dynamic Analysis of Torsional Vibrations of Thin-Walled Open Section Beams Restrained Against Warping at One End and Transversely Restrained at the Other End","authors":"Chellapilla Kameswara Rao 1, Lokavarapu Bhaskara Rao 2","doi":"10.18063/ijmp.v1i1.746","DOIUrl":"https://doi.org/10.18063/ijmp.v1i1.746","url":null,"abstract":"The present paper deals with spectral dynamic analysis of free torsional vibration of doubly symmetric thin-walled beams of open section. Spectral frequency equation is derived in this paper for the case of rotationally restrained doubly-symmetric thin-walled beam with one end rotationally restrained and transversely restrained at the other end. The resulting transcendental frequency equation with appropriate boundary conditions is derived and is solved for varying values of warping parameter and the rotational and transverse restraint parameter. The influence of rotational restraint parameter, transverse restraint parameter and warping parameter on the free torsional vibration frequencies is investigated in detail. A MATLAB computer program is developed to solve the spectral frequency equation derived in this paper. Numerical results for natural frequencies for various values of rotational and transverse restraint parameters for various values of warping parameter are obtained and presented in both tabular as well as graphical form showing the influence of these parameters on the first fundamental torsional frequency parameter.","PeriodicalId":330194,"journal":{"name":"International Journal of Mathematical Physics","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132924050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Drucker’s Associated Law of yielding and gradient plasticity cannot always adequately reflect the processes of stress-strain state in deformable continua. Aim of this issue explains these effects via theory of complex loading processes by Ilyushin.
{"title":"On Some Properties of Metals Under Complex Loading","authors":"Abirov Rustam","doi":"10.18063/ijmp.v1i1.751","DOIUrl":"https://doi.org/10.18063/ijmp.v1i1.751","url":null,"abstract":"Drucker’s Associated Law of yielding and gradient plasticity cannot always adequately reflect the processes of stress-strain state in deformable continua. Aim of this issue explains these effects via theory of complex loading processes by Ilyushin.","PeriodicalId":330194,"journal":{"name":"International Journal of Mathematical Physics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129826122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of present work is to obtain the approximate solution of fractional model for the electrical RL circuit by using quadratic Legendre multiwavelet method (QLMWM). The beauty of the paper is convergence theorem and mean square error analysis, which shows that our approximate solution converges very rapidly to the exact solution and the numerical solution is compared with the classical solution and Legendre wavelets method (LWM) solution, which is much closer to the exact solution. The fractional integration is described in the Riemann-Liouville sense. The results are shows that the method is very effective and simple. In addition, using plotting tools, we compare approximate solutions of each equation with its classical solution and LWM .
{"title":"A new approach for solving fractional RL circuit model through quadratic Legendre multi-wavelets","authors":"N. S. Chauhan","doi":"10.18063/ijmp.v1i1.724","DOIUrl":"https://doi.org/10.18063/ijmp.v1i1.724","url":null,"abstract":"The aim of present work is to obtain the approximate solution of fractional model for the electrical RL circuit by using quadratic Legendre multiwavelet method (QLMWM). The beauty of the paper is convergence theorem and mean square error analysis, which shows that our approximate solution converges very rapidly to the exact solution and the numerical solution is compared with the classical solution and Legendre wavelets method (LWM) solution, which is much closer to the exact solution. The fractional integration is described in the Riemann-Liouville sense. The results are shows that the method is very effective and simple. In addition, using plotting tools, we compare approximate solutions of each equation with its classical solution and LWM .","PeriodicalId":330194,"journal":{"name":"International Journal of Mathematical Physics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134372227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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