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# zbMATH — the first resource for mathematics

## Zhang, Tusheng S.

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 Author ID: zhang.tusheng-s Published as: Zhang, T.; Zhang, T. S.; Zhang, T.-S.; Zhang, Tu-Sheng; Zhang, Tu-sheng; Zhang, Tusheng; Zhang, Tusheng S. Homepage: http://www.maths.manchester.ac.uk/~tzhang/ External Links: ResearchGate · dblp · GND
 Documents Indexed: 169 Publications since 1989, including 6 Books
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#### Co-Authors

 24 single-authored 30 Øksendal, Bernt Karsten 20 Röckner, Michael 15 Ubœ, Jan 11 Zhai, Jianliang 10 Chen, Zhen-Qing 9 Holden, Helge 9 Lindstrøm, Tom L. 8 Mohammed, Salah-Eldin A. 7 Fang, Shizan 6 Hu, Yaozhong 5 Albeverio, Sergio A. 5 Fitzsimmons, Patrick J. 5 Lyons, Terence John 5 Sulem, Agnès 5 Xu, Tiange 4 Dong, Zhao 4 Kuwae, Kazuhiro 4 Wang, Feng-Yu 4 Yang, Xue 3 Shang, Shijie 3 Wang, Ran 3 Yue, Wen 2 Benth, Fred Espen 2 Dalang, Robert C. 2 Di Nunno, Giulia 2 Matoussi, Anis 2 Proske, Frank Norbert 2 Ran, Qikang 2 Takeda, Masayoshi 2 Wu, Jianglun 2 Xu, Lihu 2 Yan, Jia-An 2 Yang, Juan 2 Zhang, Rangrang 2 Zheng, Wuting 1 Aida, Shigeki 1 Biagini, Francesca 1 Bo, Lijun 1 Bogachev, Vladimir Igorevich 1 Doney, Ronald Arthur 1 Gjerde, Jon 1 Gjessing, Håkon K. 1 Hofmanová, Martina 1 Hu, Ying 1 Imkeller, Peter 1 Khoshnevisan, Davar 1 Ledoux, Michel 1 Liu, Kai 1 Lunt, John 1 Ma, Zhi-Ming 1 Menoukeu Pamen, Olivier 1 Meyer-Brandis, Thilo 1 Nilssen, Torstein K. 1 Nualart, David 1 Pardoux, Etienne 1 Posilicano, Andrea 1 Qian, Zhongmin M. 1 Sabbagh, Wissal 1 Wang, Chen 1 Wen, Chunhui 1 Winter, Matthias 1 Xiang, Kai-Nan 1 Xiao, Yimin 1 Xiong, Jie 1 Yang, Saisai 1 Ying, Jiangang 1 Zhang, Qi 1 Zhang, Xicheng 1 Zhao, Huaizhong 1 Zheng, Weian 1 Zhou, Xunyu
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#### Serials

 20 Stochastic Processes and their Applications 11 The Annals of Probability 11 Potential Analysis 10 Journal of Functional Analysis 7 Stochastics and Stochastics Reports 6 Infinite Dimensional Analysis, Quantum Probability and Related Topics 5 Probability Theory and Related Fields 4 Journal of Differential Equations 4 Journal of Theoretical Probability 4 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 3 Applied Mathematics and Optimization 3 Osaka Journal of Mathematics 3 Stochastic Analysis and Applications 3 Acta Mathematicae Applicatae Sinica. English Series 3 The Annals of Applied Probability 3 Journal de Mathématiques Pures et Appliquées. Neuvième Série 3 Stochastics and Dynamics 2 Journal of Mathematical Analysis and Applications 2 Proceedings of the American Mathematical Society 2 Statistics & Probability Letters 2 Communications in Partial Differential Equations 2 Bernoulli 2 Stochastics 2 Communications in Mathematics and Statistics 2 Probability and its Applications 1 Advances in Applied Probability 1 Acta Mathematica Sinica 1 Bulletin of the London Mathematical Society 1 Journal of Computational and Applied Mathematics 1 Mathematische Annalen 1 Mathematics of Operations Research 1 Memoirs of the American Mathematical Society 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 SIAM Journal on Control and Optimization 1 Comptes Rendus Mathématiques de l’Académie des Sciences 1 Journal of Mathematics. Wuhan University 1 Journal of Time Series Analysis 1 Chinese Annals of Mathematics. Series B 1 Forum Mathematicum 1 Science in China. Series A 1 SIAM Journal on Mathematical Analysis 1 Electronic Journal of Probability 1 Discrete and Continuous Dynamical Systems. Series B 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Interdisciplinary Mathematical Sciences 1 Abel Symposia 1 Communications on Stochastic Analysis 1 International Journal of Stochastic Analysis 1 Universitext
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#### Fields

 161 Probability theory and stochastic processes (60-XX) 50 Partial differential equations (35-XX) 34 Potential theory (31-XX) 16 Systems theory; control (93-XX) 11 Operator theory (47-XX) 9 Dynamical systems and ergodic theory (37-XX) 7 Fluid mechanics (76-XX) 4 Global analysis, analysis on manifolds (58-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 General and overarching topics; collections (00-XX) 3 Ordinary differential equations (34-XX) 3 Functional analysis (46-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Numerical analysis (65-XX) 1 Measure and integration (28-XX) 1 Statistics (62-XX) 1 Quantum theory (81-XX) 1 Biology and other natural sciences (92-XX)

