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# zbMATH — the first resource for mathematics

## IMA Journal of Mathematical Control and Information

 Short Title: IMA J. Math. Control Inf. Publisher: Oxford University Press, Oxford; Institute of Mathematics and its Applications (IMA), Southend-on-Sea, Essex ISSN: 0265-0754; 1471-6887/e Online: https://academic.oup.com/imamci/issue Comments: Indexed cover-to-cover
 Documents Indexed: 1,296 Publications (since 1984)
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#### Latest Issues

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#### Authors

 37 Mahmoud, Magdi Sadik Mostafa 19 Owens, David Howard 16 Tsai, Jason Sheng-Hong 15 Shieh, Leang-San 15 Xu, Gen-Qi 13 Kowalewski, Adam 13 Niculescu, Silviu-Iulian 13 Rosenblueth, Javier F. 12 Cao, Jinde 12 Karampetakis, Nicholas P. 12 Kobayashi, Toshihiro 12 Shi, Peng 10 Arvanitis, Kostas G. 9 Caines, Peter Edwin 9 de la Sen, Manuel 8 Bahaa, Gaber Mohamed 8 Boukas, El-Kébir 8 Ionescu, Vlad 7 Balachandran, Krishnan 7 Bell, David J. 7 Dragan, Vasile 7 Hu, Guangda 7 Ibrir, Salim 7 Kubrusly, Carlos S. 7 Leiva, Hugo 7 Lien, Chang-Hua 7 Otsuka, Naohisa 7 Poznyak, Aleksandr Semënovich 7 Sano, Hideki 7 Soh, Cheong Boon 7 Spurgeon, Sarah K. 7 Sun, Bing 7 Vardulakis, Antonis-Ioanis G. 7 Xu, Shengyuan 7 Yu, Ker-Wei 6 Fridman, Emilia 6 Guo, Shu-Mei 6 Gutman, Shaul 6 Kalogeropoulos, Grigoris I. 6 Karafyllis, Iasson 6 Kotsios, Stelios 6 Medina, Rigoberto 6 Teo, Kok Lay 6 Vinter, Richard B. 6 Wu, Jingbo 6 Xia, Yuanqing 5 Awad, Adnan M. 5 Banks, Stephen Paul 5 Boutoulout, Ali 5 Chen, Jenq-Der 5 Chung, Long-Yeu 5 Costa, Oswaldo Luiz V. 5 Debeljkovic, Dragutin Lj. 5 Duan, Guangren 5 Hao, Fei 5 Ilchmann, Achim 5 Ismail, Abdulla 5 Li, Zhiwu 5 Noakes, J. Lyle 5 Rogers, Eric 5 Sadek, Ibrahim S. 5 Townley, Stuart B. 5 Vũ Ngoc Phát 5 Yang, Xiaosong 4 Barb, Florin Dan 4 Benzaouia, Abdellah 4 Borzabadi, Akbar Hashemi 4 Chao, Daniel Yuh 4 Chen, Guanrong 4 Chen, Shinn-Horng 4 Chojnowski, Fabien 4 Darouach, Mohamed 4 Effati, Sohrab 4 El Bouhtouri, Abdelmoula 4 El Hadri, Kamal 4 Farahi, Mohammad Hadi 4 Goodall, David P. 4 Grabowski, Piotr 4 Grimble, Michael John 4 Hadd, Said 4 Han, Zhongjie 4 Inaba, Hiroshi 4 Karcanias, Nicos 4 Krakowiak, Anna 4 Liu, Jianzhou 4 Logemann, Hartmut 4 Mondié, Sabine 4 Nazemi, Alireza 4 Nounou, Hazem Numan 4 Oya, Masahiro 4 Pagilla, Prabhakar R. 4 Perdon, Anna-Maria 4 Prätzel-Wolters, Dieter 4 Roberts, Adrian P. 4 Sánchez Licea, Gerardo 4 Shubov, Marianna A. 4 Wang, Zidong 4 Yan, Xinggang 4 Yung, Siu Pang 4 Zeheb, Ezra ...and 1,639 more Authors
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#### Fields

 1,118 Systems theory; control (93-XX) 194 Calculus of variations and optimal control; optimization (49-XX) 131 Ordinary differential equations (34-XX) 74 Partial differential equations (35-XX) 67 Operations research, mathematical programming (90-XX) 63 Probability theory and stochastic processes (60-XX) 49 Linear and multilinear algebra; matrix theory (15-XX) 40 Computer science (68-XX) 32 Numerical analysis (65-XX) 28 Operator theory (47-XX) 28 Mechanics of particles and systems (70-XX) 27 Mechanics of deformable solids (74-XX) 24 Biology and other natural sciences (92-XX) 24 Information and communication theory, circuits (94-XX) 19 Dynamical systems and ergodic theory (37-XX) 16 Statistics (62-XX) 15 Real functions (26-XX) 11 Functional analysis (46-XX) 11 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 10 Approximations and expansions (41-XX) 8 Integral equations (45-XX) 7 Global analysis, analysis on manifolds (58-XX) 6 Functions of a complex variable (30-XX) 5 General topology (54-XX) 4 Combinatorics (05-XX) 4 Measure and integration (28-XX) 4 Integral transforms, operational calculus (44-XX) 4 Differential geometry (53-XX) 4 Manifolds and cell complexes (57-XX) 4 Fluid mechanics (76-XX) 3 General and overarching topics; collections (00-XX) 3 Number theory (11-XX) 3 Topological groups, Lie groups (22-XX) 3 Difference and functional equations (39-XX) 3 Harmonic analysis on Euclidean spaces (42-XX) 3 Quantum theory (81-XX) 2 Mathematical logic and foundations (03-XX) 2 Convex and discrete geometry (52-XX) 2 Algebraic topology (55-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 History and biography (01-XX) 1 Field theory and polynomials (12-XX) 1 Commutative algebra (13-XX) 1 Algebraic geometry (14-XX) 1 Nonassociative rings and algebras (17-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Special functions (33-XX)

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763 Publications have been cited 3,729 times in 3,175 Documents Cited by Year
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A new approach to the $$H_{\infty}$$ design of optimal digital linear filters. Zbl 0678.93063
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Strong stabilization of neutral functional differential equations. Zbl 1005.93026
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2002
On the scalar rational interpolation problem. Zbl 0637.93014
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Impulsive control of a Lotka-Volterra system. Zbl 0949.93069