#### Citations contained in zbMATH

132 Publications have been cited 1,837 times in 1,326 Documents Cited by Year
Stochastic calculus for fractional Brownian motion and applications. Zbl 1157.60002
Biagini, Francesca; Hu, Yaozhong; Øksendal, Bernt; Zhang, Tusheng
2008
Stochastic partial differential equations. A modeling, white noise functional approach. Zbl 0860.60045
Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng
1996
A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Zbl 1081.60043
Fang, Shizan; Zhang, Tusheng
2005
Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Zbl 1119.60057
Röckner, Michael; Zhang, Tusheng
2007
Parabolic SPDEs driven by Poisson white noise. Zbl 0934.60055
Albeverio, Sergio; Wu, Jiang-Lun; Zhang, Tu-Sheng
1998
Stochastic partial differential equations. A modeling, white noise functional approach. 2nd ed. Zbl 1198.60005
Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng
2010
Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Zbl 1217.93183
Øksendal, Bernt; Sulem, Agnès; Zhang, Tusheng
2011
The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations. Zbl 1169.60014
Mohammed, Salah-Eldin A.; Zhang, Tusheng; Zhao, Huaizhong
2008
Large deviations and support theorem for diffusion processes via rough paths. Zbl 1075.60510
Ledoux, M.; Qian, Z.; Zhang, T.
2002
Decomposition of Dirichlet processes and its applications. Zbl 0804.60044
Lyons, T. J.; Zhang, T. S.
1994
Absolute continuity of symmetric Markov processes. Zbl 1053.60084
Chen, Z.-Q.; Fitzsimmons, P. J.; Takeda, M.; Ying, J.; Zhang, T.-S.
2004
Local times of fractional Brownian sheets. Zbl 1009.60024
Xiao, Yimin; Zhang, Tusheng
2002
Uniqueness of generalized Schrödinger operators and applications. Zbl 0779.35028
Röckner, Michael; Zhang, Tu-Sheng
1992
Uniqueness of generalized Schrödinger operators. II. Zbl 0799.35053
Röckner, Michael; Zhang, Tu-Sheng
1994
White noise driven SPDEs with reflection: Strong Feller properties and Harnack inequalities. Zbl 1196.60121
Zhang, Tusheng
2010
White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Zbl 1175.60068
Xu, Tiange; Zhang, Tusheng
2009
Stochastic calculus for symmetric Markov processes. Zbl 1142.31005
Chen, Z.-Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T.-S.
2008
Large deviations for stochastic tamed 3D Navier-Stokes equations. Zbl 1195.60093
Röckner, Michael; Zhang, Tusheng; Zhang, Xicheng
2010
On the small time asymptotics of diffusion processes on path groups. Zbl 0993.60026
Aida, Shigeki; Zhang, Tusheng
2002
The Burgers equation with a noisy force and the stochastic heat equation. Zbl 0804.35158
Holden, Helge; Lindstrøm, Tom; Øksendal, Bernt; Ubøe, J.; Zhang, T.-S.
1994
A variational approach to the construction and Malliavin differentiability of strong solutions of SDEs. Zbl 1282.60057
Menoukeu-Pamen, Olivier; Meyer-Brandis, Thilo; Nilssen, Torstein; Proske, Frank; Zhang, Tusheng
2013
Perturbation of symmetric Markov processes. Zbl 1137.31004
Chen, Z.-Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T.-S.
2008
Stochastic partial differential equations driven by multiparameter fractional white noise. Zbl 0982.60054
Hu, Yaozhong; Øksendal, Bernt; Zhang, Tusheng
2000
Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels. Zbl 0914.60045
Lunt, John; Lyons, T. J.; Zhang, T. S.
1998
Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise. Zbl 1306.60095
Wang, Ran; Zhang, Tusheng
2015
The Itô-Ventzell formula and forward stochastic differential equations driven by Poisson random measures. Zbl 1118.60052
Øksendal, Bernt; Zhang, Tusheng
2007
Global flows for stochastic differential equations without global Lipschitz conditions. Zbl 1128.60046
Fang, Shizan; Imkeller, Peter; Zhang, Tusheng
2007
Girsanov and Feynman-Kac type transformations for symmetric Markov processes. Zbl 1004.60077
Chen, Zhen-Qing; Zhang, Tu-Sheng
2002
On the small time asymptotics of diffusion processes on Hilbert spaces. Zbl 1044.60071
Zhang, T. S.
2000
A moderate deviation principle for 2-D stochastic Navier-Stokes equations. Zbl 1310.60100
Wang, Ran; Zhai, Jianliang; Zhang, Tusheng
2015
Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Zbl 0986.60056
Øksendal, Bernt; Zhang, Tusheng
2001
Stochastic 3D tamed Navier-Stokes equations: existence, uniqueness and small time large deviation principles. Zbl 1241.60032
Röckner, Michael; Zhang, Tusheng
2012
Large deviation principles for 2-D stochastic Navier-Stokes equations driven by Lévy processes. Zbl 1210.60071
Xu, Tiange; Zhang, Tusheng
2009
Perturbed Skorohod equations and perturbed reflected diffusion processes. Zbl 1061.60056
Doney, R. A.; Zhang, Tusheng
2005
Girsanov transform for symmetric diffusions with infinite dimensional state space. Zbl 0776.60093
Albeverio, S.; Röckner, M.; Zhang, T. S.
1993
A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. Zbl 1356.60100
Dong, Zhao; Xiong, Jie; Zhai, Jianliang; Zhang, Tusheng
2017
Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. Zbl 1344.60030
Zhai, Jianliang; Zhang, Tusheng
2015
Optimal control with partial information for stochastic Volterra equations. Zbl 1214.49033
Øksendal, Bernt; Zhang, Tusheng
2010
General fractional multiparameter white noise theory and stochastic partial differential equations. Zbl 1067.35161
Hu, Yaozhong; Øksendal, Bernt; Zhang, Tusheng
2004
A class of stochastic differential equations with non-Lipschitzian coefficients: Pathwise uniqueness and no explosion. Zbl 1031.60051
Fang, Shizan; Zhang, Tusheng
2003
Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection. Zbl 1306.93078
Øksendal, Bernt; Sulem, Agnès; Zhang, Tusheng
2014
Gradient estimates for stochastic evolution equations with non-Lipschitz coefficients. Zbl 1185.60065
Wang, Feng-Yu; Zhang, Tu-Sheng
2010
Isotropic stochastic flow of homeomorphisms on $$S^{d}$$ for the critical Sobolev exponent. Zbl 1095.60020
Fang, Shizan; Zhang, Tusheng
2006
Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields. Zbl 1090.60057
Øksendal, Bernt; Proske, Frank; Zhang, Tusheng
2005
On general perturbations of symmetric Markov processes. Zbl 1176.60070
Chen, Z.-Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T.-S.
2009
On the strong solutions of one-dimensional stochastic differential equations with reflecting boundary. Zbl 0796.60062
Zhang, Tu-Sheng
1994
Absolute continuity of the law of the solution of a parabolic SPDE. Zbl 0777.60046
Pardoux, Etienne; Zhang, Tusheng
1993
Large deviations for the Brownian motion on loop groups. Zbl 1003.60028
Fang, Shizan; Zhang, Tusheng
2001
On the small time behavior of Ornstein-Uhlenbeck processes with unbounded linear drifts. Zbl 0932.60071
Fang, S.; Zhang, T. S.
1999
Markov uniqueness for a class of infinite dimensional Dirichlet operators. Zbl 0827.31007
Albeverio, Sergio; Röckner, Michael; Zhang, Tu-Sheng
1993
Characterizations of the white noise test functionals and Hida distributions. Zbl 0765.60031
Zhang, Tu-Sheng
1992
Large deviations for stochastic models of two-dimensional second grade fluids. Zbl 1370.60049
Zhai, Jianliang; Zhang, Tusheng
2017
Time-reversal and elliptic boundary value problems. Zbl 1187.60064
Chen, Zhen-Qing; Zhang, Tusheng
2009
Large deviations for stochastic systems with memory. Zbl 1136.60021
Mohammed, Salah-Eldin A.; Zhang, Tusheng
2006
Generalized Feynman-Kac semigroups, associated quadratic forms and asymptotic properties. Zbl 0984.31006
Zhang, T. S.
2001
Large deviations for SPDEs of jump type. Zbl 1327.60074
Yang, Xue; Zhai, Jianliang; Zhang, Tusheng
2015
Strong convergence of Wong-Zakai approximations of reflected SDEs in a multidimensional general domain. Zbl 1311.60066
Zhang, Tusheng
2014
Log-Harnack inequality for mild solutions of SPDEs with multiplicative noise. Zbl 1285.60065
Wang, Feng-Yu; Zhang, Tusheng
2014
Improved rectangular method on stochastic Volterra equations. Zbl 1221.65023
Wen, C. H.; Zhang, T. S.
2011
SPDEs driven by space-time white noise in high dimensions: Absolute continuity of the law and convergence of solutions. Zbl 1030.60042
Zhang, T. S.; Zheng, W. A.
2002
Existence and uniqueness of invariant measures: An approach via sectorial forms. Zbl 0953.31003
Bogachev, V.; Röckner, M.; Zhang, T. S.
2000
Convergence of non-symmetric Dirichlet processes. Zbl 0888.31005
Lyons, Terry; Zhang, Tusheng
1996
The pressure equation for fluid flow in a stochastic medium. Zbl 0834.60068
Holden, H.; Lindstrøm, T.; Øksendal, B.; Ubøe, J.; Zhang, T.-S.
1995
Exponential mixing for stochastic model of two-dimensional second grade fluids. Zbl 1329.60223
Wang, Ran; Zhai, Jianliang; Zhang, Tusheng
2016
SPH simulation of acoustic waves: effects of frequency, sound pressure, and particle spacing. Zbl 1394.76105
Zhang, Y. O.; Zhang, T.; Ouyang, H.; Li, T. Y.
2015
Large deviations for stochastic nonlinear beam equations. Zbl 1118.60021
Zhang, Tusheng
2007
Approximation of arbitrary Dirichlet processes by Markov chains. Zbl 0898.31009
Ma, Zhiming; Röckner, Michael; Zhang, Tu-Sheng
1998
Finite dimensional approximation of diffusion processes on infinite dimensional spaces. Zbl 0885.60066
Röckner, M.; Zhang, T. S.
1996
Stochastic boundary value problems: A white noise functional approach. Zbl 0792.60055
Holden, Helge; Lindstrøm, Tom; Øksendal, Bernt; Ubøe, Jan; Zhang, Tu-Sheng
1993
Markov uniqueness and its applications to martingale problems, stochastic differential equations and stochastic quantization. Zbl 0767.31009
Albeverio, Sergio; Röckner, Michael; Zhang, Tu-Sheng
1993
Quasilinear parabolic stochastic partial differential equations: existence, uniqueness. Zbl 1372.60091
Hofmanová, Martina; Zhang, Tusheng
2017
White noise driven SPDEss with two reflecting walls. Zbl 1235.60081
Zhang, Tusheng; Yang, Juan
2011
The Wick product. Zbl 0820.60048
Gjessing, H.; Holden, Helge; Lindstrøm, Tom; Øksendal, Bernt; Ubøe, J.; Zhang, T.-S.
1993
Stochastic Burgers equation with random initial velocities: a Malliavin calculus approach. Zbl 1274.60205
Mohammed, Salah-Eldin A.; Zhang, Tusheng
2013
On the small time asymptotics of the two-dimensional stochastic Navier-Stokes equations. Zbl 1196.60119
Xu, Tiange; Zhang, Tusheng
2009
Small time asymptotics for Fleming-Viot processes. Zbl 1080.60504
Xiang, Kai-Nan; Zhang, Tu-Sheng
2005
The stochastic Wick-type Burgers equation. Zbl 0823.60050
Holden, Helge; Lindstrøm, Tom; Øksendal, Bernt; Ubøe, J.; Zhang, T.-S.
1995
Stability properties of stochastic partial differential equations. Zbl 0829.60053
Lindstrøm, Tom; Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng
1995
Efficient SPH simulation of time-domain acoustic wave propagation. Zbl 1403.76173
Zhang, Y. O.; Zhang, T.; Ouyang, H.; Li, T. Y.
2016
Absolute continuity of the laws of perturbed diffusion processes and perturbed reflected diffusion processes. Zbl 1325.60131
Yue, Wen; Zhang, Tusheng
2015
Existence and uniqueness of invariant measures for SPDEs with two reflecting walls. Zbl 1308.60081
Yang, Juan; Zhang, Tusheng
2014
Hölder continuity of solutions of SPDEs with reflection. Zbl 1277.60107
Dalang, Robert C.; Zhang, Tusheng
2013
Backward stochastic differential equations with respect to general filtrations and applications to insider finance. Zbl 1331.91209
Øksendal, Bernt; Zhang, Tusheng
2012
Large deviations for invariant measures of SPDEs with two reflecting walls. Zbl 1248.60077
Zhang, Tusheng
2012
A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients. Zbl 1242.60072
Zhang, Tusheng
2011
Systems of stochastic partial differential equations with reflection: existence and uniqueness. Zbl 1218.60059
Zhang, Tusheng
2011
Existence and uniqueness of bounded weak solutions of a semilinear parabolic PDE. Zbl 1205.60116
Ran, Qikang; Zhang, Tusheng
2010
Dynamics of stochastic 2D Navier-Stokes equations. Zbl 1213.60107
Mohammed, Salah; Zhang, Tusheng
2010
Nonrobustness of some impulse control problems with respect to intervention costs. Zbl 1012.93066
Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng
2002
On the small time large deviations of diffusion processes on configuration spaces. Zbl 1047.60081
Zhang, T. S.
2001
Convergence of operator semigroups generated by elliptic operators. Zbl 0906.47027
Röckner, Michael; Zhang, Tu-Sheng
1997
Note on convergence of Dirichlet processes. Zbl 0793.31006
Lyons, T. J.; Zhang, T. S.
1993
The stochastic Volterra equation. Zbl 0783.60060
Øksendal, Bernt; Zhang, Tu-Sheng
1993
Moderate deviations for stochastic models of two-dimensional second grade fluids. Zbl 1396.60075
Zhai, Jianliang; Zhang, Tusheng; Zheng, Wuting
2018
Backward doubly SDEs and semilinear stochastic PDEs in a convex domain. Zbl 1372.60085
Matoussi, Anis; Sabbagh, Wissal; Zhang, Tusheng
2017
Estimates of heat kernels with Neumann boundary conditions. Zbl 1262.35115
Yang, Xue; Zhang, Tusheng
2013
Probabilistic representations and hyperbound estimates for semigroups. Zbl 1043.47508
Röckner, M.; Zhang, T. S.
1999
Asymptotic properties of additive functionals of Brownian motion. Zbl 0887.60077
Takeda, Masayoshi; Zhang, Tusheng
1997
A stability property of the stochastic heat equation. Zbl 0847.60042
Ubøe, Jan; Zhang, Tusheng
1995
Global solutions to stochastic reaction-diffusion equations with super-linear drift and multiplicative noise. Zbl 1418.60072
Dalang, Robert C.; Khoshnevisan, Davar; Zhang, Tusheng
2019
Large deviation principles for first-order scalar conservation laws with stochastic forcing. Zbl 07200530
Dong, Zhao; Wu, Jiang-Lun; Zhang, Rangrang; Zhang, Tusheng
2020
Global solutions to stochastic reaction-diffusion equations with super-linear drift and multiplicative noise. Zbl 1418.60072
Dalang, Robert C.; Khoshnevisan, Davar; Zhang, Tusheng
2019
Strong solutions for a stochastic model of two-dimensional second grade fluids driven by Lévy noise. Zbl 1403.60058
Shang, Shijie; Zhai, Jianliang; Zhang, Tusheng
2019
Moderate deviations for stochastic models of two-dimensional second grade fluids. Zbl 1396.60075
Zhai, Jianliang; Zhang, Tusheng; Zheng, Wuting
2018
Large deviations for stochastic heat equation with rough dependence in space. Zbl 1379.60069
Hu, Yaozhong; Nualart, David; Zhang, Tusheng
2018
A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. Zbl 1356.60100
Dong, Zhao; Xiong, Jie; Zhai, Jianliang; Zhang, Tusheng
2017
Large deviations for stochastic models of two-dimensional second grade fluids. Zbl 1370.60049
Zhai, Jianliang; Zhang, Tusheng
2017
Quasilinear parabolic stochastic partial differential equations: existence, uniqueness. Zbl 1372.60091
Hofmanová, Martina; Zhang, Tusheng
2017
Backward doubly SDEs and semilinear stochastic PDEs in a convex domain. Zbl 1372.60085
Matoussi, Anis; Sabbagh, Wissal; Zhang, Tusheng
2017
Exponential mixing for stochastic model of two-dimensional second grade fluids. Zbl 1329.60223
Wang, Ran; Zhai, Jianliang; Zhang, Tusheng
2016
Efficient SPH simulation of time-domain acoustic wave propagation. Zbl 1403.76173
Zhang, Y. O.; Zhang, T.; Ouyang, H.; Li, T. Y.
2016
Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses. Zbl 07162933
Zhang, T.; Ouyang, H.; Zhang, Y. O.; Lv, B. L.
2016
Lattice approximations of reflected stochastic partial differential equations driven by space-time white noise. Zbl 1358.60075
Zhang, Tusheng
2016
A stochastic HJB equation for optimal control of forward-backward SDEs. Zbl 1354.60061
Øksendal, Bernt; Sulem, Agnès; Zhang, Tusheng
2016
The dynamics of the stochastic shadow Gierer-Meinhardt system. Zbl 1337.60139
Winter, Matthias; Xu, Lihu; Zhai, Jianliang; Zhang, Tusheng
2016
Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise. Zbl 1306.60095
Wang, Ran; Zhang, Tusheng
2015
A moderate deviation principle for 2-D stochastic Navier-Stokes equations. Zbl 1310.60100
Wang, Ran; Zhai, Jianliang; Zhang, Tusheng
2015
Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. Zbl 1344.60030
Zhai, Jianliang; Zhang, Tusheng
2015
Large deviations for SPDEs of jump type. Zbl 1327.60074
Yang, Xue; Zhai, Jianliang; Zhang, Tusheng
2015
SPH simulation of acoustic waves: effects of frequency, sound pressure, and particle spacing. Zbl 1394.76105
Zhang, Y. O.; Zhang, T.; Ouyang, H.; Li, T. Y.
2015
Absolute continuity of the laws of perturbed diffusion processes and perturbed reflected diffusion processes. Zbl 1325.60131
Yue, Wen; Zhang, Tusheng
2015
Wong-Zakai approximations of backward doubly stochastic differential equations. Zbl 1325.60089
Hu, Ying; Matoussi, Anis; Zhang, Tusheng
2015
Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection. Zbl 1306.93078
Øksendal, Bernt; Sulem, Agnès; Zhang, Tusheng
2014
Strong convergence of Wong-Zakai approximations of reflected SDEs in a multidimensional general domain. Zbl 1311.60066
Zhang, Tusheng
2014
Log-Harnack inequality for mild solutions of SPDEs with multiplicative noise. Zbl 1285.60065
Wang, Feng-Yu; Zhang, Tusheng
2014
Existence and uniqueness of invariant measures for SPDEs with two reflecting walls. Zbl 1308.60081
Yang, Juan; Zhang, Tusheng
2014
A probabilistic approach to mixed boundary value problems for elliptic operators with singular coefficients. Zbl 1296.60164
Chen, Zhen-Qing; Zhang, Tusheng
2014
A large deviation principle of retarded Ornstein-Uhlenbeck processes driven by Lévy noise. Zbl 1308.60029
Liu, Kai; Zhang, Tusheng
2014
A variational approach to the construction and Malliavin differentiability of strong solutions of SDEs. Zbl 1282.60057
Menoukeu-Pamen, Olivier; Meyer-Brandis, Thilo; Nilssen, Torstein; Proske, Frank; Zhang, Tusheng
2013
Stochastic Burgers equation with random initial velocities: a Malliavin calculus approach. Zbl 1274.60205
Mohammed, Salah-Eldin A.; Zhang, Tusheng
2013
Hölder continuity of solutions of SPDEs with reflection. Zbl 1277.60107
Dalang, Robert C.; Zhang, Tusheng
2013
Estimates of heat kernels with Neumann boundary conditions. Zbl 1262.35115