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1998
Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systems via continuous and discontinuous feedback control. Zbl 0662.93059
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1984
Observers for one-sided Lipschitz nonlinear systems. Zbl 1113.93021
Hu, Guangda
2006
Stability of linear systems with general sawtooth delay. Zbl 1206.93080
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2010
The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control. Zbl 1049.93017
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2001
Algebraic analysis of linear multidimensional control systems. Zbl 1158.93319
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1999
Stabilization and $${\mathcal H}_\infty$$ control of two-dimensional Markovian jump systems. Zbl 1069.93007
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2004
Splines of class $$C^ k$$ on non-Euclidean spaces. Zbl 0860.58013
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1995
Robust $${\mathcal H}_\infty$$ constant gain feedback stabilization of stochastic systems. Zbl 1108.93074
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2005
Exponential decay of Timoshenko beam with locally distributed feedback. Zbl 0990.93055
Shi, Dong-Hua; Feng, De-Xing
2001
The Salamon-Weiss class of well-posed infinite-dimensional linear systems: A survey. Zbl 0880.93021
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1997
On the spectral-Lyapunov approach to parametric optimization of distributed-parameter systems. Zbl 0721.49006
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1990
Optimal control of nonlinear Volterra integral equations via Legendre polynomials. Zbl 1275.49056
Tohidi, E.; Samadi, O. R. N.
2013
A vector small-gain theorem for general non-linear control systems. Zbl 1228.93108
Karafyllis, Iasson; Jiang, Zhong-Ping
2011
Static output feedback design for uncertain linear discrete time systems. Zbl 1049.93069
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2004
Impulsive stabilization of nonlinear systems. Zbl 0789.93101
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1993
Mechanical control systems on Lie algebroids. Zbl 1106.93020
Cortés, Jorge; Martínez, Eduardo
2004
Non-identifier-based adaptive control of dynamical systems: A survey. Zbl 0824.93036
Ilchmann, Achim
1991
Adaptive control of infinite-dimensional systems without parameter estimation: An overview. Zbl 0880.93023
Logemann, Hartmut; Townley, Stuart
1997
Analytical foundations of Volterra series. Zbl 0668.93042
Boyd, Stephen; Chua, L. O.; Desoer, C. A.
1984
A delay fractioning approach to robust sliding mode control for discrete-time stochastic systems with randomly occurring non-linearities. Zbl 1228.93025
Hu, Jun; Wang, Zidong; Gao, Huijun
2011
The linear-quadratic control problem for retarded systems with delays in control and observation. Zbl 0646.34078
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1985
An algebraic analysis approach to linear time-varying systems. Zbl 1106.93019
Zerz, Eva
2006
A maximum principle for nonconservative self-adjoint systems. Zbl 0704.73065
Sloss, James M.; Bruch, John C. jun.; Sadek, Ibrahim S.
1989
New results on the operator Carleson measure criterion. Zbl 0874.93031
Hansen, Scott; Weiss, George
1997
A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control. Zbl 1050.49002
Rampazzo, F.; Vinter, R. B.
1999
An upwind finite-difference method for the approximation of viscosity solutions to Hamilton-Jacobi-Bellman equations. Zbl 0952.49025
Wang, S.; Gao, F.; Teo, K. L.
2000
Lag synchronization of chaotic delayed neural networks via impulsive control. Zbl 1242.93079
Li, Xiaodi; Fu, Xilin
2012
Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. Zbl 1279.93090
Guesmia, Aissa
2013
On controllability of semilinear stochastic systems in Hilbert spaces. Zbl 1138.93313
Mahmudov, Nazim I.
2002
Oscillations in lossless propagation models: A Lyapunov-Krasovskij approach. Zbl 1020.93010
Răsvan, Vladimir; Niculescu, Silviu-Iulian
2002
Reduced-order observer design for one-sided Lipschitz nonlinear systems. Zbl 1179.93049
Xu, Mingyue; Hu, Guang-Da; Zhao, Yanbin
2009
An analytical theory for Riemannian cubic polynomials. Zbl 1138.58307
Giambò, Roberto; Giannoni, Fabio; Piccione, Paolo
2002
A new Lyapunov technique for robust control of systems with uncertain non-small delays. Zbl 1095.93009
Fridman, Emilia
2006
Existence and optimal control for periodic parabolic equations with nonlocal terms. Zbl 0926.49002
Allegretto, W.; Nistri, P.
1999
Delay-dependent robust stabilization and $$H_\infty$$-control of uncertain stochastic systems with time-varying delay. Zbl 1053.93012
Chen, Wu-Hua; Guan, Zhi-Hong; Lu, Xiaomei
2004
Stability and hyperbolicity of linear systems with dalayed state: A matrix-pencil approach. Zbl 0918.93046
Niculescu, Silviu-Iulian
1998
Stability criteria for a class of linear neutral systems with time-varying discrete and distributed delays. Zbl 1046.93039
Han, Qing-Long
2003
Finite time stability of differential inclusions. Zbl 1115.93351
Moulay, E.; Perruquetti, W.