Yang, Xue; Zhang, Tusheng
2013
Stochastic generalized porous media equations with reflection. Zbl 1328.60152
Röckner, Michael; Wang, Feng-Yu; Zhang, Tusheng
2013
Anticipating stochastic 2D Navier-Stokes equations. Zbl 1282.35281
Mohammed, Salah; Zhang, Tusheng
2013
Stochastic 3D tamed Navier-Stokes equations: existence, uniqueness and small time large deviation principles. Zbl 1241.60032
Röckner, Michael; Zhang, Tusheng
2012
Backward stochastic differential equations with respect to general filtrations and applications to insider finance. Zbl 1331.91209
Øksendal, Bernt; Zhang, Tusheng
2012
Large deviations for invariant measures of SPDEs with two reflecting walls. Zbl 1248.60077
Zhang, Tusheng
2012
Optimal partial information control of SPDEs with delay and time-advanced backward SPDEs. Zbl 1283.93316
Øksendal, Bernt; Sulem, Agnès; Zhang, Tusheng
2012
The Burgers equation with affine linear noise: dynamics and stability. Zbl 1248.60072
Mohammed, Salah; Zhang, Tusheng
2012
Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Zbl 1217.93183
Øksendal, Bernt; Sulem, Agnès; Zhang, Tusheng
2011
Improved rectangular method on stochastic Volterra equations. Zbl 1221.65023
Wen, C. H.; Zhang, T. S.
2011
White noise driven SPDEss with two reflecting walls. Zbl 1235.60081
Zhang, Tusheng; Yang, Juan
2011
A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients. Zbl 1242.60072
Zhang, Tusheng
2011
Systems of stochastic partial differential equations with reflection: existence and uniqueness. Zbl 1218.60059
Zhang, Tusheng
2011
Stochastic evolution equations driven by Lévy processes. Zbl 1237.60049
Chen, Zhen-Qing; Zhang, Tusheng
2011
Backward SDEs and Sobolev solutions for semilinear parabolic PDEs with singular coefficients. Zbl 1227.60089
Zhang, Tusheng; Ran, Qikang
2011
Stochastic partial differential equations. A modeling, white noise functional approach. 2nd ed. Zbl 1198.60005
Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng
2010
White noise driven SPDEs with reflection: Strong Feller properties and Harnack inequalities. Zbl 1196.60121
Zhang, Tusheng
2010
Large deviations for stochastic tamed 3D Navier-Stokes equations. Zbl 1195.60093
Röckner, Michael; Zhang, Tusheng; Zhang, Xicheng
2010
Optimal control with partial information for stochastic Volterra equations. Zbl 1214.49033
Øksendal, Bernt; Zhang, Tusheng
2010
Gradient estimates for stochastic evolution equations with non-Lipschitz coefficients. Zbl 1185.60065
Wang, Feng-Yu; Zhang, Tu-Sheng
2010
Existence and uniqueness of bounded weak solutions of a semilinear parabolic PDE. Zbl 1205.60116
Ran, Qikang; Zhang, Tusheng
2010
Dynamics of stochastic 2D Navier-Stokes equations. Zbl 1213.60107
Mohammed, Salah; Zhang, Tusheng
2010
Large deviation principles for isotropic stochastic flow of homeomorphisms on $$S^{d}$$. Zbl 1205.60058
Xu, Tiange; Zhang, Tusheng
2010
White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Zbl 1175.60068
Xu, Tiange; Zhang, Tusheng
2009
Large deviation principles for 2-D stochastic Navier-Stokes equations driven by Lévy processes. Zbl 1210.60071
Xu, Tiange; Zhang, Tusheng
2009
On general perturbations of symmetric Markov processes. Zbl 1176.60070
Chen, Z.-Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T.-S.
2009
Time-reversal and elliptic boundary value problems. Zbl 1187.60064
Chen, Zhen-Qing; Zhang, Tusheng
2009
On the small time asymptotics of the two-dimensional stochastic Navier-Stokes equations. Zbl 1196.60119
Xu, Tiange; Zhang, Tusheng
2009
Large deviations for perturbed reflected diffusion processes. Zbl 1187.60018
Bo, Lijun; Zhang, Tusheng
2009
Invariant measures for stochastic evolution equations of pure jump type. Zbl 1157.60062
Dong, Zhao; Xu, Tiange; Zhang, Tusheng
2009
Anticipating stochastic differential systems with memory. Zbl 1196.60103
Mohammed, Salah; Zhang, Tusheng
2009
Stochastic calculus for fractional Brownian motion and applications. Zbl 1157.60002
Biagini, Francesca; Hu, Yaozhong; Øksendal, Bernt; Zhang, Tusheng
2008
The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations. Zbl 1169.60014
Mohammed, Salah-Eldin A.; Zhang, Tusheng; Zhao, Huaizhong
2008
Stochastic calculus for symmetric Markov processes. Zbl 1142.31005
Chen, Z.-Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T.-S.
2008
Perturbation of symmetric Markov processes. Zbl 1137.31004
Chen, Z.-Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T.-S.
2008
Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Zbl 1119.60057
Röckner, Michael; Zhang, Tusheng
2007
The Itô-Ventzell formula and forward stochastic differential equations driven by Poisson random measures. Zbl 1118.60052
Øksendal, Bernt; Zhang, Tusheng
2007
Global flows for stochastic differential equations without global Lipschitz conditions. Zbl 1128.60046
Fang, Shizan; Imkeller, Peter; Zhang, Tusheng
2007
Large deviations for stochastic nonlinear beam equations. Zbl 1118.60021
Zhang, Tusheng
2007
The substitution theorem for semilinear stochastic partial differential equations. Zbl 1138.60048
Mohammed, Salah-Eldin A.; Zhang, Tusheng
2007
Isotropic stochastic flow of homeomorphisms on $$S^{d}$$ for the critical Sobolev exponent. Zbl 1095.60020
Fang, Shizan; Zhang, Tusheng
2006
Large deviations for stochastic systems with memory. Zbl 1136.60021
Mohammed, Salah-Eldin A.; Zhang, Tusheng
2006
A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Zbl 1081.60043
Fang, Shizan; Zhang, Tusheng
2005
Perturbed Skorohod equations and perturbed reflected diffusion processes. Zbl 1061.60056
Doney, R. A.; Zhang, Tusheng
2005
Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields. Zbl 1090.60057
Øksendal, Bernt; Proske, Frank; Zhang, Tusheng
2005
Small time asymptotics for Fleming-Viot processes. Zbl 1080.60504
Xiang, Kai-Nan; Zhang, Tu-Sheng
2005
Absolute continuity of symmetric Markov processes. Zbl 1053.60084
Chen, Z.-Q.; Fitzsimmons, P. J.; Takeda, M.; Ying, J.; Zhang, T.-S.
2004
General fractional multiparameter white noise theory and stochastic partial differential equations. Zbl 1067.35161
Hu, Yaozhong; Øksendal, Bernt; Zhang, Tusheng
2004
Convergence of symmetric diffusions on Wiener spaces. Zbl 1065.60111
Posilicano, A.; Zhang, T. S.
2004
A class of stochastic differential equations with non-Lipschitzian coefficients: Pathwise uniqueness and no explosion. Zbl 1031.60051
Fang, Shizan; Zhang, Tusheng
2003
Lower order perturbations of Dirichlet processes. Zbl 1021.60058
Röckner, M.; Zhang, T. S.
2003
A comparison theorem for solutions of backward stochastic differential equations with two reflecting barriers and its applications. Zbl 1062.60058
Zhang, T. S.
2003
Large deviations and support theorem for diffusion processes via rough paths. Zbl 1075.60510
Ledoux, M.; Qian, Z.; Zhang, T.
2002
Local times of fractional Brownian sheets. Zbl 1009.60024
Xiao, Yimin; Zhang, Tusheng
2002
On the small time asymptotics of diffusion processes on path groups. Zbl 0993.60026
Aida, Shigeki; Zhang, Tusheng
2002
Girsanov and Feynman-Kac type transformations for symmetric Markov processes. Zbl 1004.60077
Chen, Zhen-Qing; Zhang, Tu-Sheng
2002
SPDEs driven by space-time white noise in high dimensions: Absolute continuity of the law and convergence of solutions. Zbl 1030.60042
Zhang, T. S.; Zheng, W. A.
2002
Nonrobustness of some impulse control problems with respect to intervention costs. Zbl 1012.93066
Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng
2002
Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Zbl 0986.60056
Øksendal, Bernt; Zhang, Tusheng
2001
Large deviations for the Brownian motion on loop groups. Zbl 1003.60028
Fang, Shizan; Zhang, Tusheng
2001
Generalized Feynman-Kac semigroups, associated quadratic forms and asymptotic properties. Zbl 0984.31006
Zhang, T. S.
2001
On the small time large deviations of diffusion processes on configuration spaces. Zbl 1047.60081
Zhang, T. S.
2001
Stochastic partial differential equations driven by multiparameter fractional white noise. Zbl 0982.60054
Hu, Yaozhong; Øksendal, Bernt; Zhang, Tusheng
2000
On the small time asymptotics of diffusion processes on Hilbert spaces. Zbl 1044.60071
Zhang, T. S.
2000
Existence and uniqueness of invariant measures: An approach via sectorial forms. Zbl 0953.31003
Bogachev, V.; Röckner, M.; Zhang, T. S.
2000
On the small time behavior of Ornstein-Uhlenbeck processes with unbounded linear drifts. Zbl 0932.60071
Fang, S.; Zhang, T. S.
1999
Probabilistic representations and hyperbound estimates for semigroups. Zbl 1043.47508
Röckner, M.; Zhang, T. S.
1999
Parabolic SPDEs driven by Poisson white noise. Zbl 0934.60055
Albeverio, Sergio; Wu, Jiang-Lun; Zhang, Tu-Sheng
1998
Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels. Zbl 0914.60045
Lunt, John; Lyons, T. J.; Zhang, T. S.
1998
...and 32 more Documents
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#### Cited by 1,439 Authors

 87 Zhang, Tusheng S. 31 Øksendal, Bernt Karsten 31 Röckner, Michael 27 Chen, Zhen-Qing 27 Yan, Litan 25 Wang, Feng-Yu 20 Xie, Yingchao 17 Proske, Frank Norbert 16 Zhai, Jianliang 16 Zhang, Xicheng 15 Kuwae, Kazuhiro 14 Alpay, Daniel Aron 13 Albeverio, Sergio A. 11 Duan, Jinqiao 11 Hu, Yaozhong 11 Liu, Wei 11 Sun, Wei 11 Wang, Ran 11 Wu, Fuke 11 Xiao, Yimin 11 Yuan, Chenggui 10 Gao, Hongjun 10 Lanconelli, Alberto 10 Luo, Dejun 10 Wang, Yongjin 10 Zhao, Huaizhong 9 Fang, Shizan 9 Fitzsimmons, Patrick J. 9 Inahama, Yuzuru 9 Menoukeu Pamen, Olivier 9 Meyer-Brandis, Thilo 9 Ubœ, Jan 9 Xiong, Jie 8 Dong, Zhao 8 Friz, Peter Karl 8 Kondrat’yev, Yuriĭ Grygorovych 8 Nualart, David 8 Seleši, Dora 8 Takeda, Masayoshi 8 Xie, Bin 8 Xu, Lihu 8 Xu, Yong 8 Yang, Xue 7 Agram, Nacira 7 Aida, Shigeki 7 Bo, Lijun 7 Chen, Zhenlong 7 Hairer, Martin 7 Hu, Shigeng 7 Leonenko, Nikolai N. 7 Liu, Junfeng 7 Lu, Kening 7 Ma, Zhi-Ming 7 Maslowski, Bohdan 7 Mohammed, Salah-Eldin A. 7 Pei, Bin 7 Priola, Enrico 7 Rozovskii, Boris L. 7 Wu, Dongsheng 7 Wu, Jianglun 7 Yang, Juan 7 Zeng, Caibin 6 Baños, David R. 6 Bao, Jianhai 6 Brzeźniak, Zdzisław 6 Chen, Bin 6 Cui, Jing 6 Fan, Xiliang 6 Huang, Jianhua 6 Jiang, Yiming 6 Jørgensen, Palle E. T. 6 Kim, Daehong 6 Levajković, Tijana 6 Liang, Zongxia 6 Liu, Qing 6 Lototsky, Sergey V. 6 Lv, Guangying 6 Melnikova, Irina V. 6 Shen, Guangjun 6 Shi, KeHua 6 Sun, Xichao 6 Tachim Medjo, Theodore 6 Yang, Xiaoyuan 6 Yin, Xiuwei 5 Bonaccorsi, Stefano 5 Chen, Chao 5 Chen, Chuanzhong 5 Holden, Helge 5 Li, Yumeng 5 Lindstrøm, Tom L. 5 Manouzi, Hassan 5 Mao, Xuerong 5 Matoussi, Anis 5 Matthies, Hermann Georg 5 Mishura, Yuliya Stepanivna 5 Nilssen, Torstein K. 5 Pilipović, Stevan 5 Ren, Jiagang 5 Russo, Francesco 5 Shao, Jinghai ...and 1,339 more Authors
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 104 Stochastic Processes and their Applications 63 Journal of Functional Analysis 48 The Annals of Probability 43 Statistics & Probability Letters 43 Potential Analysis 42 Journal of Mathematical Analysis and Applications 42 Stochastic Analysis and Applications 35 Journal of Differential Equations 33 Stochastics and Dynamics 30 Journal of Theoretical Probability 25 Chaos, Solitons and Fractals 25 Stochastics 24 Infinite Dimensional Analysis, Quantum Probability and Related Topics 22 Probability Theory and Related Fields 19 Journal of Mathematical Physics 18 Journal of Computational and Applied Mathematics 18 Advances in Difference Equations 17 Bernoulli 17 Science China. Mathematics 15 Applied Mathematics and Optimization 14 Bulletin des Sciences Mathématiques 13 Abstract and Applied Analysis 12 Transactions of the American Mathematical Society 12 Acta Mathematica Sinica. English Series 12 Frontiers of Mathematics in China 11 Applied Mathematics and Computation 11 Discrete and Continuous Dynamical Systems. Series B 10 Acta Applicandae Mathematicae 9 Journal of Optimization Theory and Applications 9 Proceedings of the American Mathematical Society 8 Journal of Statistical Physics 8 Acta Mathematicae Applicatae Sinica. English Series 8 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 8 Discrete and Continuous Dynamical Systems 8 Mathematical Control and Related Fields 7 Advances in Applied Probability 7 Computers & Mathematics with Applications 7 Computer Methods in Applied Mechanics and Engineering 7 Journal of Computational Physics 7 SIAM Journal on Control and Optimization 7 Tohoku Mathematical Journal. Second Series 7 The Annals of Applied Probability 6 Chinese Annals of Mathematics. Series B 6 Fractional Calculus & Applied Analysis 6 Comptes Rendus. Mathématique. Académie des Sciences, Paris 6 International Journal of Stochastic Analysis 6 Stochastic and Partial Differential Equations. Analysis and Computations 5 Physics Letters. A 5 Journal of Applied Probability 5 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 5 Journal of Economic Dynamics & Control 5 Electronic Journal of Probability 5 Journal of Evolution Equations 5 Journal of Systems Science and Complexity 5 Afrika Matematika 4 Communications in Mathematical Physics 4 ZAMP. Zeitschrift für angewandte Mathematik und Physik 4 Automatica 4 BIT 4 Czechoslovak Mathematical Journal 4 Osaka Journal of Mathematics 4 Systems & Control Letters 4 Forum Mathematicum 4 Journal de Mathématiques Pures et Appliquées. Neuvième Série 4 Theory of Probability and Mathematical Statistics 4 NoDEA. Nonlinear Differential Equations and Applications 4 Engineering Analysis with Boundary Elements 4 Chaos 4 Nonlinear Analysis. Real World Applications 4 Quantitative Finance 4 Mediterranean Journal of Mathematics 4 Communications in Mathematics and Statistics 3 International Journal of Control 3 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 3 Integral Equations and Operator Theory 3 Journal of the Mathematical Society of Japan 3 Mathematische Annalen 3 Quarterly of Applied Mathematics 3 Applied Numerical Mathematics 3 Science in China. Series A 3 Communications in Partial Differential Equations 3 SIAM Journal on Mathematical Analysis 3 Stochastics and Stochastics Reports 3 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 3 Journal of Mathematical Sciences (New York) 3 Random Operators and Stochastic Equations 3 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 3 European Series in Applied and Industrial Mathematics (ESAIM): Probability and Statistics 3 Nonlinear Dynamics 3 Journal of Inequalities and Applications 3 Methodology and Computing in Applied Probability 3 Differential Equations 3 Communications on Pure and Applied Analysis 3 Journal of the Korean Statistical Society 3 Mathematics and Financial Economics 3 Carpathian Mathematical Publications 3 SIAM/ASA Journal on Uncertainty Quantification 3 Journal of Function Spaces 2 Communications on Pure and Applied Mathematics 2 Indian Journal of Pure & Applied Mathematics ...and 155 more Serials
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#### Cited in 50 Fields