2005
On the computation of the generalized inverse of a polynomial matrix. Zbl 0987.93023
Karampetakis, N. P.; Tzekis, P.
2001
Boundary feedback stabilization by time delay for one-dimensional wave equations. Zbl 1198.93177
Gugat, Martin
2010
Observability and detectability of a class of discrete-time stochastic linear systems. Zbl 1095.93004
Dragan, Vasile; Morozan, Toader
2006
Wave energy decay under fractional derivative controls. Zbl 1095.93015
Mbodje, Brahima
2006
Optimal control and stabilization for nonlinear systems. Zbl 0773.49018
Banks, S. P.; Mhana, K. J.
1992
Consistency and Liapunov stability of linear descriptor systems: A geometric analysis. Zbl 0637.93051
Owens, David H.; Debeljkovic, Dragutin Lj.
1985
Model-reference adaptive control. Stability, parameter convergence, and robustness. Zbl 0622.93033
Sastry, S. Shankar
1984
Active disturbance rejection control and sliding mode control of one-dimensional unstable heat equation with boundary uncertainties. Zbl 1308.93057
Liu, Jun-Jun; Wang, Jun-Min
2015
Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces. Zbl 1279.93024
Yan, Zuomao
2013
A class of predefined-time stable dynamical systems. Zbl 1402.93192
Sánchez-Torres, Juan Diego; Gómez-Gutiérrez, David; López, Esteban; Loukianov, Alexander G.
2018
Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term. Zbl 1458.93082
Liu, Wenjun; Chen, Kewang; Yu, Jun
2017
Stability radii for some propagation models. Zbl 0873.93067
Halanay, A.; Rasvan, Vl.
1997
Analysis and synthesis of uncertain switched discrete-time systems. Zbl 1127.93050
Mahmoud, Magdi S.; Nounou, Mohamed N.; Nounou, Hazem N.
2007
Solving a class of linear and non-linear optimal control problems by homotopy perturbation method. Zbl 1251.49037
Effati, S.; Saberi Nik, H.
2011
Switched exponential state estimation and robust stability for interval neural networks with the average dwell time. Zbl 1328.93252
Li, Ning; Cao, Jinde
2015
Dynamic routing in fully connected networks. Zbl 0697.94017
Gibbens, R. J.; Kelly, F. P.
1990
Existence of almost periodic solutions for strong stable impulsive differential equations. Zbl 0987.34040
Stamov, Gani T.
2001
Non-uniform stabilization of control systems. Zbl 1016.93059
Karafyllis, Iasson
2002
Finite-time stability of delayed systems. Zbl 0979.93095
Lazarević, M. P.; Debeljković, D. Lj.; Nenadić, Z. Lj.; Milinković, S. A.
2000
Existence and generic properties of $$L^ 2$$ approximants for linear systems. Zbl 0637.93027
Baratchart, Laurent
1986
An ergodic control problem for reflected diffusion with jump. Zbl 0638.93079
Menaldi, José Luis; Robin, Maurice
1984
On stability radii of infinite-dimensional time-varying discrete-time systems. Zbl 0812.93055
Wirth, Fabian; Hinrichsen, Diederich
1994
A predictive least-squares principle. Zbl 0626.93069
Rissanen, Jorma
1986
Stability result of the Timoshenko system with delay and boundary feedback. Zbl 1252.93110
Said-Houari, Belkacem; Soufyane, Abdelaziz
2012
Consensus of multi-agent systems on time scales. Zbl 1256.93068
Shen, Jun; Cao, Jinde
2012
Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems. Zbl 1256.93033
Polyakov, Andrey; Poznyak, Alexander
2012
Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces. Zbl 1297.93038
Liu, Zhenhai; Lv, Jingyun; Sakthivel, R.
2014
Identifiability analysis of linear delay-differential systems. Zbl 1009.93015
Belkoura, L.; Orlov, Y.
2002
Fast sampling and stability of nonlinear sampled-data systems. I: Existence theorems. Zbl 0717.93041
Owens, D. H.; Zheng, Y.; Billings, S. A.
1990
Decentralized output feedback control of a class of large-scale interconnected systems. Zbl 1137.93007
Zhu, Yongliang; Pagilla, Prabhakar R.
2007
Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion. Zbl 1328.93056
Ahmed, Hamdy M.
2015
Adaptation of sliding modes. Zbl 1273.93035
Bartolini, Giorgio; Levant, Arie; Plestan, Franck; Taleb, Mohammed; Punta, Elisabetta
2013
Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay. Zbl 1044.49018
Kotarski, W.; El-Saify, H. A.; Bahaa, G. M.
2002
Positive-real structure and high-gain adaptive stabilization. Zbl 0631.93058
Owens, David H.; Prätzel-Wolters, Dieter; Ilchmann, Achim
1987
A note on observer for one-sided Lipschitz non-linear systems. Zbl 1160.93006
Hu, Guang-Da
2008
Systems with time delay in the calculus of variations: A variational approach. Zbl 0655.49017
Rosenblueth, Javier F.
1988
Solving fractional optimal control problems with fixed or free final states by Haar wavelet collocation method. Zbl 1397.93104
Hosseinpour, Soleiman; Nazemi, Alireza
2016
Recursive maximum likelihood identification method for a multivariable controlled autoregressive moving average system. Zbl 1397.93213
Wang, Dongqing; Gao, Yanpu
2016
Riesz basis property of the generalized eigenvector system of a Timoshenko beam. Zbl 1066.93027
Xu, Gen-Qi; Feng, De-Xing; Yung, Siu Pang
2004
Shapes and measures. Zbl 0935.49025
Fakharzadeh, A.; Rubio, J. E.