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text:
# migrate

## Using migrate

This package is intended to serve as a set of tools to help convert credit risk data at two time points into traditional credit state migration (aka, “transition”) matrices. At a higher level, migrate is intended to help an analyst understand how risk moved in their credit portfolio over a time interval.

## Background

One of the more difficult aspects of making a credit state migration matrix in R (or Python, for that matter) is the fact that the output doesn’t satisfy the structure of a traditional data frame object. Rather, the output needs to be a matrix, which is a data structure that R does support. In the past, there has been difficulty converting a matrix to something more visual-friendly. More recently, however, tools like the kableExtra and gt packages allow us to present visually appealing output that extends the structure of a data frame. Using the matrix-style output of migrate’s functions with a visual formatting package such as the two mentioned above will hopefully help analysts streamline the presentation of their credit portfolio’s state migration matrices to an audience.

## Getting Started

If you haven’t done so already, first install migrate with the instructions in the README section.

First, load the package & the mock dataset (as a data frame) using library()

library(migrate)

The package has a built-in mock dataset, which can be loaded into the environment like so:

data("mock_credit")

head(mock_credit[order(mock_credit$customer_id), ]) # sort by 'customer_id' customer_id date risk_rating principal_balance Customer_1001 2020-06-30 A 915000 Customer_1001 2020-09-30 A 1328000 Customer_1002 2020-06-30 AAA 979000 Customer_1002 2020-09-30 AAA 354000 Customer_1003 2020-06-30 BBB 1400000 Customer_1003 2020-09-30 BBB 356000 Note that an important feature of the mock_credit dataset is that there are exactly two (2) unique values in the date column variable; if the time argument passed to migrate has more than two (2) unique values, the function will throw an error. unique(mock_credit$date)
#> [1] "2020-06-30" "2020-09-30"

To summarize the migration within the data, use the migrate() function

migrated_df <- migrate(
data = mock_credit,
id = customer_id,
time = date,
state = risk_rating,
)
#> === Migrating from: 2020-06-30 --> 2020-09-30 ===
head(migrated_df)
#> # A tibble: 6 x 3
#>   risk_rating_start risk_rating_end   prop
#>   <ord>             <ord>            <dbl>
#> 1 AAA               AAA             0.774
#> 2 AAA               AA              0.194
#> 3 AAA               A               0.0323
#> 4 AAA               BBB             0
#> 5 AAA               BB              0
#> 6 AAA               B               0

To create the state migration matrix, use the build_matrix() function

build_matrix(migrated_df)
#> Using risk_rating_start as the 'state_start' column variable
#> Using risk_rating_end as the 'state_end' column variable
#> Using prop as the 'metric' column variable
#>             AAA         AA          A        BBB         BB          B        CCC
#> AAA 0.774193548 0.19354839 0.03225806 0.00000000 0.00000000 0.00000000 0.00000000
#> AA  0.101123596 0.66292135 0.15730337 0.07865169 0.00000000 0.00000000 0.00000000
#> A   0.008333333 0.06666667 0.72500000 0.16666667 0.03333333 0.00000000 0.00000000
#> BBB 0.000000000 0.00000000 0.11363636 0.68181818 0.14772727 0.05681818 0.00000000
#> BB  0.000000000 0.00000000 0.00000000 0.11392405 0.63291139 0.16455696 0.08860759
#> B   0.000000000 0.00000000 0.00000000 0.01388889 0.09722222 0.62500000 0.26388889
#> CCC 0.000000000 0.00000000 0.00000000 0.00000000 0.00000000 0.14285714 0.85714286

Or, to do it all in one shot, use the %>%

mock_credit %>%
migrate(
id = customer_id,
time = date,
state = risk_rating,
metric = principal_balance,
percent = FALSE,
verbose = FALSE
) %>%
build_matrix(
state_start = risk_rating_start,
state_end = risk_rating_end,
metric = principal_balance
)
#>          AAA       AA        A      BBB       BB        B      CCC
#> AAA 29042000  6575000    20000        0        0        0        0
#> AA   6445000 58095000 13045000 14467000        0        0        0
#> A     804000  7898000 85330000 21015000  5829000        0        0
#> BBB        0        0 12461000 65315000 13911000  8140000        0
#> BB         0        0        0 11374000 45986000 14057000  5723000
#> B          0        0        0   413000  6700000 47402000 17132000
#> CCC        0        0        0        0        0  2094000 14843000

format: plain


text:
Free Arizona CDL Combination Vehicles Practice Test 2023

Our CDL practice tests:
Based on 2021 AZ commercial driver's license manual
Perfect for first-time, renewal applicants
AZ CDL Combination Vehicles Test format:
20 questions
80% passing score
List of questions
1.
The air lines on a combination vehicle are often colored to prevent them from mixing up. The emergency line is _____, the service line is _________.
2.
A tractor with a _______ trailer, requires the most amount of stopping distance.
3.
The trailer air supply control knob is _____.
4.
The higher the center of gravity of your truck, the:
5.
To unlock the fifth wheel, the locking lever should be placed in the ______ position.
6.
A function of the tractor protection valve is:
7.
Spring brakes are not required on trailers and dollies built before:
8.
The shut-off valves at the back of the last trailer should be:
9.
The tractor protection valve will close, and the trailer emergency brakes will come on when there is a major leak in the _____ brake line.
10.
The trailer service brakes should be tested with the hand control, and controlled during normal operation by:
11.
If you have to enter another traffic lane to complete a wide turn, you should:
12.
When driving a combination vehicle with multiple trailers or dollies, all air shut-off valves except the last one at the back must be:
13.
When checking the trailer emergency brakes, the tractor protection control valve should be placed in what position?
14.
Trailers will have an ABS malfunction lamp:
15.
The emergency air line is usually what color?
16.
When uncoupling, the tractor should:
17.
Crack-the-whip effect occurs when:
18.
Your emergency air line breaks or gets pulled apart while you are driving. The loss of pressure will cause the:
19.
The fifth wheel locking lever is not locked after the jaws close around the kingpin. This means that:
20.
Where should the tractor be when you inspect the landing gear after uncoupling the trailer?


text:
# Falling through the earth

The 2012 remake of the classic film Total Recall features a giant elevator that plunges through the earth from Australia to England. This trip is called the “fall”, which I presume to mean it is propelled by gravity alone in an evacuated tube. The film states that the trip takes 17 minutes (I don’t remember if this is to get to the center of the earth or the other side). It also made some goofy point that the seats flip around in the vehicle when you cross the center because gravity reverses. This makes no sense because when you fall you are weightless and if you are strapped in, what difference does it make what direction you are in. In any case, I was still curious to know if 17 minutes was remotely accurate and the privilege of a physics education is that one is given the tools to calculate the transit time through the earth due to gravity.

The first thing to do is to make an order of magnitude estimate to see if the time is even in the ballpark. For this you only need middle school physics. The gravitational acceleration for a mass at the surface of the earth is $g = 9.8 m/s^2$. The radius of the earth is 6.371 million metres. Using the formula that distance $r = 1/2 g t^2$ (which you get by integrating twice over time), you get $t = \sqrt{2 r / g}$. Plugging in the numbers gives 1140 seconds or 19 minutes. So it would take 19 minutes to get to the center of the earth if you constantly accelerated at $9.8 m/s^2$. It would take the same amount of time to get back to the surface. Given that the gravitational acceleration at the surface should be an upper bound, the real transit time should be slower. I don’t know who they consulted but 17 minutes is not too far off.

We can calculate a more accurate time by including the effect of the gravitational force changing as you transit through the earth but this will require calculus. It’s a beautiful calculation so I’ll show it here. Newton’s law for the gravitational force between a point mass m and a point mass M separated by a distance r is

$F = - \frac{G Mm}{r^2}$

where $G = 6.67\times 10^{-11} m^3 kg^{-1} s^{-2}$ is the gravitational constant. If we assume that mass M (i.e. earth) is fixed then Newton’s 2nd law of motion for the mass m is given by $m \ddot r = F$. The equivalence of inertial mass and gravitational mass means you can divide m from both sides. So, if a mass m were outside of the earth, then it would accelerate towards the earth as

$\ddot r = F/m = - \frac{G M}{r^2}$

and this number is g when r is the radius of the earth. This is the reason that all objects fall with the same acceleration, apocryphally validated by Galileo on the Tower of Pisa. (It is also the basis of the theory of general relativity).

However, the earth is not a point but an extended ball where each point of the ball exerts a gravitational force on any mass inside or outside of the ball. (Nothing can shield gravity). Thus to compute the force acting on a particle we need to integrate over the contributions of each point inside the earth. Assume that the density of the earth $\rho$ is constant so the mass $M = \frac{4}{3} \pi R^3\rho$ where $R$ is the radius of the earth. The force between two point particles acts in a straight line between the two points. Thus for an extended object like the earth, each point in it will exert a force on a given particle in a different direction. So the calculation involves integrating over vectors. This is greatly simplified because a ball is highly symmetric. Consider the figure below, which is a cross section of the earth sliced down the middle.

The particle/elevator is the red dot, which is located a distance r from the center. (It can be inside or outside of the earth). We will assume that it travels on an axis through the center of the earth. We want to compute the net gravitational force on it from each point in the earth along this central axis. All distances (positive and negative) are measured with respect to the center of the earth. The blue dot is a point inside the earth with coordinates (x,y). There is also the third coordinate coming out of the page but we will not need it. For each blue point on one side of the earth there is another point diametrically opposed to it. The forces exerted by the two blue points on the red point are symmetrical. Their contributions in the y direction are exactly opposite and cancel leaving the net force only along the x axis. In fact there is an entire circle of points with radius y (orthogonal to the page) around the central axis where each point on the circle combines with a partner point on the opposite side to yield a force only along the x axis. Thus to compute the net force on the elevator we just need to integrate the contribution from concentric rings over the volume earth. This reduces an integral over three dimensions to just two.