1999
Mixed constraints in optimal control: An implicit function theorem approach. Zbl 1124.49013
de Pinho, Maria do Rosário; Rosenblueth, Javier F.
2007
Stability analysis of heat-exchanger equations with boundary feedbacks. Zbl 0917.93034
Kunimatsu, Noboru; Sano, Hideki
1998
A computational method for a class of dynamical optimization problems in which the terminal time is conditionally free. Zbl 0673.49013
Teo, K. L.; Goh, C. J.; Lim, C. C.
1989
Observer-based event-triggered control for certain and uncertain linear systems. Zbl 1279.93073
Chen, Xia; Hao, Fei
2013
A broad class of evolution equations are approximately controllable, but never exactly controllable. Zbl 1108.93014
Bárcenas, Diomedes; Leiva, Hugo; Sívoli, Zoraida
2005
Stability of non-linear systems by means of event-triggered sampling algorithms. Zbl 1297.93125
Seuret, Alexandre; Prieur, Christophe; Marchand, Nicolas
2014
Exponential stability and robust $$H_\infty$$ control for uncertain discrete switched systems with interval time-varying delay. Zbl 1216.93041
Lien, Chang-Hua; Yu, Ker-Wei; Chung, Yeong-Jay; Chang, Hao-Chin; Chung, Long-Yeu; Chen, Jenq-Der
2011
Delay-dependent dissipativity of singular time-delay systems. Zbl 1159.93333
Mahmoud, Magdi S.
2009
A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix. Zbl 1400.49033
Nemati, Ali; Yousefi, Sohrab Ali
2017
Using delays and time-varying gains to improve the static output feedback stabilizability of linear systems: a comparison. Zbl 1083.93047
Michiels, Wim; Niculescu, Silviu-Iulian; Moreau, Luc
2004
Controllability of neutral functional evolution integrodifferential systems with infinite delay. Zbl 1146.93006
Balachandran, K.; Leelamani, A.; Kim, J.-H.
2008
Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis. Zbl 1106.93043
Karafyllis, Iasson
2006
Vector-valued integration with applications to the operator-valued $$H^ \infty$$ space. Zbl 0880.28008
Thomas, E. G. F.
1997
Exact controllability of damped Timoshenko beam. Zbl 0991.93016
Shubov, Marianna A.
2000
Regular linear systems governed by systems with state, input and output delays. Zbl 1115.93037
Hadd, Said; Idrissi, Abdelali
2005
On a sufficient condition for finite-time partial stability and stabilization: applications. Zbl 1193.93151
Jammazi, Chaker
2010
Reconstructing the potential for the one-dimensional Schrödinger equation from boundary measurements. Zbl 1286.49039
Avdonin, Sergei A.; Mikhaylov, Victor S.; Ramdani, Karim
2014
On the explicit feedback stabilization of one-dimensional linear nonautonomous parabolic equations via oblique projections. Zbl 1436.93109
Rodrigues, Sérgio S.; Sturm, Kevin
2020
On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network. Zbl 1436.93019
Cerpa, Eduardo; Crépeau, Emmanuelle; Moreno, Claudia
2020
Existence, stability and controllability results of a Volterra integro-dynamic system with non-instantaneous impulses on time scales. Zbl 1436.93098
Malik, Muslim; Kumar, Vipin
2020
Well-posedness and stability results for laminated Timoshenko beams with interfacial slip and infinite memory. Zbl 1436.93116
Guesmia, Aissa
2020
Control of bounded solutions for first-order singular differential equations with impulses. Zbl 1451.93026
Kong, Fanchao; Nieto, Juan J.
2020
A weak maximum principle for optimal control problems with mixed constraints under a constant rank condition. Zbl 1454.49026
Andreani, Roberto; Antunes de Oliveira, Valeriano; Tomaz Pereira, Jamielli; Nunes Silva, Geraldo
2020
Approximate controllability of semilinear impulsive functional differential systems with non-local conditions. Zbl 07411959
Arora, Sumit; Singh, Soniya; Dabas, Jaydev; Mohan, Manil T.
2020
Optimal scheduling of discrete-time switched linear systems. Zbl 1436.93080
Xu, Wei; Feng, Zhi Guo; Lin, Gui-Hua; Yu, Liying
2020
Stability of linear equations with differentiable operators in a Hilbert space. Zbl 1436.93115
Gil’, Michael
2020
Mild solution and controllability of second-order non-local retarded semilinear systems. Zbl 1436.93022
Kumar, Suman; Kumar Tomar, Nutan
2020
On event-triggered tracking for non-linear SISO systems via sliding mode control. Zbl 1436.93088
Zarei, Farzad; Shafiei, Mohammad Hossein
2020
A dissipative approach to the stability of multi-order fractional systems. Zbl 1436.93097
Gallegos, Javier A.; Duarte-Mermoud, Manuel A.
2020
Chaotic dynamics from a pseudo-linear system. Zbl 1447.93136
Ghane, Hamed; Sterk, Alef E.; Waalkens, Holger
2020
On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming. Zbl 1447.93279
Ferrante, Francesco; Gouaisbaut, Frédéric; Tarbouriech, Sophie
2020
Constrained controllability of second order retarded nonlinear systems with nonlocal condition. Zbl 1447.93028
Kumar, Suman; Sakthivel, R.
2020
Generalized fractional-order Bernoulli-Legendre functions: an effective tool for solving two-dimensional fractional optimal control problems. Zbl 1418.49017
Rahimkhani, Parisa; Ordokhani, Yadollah
2019
Optimal control for a time delay multi-strain tuberculosis fractional model: a numerical approach. Zbl 1417.92198
Sweilam, N. H.; AL-Mekhlafi, S. M.
2019
Approximate controllability for a retarded semilinear stochastic evolution system. Zbl 1417.93075
Mokkedem, Fatima Zahra; Fu, Xianlong
2019
Impulsive discrete-time GRNs with probabilistic time delays, distributed and leakage delays: an asymptotic stability issue. Zbl 1417.92057
Pandiselvi, S.; Raja, R.; Cao, Jinde; Li, Xiaodi; Rajchakit, G.
2019
An output stabilization of infinite dimensional semilinear systems. Zbl 1417.93244
Zerrik, El Hassan; Ezzaki, Lahcen
2019
Multi-switching combination synchronization of discrete-time hyperchaotic systems for encrypted audio communication. Zbl 07421075
Hammami, Sonia
2019
Finite-time robust decentralized control for uncertain singular large-scale systems with exogenous disturbances. Zbl 07421101
Wo, Songlin; Han, Xiaoxin
2019
Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects. Zbl 07421105
Biccari, Umberto; Hernández-Santamaría, Víctor
2019
Non-fragile $$H_{\infty}$$ state estimation for discrete-time complex networks with randomly occurring time-varying delays and channel fadings. Zbl 1417.93129
Zhang, Sunjie; Wang, Zidong; Ding, Derui; Wei, Guoliang; Alsaadi, Fuad E.
2019
Stabilization for a joint string equation with input disturbance. Zbl 1417.93250
Chen, Yunlan; Xu, Genqi
2019
Sliding mode control for quantized semi-Markovian switching systems with bounded disturbances. Zbl 1417.93101
Liu, Xinghua; Ma, Guoqi
2019
A class of predefined-time stable dynamical systems. Zbl 1402.93192
Sánchez-Torres, Juan Diego; Gómez-Gutiérrez, David; López, Esteban; Loukianov, Alexander G.
2018
Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces. Zbl 1402.93056
Vijayakumar, V.
2018
Fractional optimal control problem for variational inequalities with control constraints. Zbl 1403.49005
Bahaa, G. M.
2018
Wirtinger-like Lyapunov-Krasovskii functionals for discrete-time delay systems. Zbl 1402.93173
Seuret, Alexandre; Fridman, Emilia
2018
Fractional optimal control problems: optimality conditions and numerical solution. Zbl 1403.49026
Sayevand, Khosro; Rostami, Mohammadreza
2018
Controllability of retarded semilinear fractional system with non-local conditions. Zbl 1402.93052
Arora, Urvashi; Sukavanam, N.
2018
Distributed adaptive consensus of heterogeneous multi-agent systems with unknown coupling weights. Zbl 1402.93156
Li, Hongxia; Li, Junmin; Zhu, Xiaojun; Wu, Haibin; Gan, Zhongxue
2018
Error feedback regulation problem for regular linear systems. Zbl 1402.93201
Boulite, Said; Hadd, Said; Saij, Rachid
2018
Controllability of the wave equation via multiplicative controls. Zbl 1402.93053
El Harraki, I.; Boutoulout, A.
2018
Controllability of semilinear stochastic control system with finite delay. Zbl 1402.93055
Shukla, Anurag; Sukavanam, N.; Pandey, D. N.
2018
Influence of discretization step on positivity of a certain class of two-dimensional continuous-discrete fractional linear systems. Zbl 1402.93111
Ghezzar, Mohammed Amine; Bouagada, Djillali; Chadli, Mohammed
2018
Some results on finite-time stabilizability: application to triangular control systems. Zbl 1402.93206
Jammazi, Chaker
2018
Dynamic time-delayed feedback control of Westwood+ TCP flow control model with communication delay. Zbl 1402.93128
Yu, Hongyan; Guo, Songtao; Wang, Fei; Yang, Yang
2018
Containment control of fractional-order nonlinear multi-agent systems under fixed topologies. Zbl 1402.93044
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2018
Optimal control of linear multi-delay systems with piecewise constant delays. Zbl 1403.49024
Marzban, Hamid Reza; Hoseini, Sayyed Mohammad
2018
Approximate controllability of Caputo fractional neutral stochastic differential inclusions with state-dependent delay. Zbl 1417.93080
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Finite-time distributed formation tracking control of multi-UAVs with a time-varying reference trajectory. Zbl 1417.93228
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2018
Some stability proofs on proxy-based sliding mode control. Zbl 1417.93239
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2018
$$H_\infty$$ dynamical observer-based control for descriptor systems. Zbl 1402.93103
Osorio-Gordillo, Gloria-Lilia; Darouach, Mohamed; Boutat-Baddas, Latifa; Astorga-Zaragoza, Carlos-Manuel
2018
Adaptive nonlinear gain based composite nonlinear feedback controller with input saturation. Zbl 1402.93155
Kulkarni, A.; Purwar, S.
2018
On the exponential stability of nonlinear delay systems with impulses. Zbl 1402.93217
You, Zhongli; Wang, JinRong
2018
Frequency weighted passivity preserving model reduction technique. Zbl 1402.93070
Imran, Muhammad; Ghafoor, Abdul
2018
Some new results on the global uniform asymptotic stability of time-varying dynamical systems. Zbl 1402.93216
Taieb, Nizar Hadj; Hammami, Mohamed Ali
2018
Exponential stability and stabilization for uncertain discrete-time periodic systems with time-varying delay. Zbl 1402.93195
Dong, Yali; Liang, Shuang; Guo, Liangliang; Wang, Wenqin
2018
Criteria for the exponential stability of linear evolution difference equations. Zbl 1402.93218
Zada, Akbar; Li, Tongxing; Arif, Muhammad; Lassoued, Dhaou
2018
Synchronization for coupled networks with Markov switching: ergodicity and $$M$$-matrix method. Zbl 1402.93048
Pan, Lijun; Cao, Jinde
2018
Regional boundary gradient observability of semilinear hyperbolic systems via HUM approach. Zbl 1402.93062
Khazari, A.; Boutoulout, A.; El Alaoui, F. Z.