The magnitude of the force (density) between the blue and red dot is given by

$\frac{G m \rho}{(r-x)^2+y^2}$

To get the component of the force along the x direction we need to multiple by the cosine of the angle between the central axis and the blue dot, which is

$\frac{r-x}{((r-x)^2+y^2)^{1/2}}$

(i.e. ratio of the adjacent to the hypotenuse of the relevant triangle). Now, to capture the contributions for all the pairs on the circle we multiple by the circumference which is $2\pi y$. Putting this together gives

$F/m = -G\rho \int_{-R}^{R}\int_{0}^{\sqrt{R^2-x^2}}\frac{r-x}{((r-x)^2+y^2)^{3/2}} 2\pi y dy dx$

The y integral extends from zero to the edge of the earth, which is $\sqrt{R^2-x^2}$. (This is R at $x=0$ (the center) and zero at $x=\pm R$ (the poles) as expected). The x integral extends from one pole to the other, hence -R to R. Completing the y integral gives

$2\pi G\rho \int_{-R}^{R}\left. \frac{r-x}{((r-x)^2+y^2)^{1/2}} \right|_{0}^{\sqrt{R^2-x^2}}dx$

$= 2\pi G\rho \int_{-R}^{R}\left[ \frac{r-x}{((r-x)^2+R^2-x^2)^{1/2}} - \frac{r-x}{|r-x|} \right]dx$ (*)

The second term comes from the 0 limit of the integral, which is $\frac{r-x}{((r-x)^2)^{1/2}}$. The square root of a number has positive and negative roots but the denominator here is a distance and thus is always a positive quantity and thus must include the absolute value. The first term of the above integral can be completed straightforwardly (I’ll leave it as an exercise) but the second term must be handled with care because r-x can change sign depending on whether r is greater or less than x. For a particle outside of the earth r-x is always positive and we get

$\int_{-R}^{R} \frac{r-x}{|r-x|} dx = \int_{-R}^{R} dx = 2R, r > R$

Inside the earth, we must break the integral up into two parts

$\int_{-R}^{R} \frac{r-x}{|r-x|} dx = \int_{-R}^{r} dx - \int_{r}^{R} dx = r+R - R + r = 2r, -R \le r\le R$

The first term of (*) integrates to

$\left[ \frac{(r^2-2rx+R^2)^{1/2}(-2r^2+rx+R^2)}{3r^2} \right]_{-R}^{R}$

$= \frac{(r^2-2rR+R^2)^{1/2}(-2r^2+rR+R^2)}{3r^2} - \frac{(r^2+2rR+R^2)^{1/2}(-2r^2-rR+R^2)}{3r^2}$

Using the fact that $(r \pm R)^2 = r^2 \pm 2rR + R^2$, we get

$= \frac{|R-r|(-2r^2+rR+R^2)}{3r^2} - \frac{|R+r|(-2r^2-rR+R^2)}{3r^2}$

(We again need the absolute value sign). For $r > R$, the particle is outside of the earth) and $|R-r| = r-R, |R+r| = r + R$. Putting everything together gives

$F/m = 2\pi G\rho \left[ \frac{6r^2R-2R^3}{3r^2} - 2 R\right] = -\frac{4}{3}\pi R^3\rho G\frac{1}{r^2} = - \frac{MG}{r^2}$

Thus, we have explicitly shown that the gravitational force exerted by a uniform ball is equivalent to concentrating all the mass in the center. This formula is true for $r < - R$ too.

For $-R \le r \le R$ we have

$F/m = 2\pi G\rho \left[ \frac{4}{3} r - 2r\right] =-\frac{4}{3}\pi\rho G r = -\frac{G M}{R^3}r$

Remarkably, the gravitational force on a particle inside the earth is just the force on the surface scaled by the ratio r/R. The equation of motion of the elevator is thus

$\ddot r = - \omega^2 r$ with $\omega^2 = GM/R^3 = g/R$

(Recall that the gravitational acceleration at the surface is $g = GM/R^2 = 9.8 m/s^2$). This is the classic equation for a harmonic oscillator with solutions of the form $\sin \omega t$. Thus, a period (i.e. round trip) is given by $2\pi/\omega$. Plugging in the numbers gives 5062 seconds or 84 minutes. A transit through the earth once would be half that at 42 minutes and the time to fall to the center of the earth would be 21 minutes, which I find surprisingly close to the back of the envelope estimate.

Now Australia is not exactly antipodal to England so the tube in the movie did not go directly through the center, which would make the calculation much harder. This would be a shorter distance but the gravitational force would be at an angle to the tube so there would be less acceleration and something would need to keep the elevator from rubbing against the walls (wheels or magnetic levitation). I actually don’t know if it would take a shorter or longer time than going through the center. If you calculate it, please let me know.

## 3 thoughts on “Falling through the earth”

1. Rick G says:

I’ve always loved this problem and rederived this every few years for fun. But I always just said “Gauss’s Law” and skipped the first 2/3 of the work, getting right to the part where all shells outside the current position are irrelevant, and all mass in the interior can be considered a point at the center.

Is it only a coincidence that the result is the same as for the equivalent e&m problem? Is it just true of all inverse square forces?

As for the alternate paths, I seem to recall (but not totally, pun intended) a result that all chords through the Earth have the same transit time by gravity. I don’t know how to prove this but maybe you do, or at least you can pick your favorite chord, see if you get the same result, and consider it an as yet unrefuted conjecture.

Like

2. Jc says:

Divergence theorem. Reminds me of undergrad physics when we had a midterm where the answer to all the questions was 0 and one could either do the calculation the hard way or invoke green’s theorem

Like

3. @Rick G I think you’re right. The time through any chord is the same. This is because the force component along any chord is reduced by the exact same amount as the distance to the mid point of the chord. I’ll post the full solution shortly.

Like


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## proof identity

Can someone help me to proof this identity

$(\vec{a} \times \vec{b})^2(\vec{a} \times \vec{b})^2-[(\vec{a} \times \vec{b})(\vec{a} \times \vec{b})]^2=[(\vec{a} \times \vec{b})\vec{c}]^2 \vec{a}^2$


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Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# How do you calculate formal charge from an equation?

The formal charge of any atom in a molecule can be calculated by the following equation:

FC = V - N - B/2

where V is the number of of the neutral atom in isolation (in its ground state); N is the number of non- on this atom in the molecule; and B is the total number of electrons shared in bonds with other atoms in the molecule.


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6.8 Scientific notation

Exponents are always used in mathematical expressions. Exponents tells us how many times a number is multiplied to itself, like if we were ask to solved $5^7$, this would be just mean to multiply 5 to itself seven times. In the given expression $5^7$m 5 is the base, and 7 is the exponent. Through the help of exponents calculator you could easily compute the solution by just plugging in the given numbers. Buy If you were given $(5^3+ 2b)^7$ to evaluate, or an expression that would involve more exponents, it would be best to acquaint yourself with the exponent rules. These rules would guide us in simplifying expressions that contains exponents.

There a lot of different rules like the product rule where the exponents are added to each other provided that the bases are the same. Then, there is the quotient rule which states that if the bases are divided with each other you can just subtract the exponents, provided again that the bases are the same. There is also the power rule, where a base and an exponent are raised to another exponent where you would only need to multiply the exponents. Lastly, we would also look at the negative exponent rule which suggests that any base raised to a negative exponent will basically mean that you need to divide the base by the number denoted by the exponent. Each of the sections of this chapter will discuss each rule in depth and session 7 will be teaching us to integrate these rules and combine them all together. There’s also going to be a formula sheet for the exponent rules for you to easily understand and remember them.

This chapter would also discuss scientific notation, a technique used to express very large or very small numbers through the use of the base 10, raised to an exponent. We would also be converting rational exponents into radical exponents and vice versa and solving problems that involve exponents.

Scientific notation

Scientific notation is a way of writing number. It is especially useful when we want to express very large and small numbers. There are two parts in scientific notation. The first part consists of digits, and the second part is x 10 to a power.


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# The meaning of a “subcomplex” of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^*$ be the dual vector space. We have the standard Cartan-Eilenberg complex $(\wedge^{\cdot} \mathfrak{g}^* ,\text{d}_{CE})$ where the differential is given by the dual of the Lie bracket.

Now assume that we have a Lie subalgebra $\mathfrak{k}\subset \mathfrak{g}$ and a $\mathfrak{k}$ invariant complement $\mathfrak{p}\subset \mathfrak{g}$. That is: $$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$$ as vector spaces, $[\mathfrak{k},\mathfrak{k}]\subset \mathfrak{k}$ and $[\mathfrak{k},\mathfrak{p}]\subset \mathfrak{p}$. Remember that in general $[\mathfrak{p},\mathfrak{p}]\nsubseteq$$\mathfrak{p}. We can construct \wedge^{\cdot} \mathfrak{p}^* and define a derivative on it as follows. For any \theta \in \wedge^k \mathfrak{p}^* , use the split \mathfrak{g}^* =\mathfrak{k}^* \oplus\mathfrak{p}^* we can treat \theta as an element in \wedge^k \mathfrak{g}^* , take the Cartan-Eilenberg differential, we get \text{d}_{CE}\theta \in \wedge^{k+1} \mathfrak{g}^* . Again use \mathfrak{g}^* =\mathfrak{k}^* \oplus\mathfrak{p}^* we get$$ \wedge^{k+1} \mathfrak{g}^* = \bigoplus_{i=0}^{k+1} \wedge^i \mathfrak{k}^* \otimes \wedge^{k+1-i} \mathfrak{p}^*$$Let$\text{pr}$denote the projection onto the$i=0$component of the above decomposition. Then our derivation$\overline{\text{d}}$is defined to be$\text{pr}\circ \text{d}_{CE}: \wedge^{\cdot} \mathfrak{p}^* \rightarrow \wedge^{\cdot+1}\mathfrak{p}^* $. It is not difficult to check that$\overline{\text{d}}$is a derivation but$\overline{\text{d}}\circ \overline{\text{d}}\neq 0$. It measures the failure of$[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{p}$. My question is: what does this$\overline{\text{d}}\circ \overline{\text{d}}$stand for? Is it a kind of curvature? Is there any nice expression of it? In particular, does it have anything to do with the connection on the principle bundle$K\rightarrow G \rightarrow G/K\$?

I have the impression that Kostant has done some work of this kind but I just cannot find them.

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Take a look at Chapter X in the third volume of "Connections, curvature and cohomology" by Greub, Halperin and Vanstone. You will find that indeed what you have can be interpreted as a curvature of an algebraic connection. –  José Figueroa-O'Farrill Aug 8 '12 at 14:11


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# Homework policy

The accepted answer on this question is our homework policy, adopted by a community poll.

The poll rules and outcome are retained below for historical purposes.

### Rules

Below, you will find three proposed policies regarding homework questions. Please upvote the one(s) you agree with and downvote the one(s) you disagree with (or don't vote if you're ambivalent or indifferent).

The voting period runs for 1 week, until Thursday, July 5, 2012 at 23:59:59 GMT. After that:

• If the dominant option has a score of at least 5 and at least 2 more than the next contender, that policy will be adopted.
• Otherwise, the moderators will meet and decide on a provisional policy, which will last until meta participation is sufficient to achieve a consensual improvement.

### Results

• Raphael's proposal: 10 (+14/-4)
• David Lewis's proposal: 7 (+10/-3)
• Patrick87's/Gilles's proposal: 2 (+4/-2)

Since Raphael's proposal has a score of at least 5, and has at least 2 more votes than the next runner-up, this proposal is adopted, effective immediately.

• Moderator notice: this thread represents the poll that led to the present policy and has been archived. If you wish to reopen the subject, open a new meta thread and expose why you wish the policy to change. – Gilles 'SO- stop being evil' Feb 6 '13 at 10:39

## Policy

We do not try to decide which questions are homework, and we don't use a special tag. Users that want to receive hints rather than full answers should say so, and answerers should honor such requests (by not posting details or hiding them in spoiler tags). Other than that, the usual quality control applies.

## Guideline

All users should be aware that askers might use cs.SE to cheat and are asked to act responsibly (e.g. by not answering, answering after a delay or providing only hints) if they suspect a question is a mere restatement of a homework exercise without own effort.

## Rationale

We have no way of identifying homework questions reliably and in general (with notable exceptions). Therefore, we have to take what users post at face value. However, from a teaching point-of-view, it is often more helpful to a student to be led to an answer instead of told the answer. That is why we should work with helpful hints rather than detailed answers where appropriate.

has no use given such a policy.

On the other hand, we want to prevent students from exploiting cs.SE to do their homework (i.e. cheat the system and themselves), as that would not help them and might gain the site a bad reputation among teachers. Most homework-dump-style questions should be dealt with by usual quality-control, but that response might come with a delay or, in special cases, never. In these cases, it is up to the users to decide how to act.

## Source

Raphael's proposal

(Note: if adopted, references to mathematics and Math.SE shall be adapted to computer science and CS.SE.)

## Policy

### Can I ask a homework question here?

There are actually two factors involved here, one is on your end, and one over here on this website.

As a general rule, we do not discourage any specific category of questions, as long as it is mathematical. But please do make an effort to search through the list of previously asked questions. In our experience homework questions are usually not very imaginative, and tend to fall in one of the abstract categories of commonly asked questions. We will close duplicate questions, especially if they are of homework type.

On the other hand, whether your learning institution (middle school, high school, college, etc.) and your teacher or professor allows you to consult other people, or to post the exact question on the internet, is something that is usually addressed by your institution's honor code or rules and regulations, and any specific class policies. You should ask your teacher whether asking a homework question here is appropriate before posting your question.

### How do I ask a homework question on this website?

Please use the homework tag. This serves several purposes: the most important of which is to let the answerer know to give an answer that more clearly explains the underlying concepts. In other words, we believe in the "teach a man to fish" philosophy.

Also, please put some work into formulating your question. Please do not just copy and paste the exact question text from your homework sheet. In particular, when you are asking for help, writing in imperative mode ("Show that...", "Compute...", or "Prove or find a counterexample: ...") is at the very least impolite: you are, after all, trying to ask a question, not give an assignment. It also turns many people off.

If you feel that it is somehow just so much more straightforward to copy and paste, then it is generally good real world advice to both quote the question (because you are copying after all) and give a reference to the source (so that others can refer to the context of the question). You should also volunteer all relevant information (see next section). It is a waste of everybody's time if someone has to ask for what you should explicitly have given.

### What information should I include in a question about homework?

• The context. What kind of course it is, what textbook you use. It wouldn't help you if someone gives an answer using some sophisticated, high-powered machinery that you have never seen before. (An example would be asking for help about a step that is used to prove a big theorem in the textbook, and receiving an answer using said big theorem.)
• All the definitions. If you are asking about a question that is more advanced than basic college-level calculus, then you should consider including the definitions. A lot of homework questions are assigned to familiarize you with the definitions used in the course. For many objects in mathematics there are several equivalent definitions. What needs to be proven when starting from one set of definitions may actually be a trivial consequence in a different set of definitions and vice versa. It is important that you let us know what framework in which you work so our answers can properly address the question.
• Show your work. You should definitely include any partial work you have done. This will help bolster your claim that you are not just coming here asking other people to do your homework for you, and it will help the answerers to give more clinical responses. Showing your work will help us gauge where you are having problems: if it is a technical thing near the end, a short to the point answer will suffice; if it is some fundamental problem with understanding the subject, we will then write a longer, more detailed response. It will also prevent people from spending a lot of time going over ground that you have already covered or understand well already.

### Why don't you provide a complete answer to my question?

We've had another discussion from the point of view of the answerer, and you should see that for a more complete discussion. To quote the accepted answer in that thread, we feel that

Providing an answer that doesn't help a student learn is not in the student's own best interest, and if a solution complete enough to be copied verbatim and handed in is given immediately, it will encourage more people to use the site as a free homework service. In the spirit of creating a lasting resource of mathematical knowledge, you may come back after a suitable amount of time and edit your response to include a more complete answer. Or even better, the student can post his own correct answer!

## Rationale

This is the current policy on our sister site Mathematics Stack Exchange.

## Source

Proposed by David Lewis with the following endorsement:

I vote for the Math Area policy. As a former teacher, I am uncomfortable without at least a policy that encourages students to mark their questions as homework and responders to answer such questions with hints rather than fully worked out solutions. I know there are many problems with such a policy, but I'd rather not just capitulate in the face of them and open things up totally for homework cheating (for that's what it is). At least if there is a policy in place, some if not most of the honest folks will do the right thing. Without such a policy, honest folks are penalized.

• I don't like the policy because of the neverending struggle marking homework causes, but there is some good general advice in there we might want to copy in any case. – Raphael Jun 29 '12 at 12:39
• I like this policy because it is the same for the sites I visit, and I like the consistency. – Aryabhata Jul 3 '12 at 19:36
• Please use the homework tag. This serves several purposes: the most important of which is to let the answerer know to give an answer that more clearly explains the underlying concepts. In other words, we believe in the "teach a man to fish" philosophy. So unless there's a meta tag, poorly explained answers are acceptable? If the user needs help with concepts in an answer it might even be a good idea to post another question about those concepts. – Matthew Read Jul 4 '12 at 21:35
• @MatthewRead: No. Users can state in the question text that they prefer hints and/or need details. Usually, the way the question if phrased also allows to estimate the asker's level. Even though, follow-up questions are a good tool. – Raphael Jul 5 '12 at 6:50
• – Raphael Jul 30 '14 at 15:12

## Policy

Homework questions are not treated differently.

## Rationale

This way homework questions do not get any special treatment, they are subject to the usual quality control rules. We do not need to argue over whether a question is homework or not, and we can get rid of the meaningless tag. We may have guidelines advising how to post a good homework question and how to answer homework questions, but these will not be binding rules.

### Source

Patrick87's proposal and Gilles's proposal (merged due to similarity)