2018
Controllability of linearized systems implies local finite-time stabilizability: applications to finite-time attitude control. Zbl 1402.93054
Jammazi, Chaker; Abichou, Azgal
2018
A survey on the theory of bonds. Zbl 1404.53009
Li, Zijia; Schicho, Josef; Schröcker, Hans-Peter
2018
Controllability results for second order semilinear functional differential inclusions based on Kuratowski product measures. Zbl 1402.93057
Xiao, Jian-Zhong; Zhu, Xing-Hua; Yao, Zhong-Hao
2018
Spatial discretization error in Kalman filtering for discrete-time infinite dimensional systems. Zbl 1402.93243
Aalto, Atte
2018
Transformed Legendre spectral method for solving infinite horizon optimal control problems. Zbl 1403.49027
Shahini, M.; Mehrpouya, M. A.
2018
Time scale separation in control of a single-link flexible-joint robot manipulator. Zbl 1402.93180
Asadi, Mehrnoosh; Shandiz, Heydar Toossian; Khayatian, Alireza
2018
Feedback controls for exact remediability in disturbed dynamical systems. Zbl 1402.93123
Afifi, Larbi; Lasri, Karima; Joundi, Meriem; Amimi, Najia
2018
Dendriform-tree setting for fully non-commutative Fliess operators. Zbl 1402.93088
Duffaut Espinosa, Luis A.; Gray, W. Steven; Ebrahimi-Fard, Kurusch
2018
Position and posture control for a class of second-order nonholonomic underactuated mechanical system. Zbl 1402.93185
Xiong, Pei-yin; Lai, Xu-zhi; Wu, Min
2018
Absolute stability of Lurie systems with two additive time-varying delays. Zbl 1402.93194
Zhang, Rui; Li, Junmin; Jiao, Jianmin
2018
Adaptive full-order and reduced-order observers for one-sided Lur’e systems with set-valued mappings. Zbl 1402.93066
Shi, Min-Jie; Huang, Jun; Chen, Liang; Yu, Lei
2018
$$H_{\infty}$$ based anti-windup controller for two-dimensional discrete delayed systems in presence of actuator saturation. Zbl 1402.93104
Pal, Vipin Chandra; Negi, Richa
2018
Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term. Zbl 1458.93082
Liu, Wenjun; Chen, Kewang; Yu, Jun
2017
A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix. Zbl 1400.49033
Nemati, Ali; Yousefi, Sohrab Ali
2017
Optimal control of a class of non-linear time-delay systems via hybrid functions. Zbl 1397.93098
Dadkhah, Mahmood; Farahi, Mohammad Hadi; Heydari, Aghileh
2017
Iterative estimation for a non-linear IIR filter with moving average noise by means of the data filtering technique. Zbl 1417.93316
Wang, Yanjiao; Ding, Feng
2017
Tent-method application for minmax stabilization and maxmin testing. Zbl 1397.49036
Alexandrov, V. V.; Bugrov, D. I.; Corona Morales, G.; Tikhonova, K. V.
2017
A numerical approximation for delay fractional optimal control problems based on the method of moments. Zbl 1400.49031
Dehghan, Reza; Keyanpour, Mohammad
2017
Viscosity solutions for a system of PDEs and optimal switching. Zbl 1417.93154
El Asri, Brahim; Fakhouri, Imade
2017
Bogdanov-Takens bifurcation in a Hopfield network model with bidirectional connection and multiple delays. Zbl 1458.93098
Ben Hamed, Aws; Ben Hamed, Bassem
2017
Dissipative lag synchronization of chaotic Lur’e systems with unknown disturbances. Zbl 1397.93047
Zhao, Min; Liu, Ronghua; Gao, Yanbo
2017
Synchronization of complex networks with coupling delay via pinning control. Zbl 1397.93182
Yang, Xinsong; Cao, Jinde
2017
Continuous fractional sliding mode-like control for exact rejection of non-differentiable Hölder disturbances. Zbl 1397.93044
Muñoz-Vázquez, A. J.; Parra-Vega, V.; Sánchez-Orta, A.
2017
A collocation method via block-pulse functions for solving delay fractional optimal control problems. Zbl 1400.49032
Hosseinpour, Soleiman; Nazemi, Alireza
2017
Stabilization of an Euler-Bernoulli beam system with a tip mass subject to non-uniform bounded disturbance. Zbl 1397.93170
Li, Yanfang; Xu, Genqi; Han, Zhongjie
2017
Exponential stabilization for Timoshenko beam with different delays in the boundary control. Zbl 1397.93184
Liu, Xiu Fang; Xu, Gen Qi
2017
Stability of fractional-order periodic discrete-time linear systems. Zbl 1397.93158
Tehrani, H. A.; Esmaeili, J.
2017
Infinite-horizon non-zero-sum stochastic differential games with additive structure. Zbl 1416.91033
Jasso-Fuentes, Héctor; López-Barrientos, José Daniel; Escobedo-Trujillo, Beatris Adriana
2017
The solution bounds and fixed point iterative algorithm for the discrete coupled algebraic Riccati equation applied to automatic control. Zbl 1397.93072
Liu, Jianzhou; Wang, Li; Zhang, Juan
2017
Approximate controllability of impulsive fractional stochastic partial neutral integrodifferential inclusions with infinite delay. Zbl 1417.93081
Yan, Zuomao; Jia, Xiumei
2017
Distributed tracking control of the Schrödinger equation with internal disturbance. Zbl 1417.93235
Cui, Haoyue; Xu, Genqi; Han, Zhongjie
2017
Regional stabilization of semi-linear parabolic systems. Zbl 1417.93251
El Harraki, I.; El Alami, A.; Boutoulout, A.; Serhani, M.
2017
Optimal control problem for a class of bilinear systems via block pulse functions. Zbl 1418.49031
Balatif, Omar; Rachik, Mostafa; Labriji, El houssine; Rachik, Zineb
2017
Tracking control for uncertain fractional-order chaotic systems based on disturbance observer and neural network. Zbl 1417.93179
Shao, Shuyi; Chen, Mou; Wu, Qingxian
2017
Global asymptotic stabilization of a class of high-order feedforward systems with unknown control directions. Zbl 1397.93178
Liu, Wenhui; Xu, Shengyuan; Zhang, Baoyong; Zou, Yun
2017
Stabilization of a pendulum in dynamic boundary feedback with a memory type heat equation. Zbl 1397.93171
Lu, Lu; Wang, Jun-Min; Zhao, Dong
2017
Reliable dissipative sampled-data control for uncertain systems with actuator failures and application to vehicle dynamics. Zbl 1397.93131
Sakthivel, R.; Santra, Srimanta; Mathiyalagan, K.; Anthoni, S. Marshal
2017
Interpolation techniques for robust constrained model predictive control based on polyhedral invariant set. Zbl 1397.93057
Kheawhom, Soorathep; Bumroongsri, Pornchai
2017
A direct approach for approximate optimal control of integro-differential equations based on homotopy analysis and parametrization method. Zbl 1400.49038
Alipour, M.; Vali, M. A.; Borzabadi, A. H.
2017
Model reduction of discrete-time linear systems using Meixner-like filters. Zbl 1397.93039
Maraoui, Safa; Krifa, Abdelkader; Bouzrara, Kais
2017
Event-driven model-free control in motion control with comparisons. Zbl 1397.93148
Wang, Jing; Mounier, Hugues; Niculescu, Silviu-Iulian; Geamanu, Marcel-Stefan; Cela, Arben
2017
Exponential stabilization and motion planning of an overhead crane system. Zbl 1397.93166
Elharfi, Abdelhadi
2017
$$\mathcal{H}_{2}$$ model order reduction for bilinear systems based on the cross gramian. Zbl 1397.93040
Yang, Ping; Xu, Kang-Li; Jiang, Yao-Lin
2017
Stability and $$L_1$$-gain control for positive impulsive switched systems with mixed time-varying delays. Zbl 1397.93176
Li, Shuo; Xiang, Zhengrong
2017
The cross-motion invariant group and its application to kinematics. Zbl 1411.70005
Adorno, Bruno Vilhena; Fraisse, Philippe
2017
Iterative learning control for quasi-one-sided Lipschitz non-linear systems. Zbl 1417.93132
Gu, Weiguo; Fu, Qin; Wu, Jianrong
2017
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#### Cited in 376 Journals

 270 Automatica 207 International Journal of Control 181 Systems & Control Letters 159 Journal of the Franklin Institute 101 Applied Mathematics and Computation 80 Journal of Mathematical Analysis and Applications 72 Mathematical Problems in Engineering 68 International Journal of Robust and Nonlinear Control 65 Linear Algebra and its Applications 58 Journal of Optimization Theory and Applications 55 International Journal of Systems Science 50 SIAM Journal on Control and Optimization 49 MCSS. Mathematics of Control, Signals, and Systems 47 Circuits, Systems, and Signal Processing 46 Nonlinear Dynamics 44 European Journal of Control 40 Asian Journal of Control 34 Abstract and Applied Analysis 33 Advances in Difference Equations 31 Journal of Computational and Applied Mathematics 30 Kybernetika 29 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 29 Nonlinear Analysis. Hybrid Systems 28 Journal of Dynamical and Control Systems 27 International Journal of Systems Science. Principles and Applications of Systems and Integration 26 Computers & Mathematics with Applications 25 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 24 Journal of Differential Equations 23 Applied Mathematics Letters 23 Journal of Systems Science and Complexity 21 International Journal of Applied Mathematics and Computer Science 19 Chaos, Solitons and Fractals 19 Optimal Control Applications & Methods 19 Applied Mathematical Modelling 18 Applied Mathematics and Optimization 18 Information Sciences 18 Journal of Difference Equations and Applications 18 Nonlinear Analysis. Real World Applications 17 Applicable Analysis 17 Complexity 17 Discrete Dynamics in Nature and Society 16 Communications in Nonlinear Science and Numerical Simulation 15 Journal of Mathematical Physics 15 Stochastic Analysis and Applications 15 Mathematical and Computer Modelling 15 Multidimensional Systems and Signal Processing 15 International Journal of Adaptive Control and Signal Processing 14 Mathematical Methods in the Applied Sciences 14 Acta Applicandae Mathematicae 13 Integral Equations and Operator Theory 13 Automation and Remote Control 12 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 12 Journal of Mathematical Sciences (New York) 12 Discrete and Continuous Dynamical Systems. Series B 12 Journal of Applied Mathematics and Computing 11 Fuzzy Sets and Systems 11 Neural Networks 11 Journal of Applied Mathematics 11 Evolution Equations and Control Theory 10 Mathematics and Computers in Simulation 10 Journal of Vibration and Control 9 Numerical Functional Analysis and Optimization 9 Optimization 9 Discrete and Continuous Dynamical Systems. Series S 8 ZAMP. Zeitschrift für angewandte Mathematik und Physik 8 Signal Processing 8 Journal of Global Optimization 8 Dynamics and Control 8 Computational and Applied Mathematics 8 Boundary Value Problems 8 Nonlinear Analysis. Theory, Methods & Applications 7 Chaos 7 International Journal of Geometric Methods in Modern Physics 7 Journal of Control Theory and Applications 7 Journal of Industrial and Management Optimization 7 Journal of Control Science and Engineering 7 Mathematical Control and Related Fields 6 Journal of Symbolic Computation 6 Numerical Algorithms 6 Journal of Dynamics and Differential Equations 6 Archives of Control Sciences 6 SIAM Journal on Scientific Computing 6 Advances in Computational Mathematics 6 Nonlinear Analysis. Modelling and Control 6 Journal of Evolution Equations 6 Journal of Geometric Mechanics 5 Computer Aided Geometric Design 5 Applied Numerical Mathematics 5 Dynamics and Stability of Systems 5 International Journal of Computer Mathematics 5 Mathematical Programming. Series A. Series B 5 Journal of Nonlinear Science 5 Discrete and Continuous Dynamical Systems 5 Journal of Inequalities and Applications 5 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 5 Mediterranean Journal of Mathematics 5 Set-Valued and Variational Analysis 5 Palestine Journal of Mathematics 4 Mathematical Notes 4 Moscow University Mechanics Bulletin ...and 276 more Journals
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#### Cited in 58 Fields

 2,202 Systems theory; control (93-XX) 539 Ordinary differential equations (34-XX) 407 Calculus of variations and optimal control; optimization (49-XX) 381 Partial differential equations (35-XX) 224 Probability theory and stochastic processes (60-XX) 222 Numerical analysis (65-XX) 153 Mechanics of deformable solids (74-XX) 151 Operator theory (47-XX) 144 Biology and other natural sciences (92-XX) 137 Operations research, mathematical programming (90-XX) 106 Linear and multilinear algebra; matrix theory (15-XX) 99 Dynamical systems and ergodic theory (37-XX) 97 Mechanics of particles and systems (70-XX) 80 Computer science (68-XX) 58 Differential geometry (53-XX) 55 Integral equations (45-XX) 49 Difference and functional equations (39-XX) 45 Approximations and expansions (41-XX) 44 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 39 Real functions (26-XX) 39 Information and communication theory, circuits (94-XX) 30 Statistics (62-XX) 28 Global analysis, analysis on manifolds (58-XX) 21 Functions of a complex variable (30-XX) 16 Commutative algebra (13-XX) 15 Associative rings and algebras (16-XX) 14 Fluid mechanics (76-XX) 12 General topology (54-XX) 11 Nonassociative rings and algebras (17-XX) 11 Topological groups, Lie groups (22-XX) 10 Quantum theory (81-XX) 9 Combinatorics (05-XX) 9 Classical thermodynamics, heat transfer (80-XX) 8 Number theory (11-XX) 7 Field theory and polynomials (12-XX) 7 Several complex variables and analytic spaces (32-XX) 7 Special functions (33-XX) 7 Functional analysis (46-XX) 7 Optics, electromagnetic theory (78-XX) 5 General and overarching topics; collections (00-XX) 5 Mathematical logic and foundations (03-XX) 5 Integral transforms, operational calculus (44-XX) 5 Convex and discrete geometry (52-XX) 5 Manifolds and cell complexes (57-XX) 4 Algebraic geometry (14-XX) 4 Group theory and generalizations (20-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 4 Geometry (51-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Potential theory (31-XX) 2 Algebraic topology (55-XX) 2 Mathematics education (97-XX) 1 History and biography (01-XX) 1 Category theory; homological algebra (18-XX) 1 Measure and integration (28-XX) 1 Sequences, series, summability (40-XX) 1 Astronomy and astrophysics (85-XX)


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# The Mystery of Banking, 2nd Edition

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# MURRAY N. ROTHBARD

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Book Details
  Price 2.50 322 p 1,697 KB PDF format 978-1-933550-28-2 2008  by the Ludwig von Mises Institute

PREFACE
Although first published 25 years ago, Murray Rothbard’s
The Mystery of Banking continues to be the only book that
clearly and concisely explains the modern fractional
reserve banking system, its origins, and its devastating effects on
the lives of every man, woman, and child. It is especially appropriate
in a year that will see; a surge in bank failures, central
banks around the globe bailing out failed commercial and investment
banks, double-digit inflation rates in many parts of the
world and hyperinflation completely destroying Zimbabwe’s
economy, that a new edition of Rothbard’s classic work be republished
and made available through the efforts of Lew Rockwell
and the staff at the Ludwig von Mises Institute at an obtainable
price for students and laymen interested in the vagaries of banking
and how inflation and business cycles are created.
In the absence of central-bank intervention, the current financial
meltdown could be a healthy check on the inflation of the
banking system as Rothbard points out in his scathing review of
Lawrence H. White’s Free Banking in Britain: Theory, Experience,
and Debate, 1800–1845 that first appeared in The Review of Austrian
Economics and is included as a part of this new edition to
correct Rothbard’s initial support of White’s work in the first edition.
There have been virtually no bank failures in the United
States since the early 1990s and as Rothbard surmised during that
period where there was “an absence of failure” that “inflation of
money and credit [was] all the more rampant.” Indeed, from January
1990 to April 2008, the United States M-2 money supply
more than doubled from \$3.2 trillion to \$7.7 trillion. Bankers
were living it up, “at the expense of society and the economy faring
worse” (Rothbard’s emphasis).
Although ostensibly it is dodgy real estate loans that are bringing
the banks down this year, in the seminal book that you hold,
Rothbard shows that it is really the fraudulent nature of fractionalized
banking that is the real culprit for the bankers’ demise.
But central bankers will never learn. “We should not have a
system that’s this fragile, that causes this much risk to the economy,”
New York Federal Reserve President Tim Geithner said
after engineering J.P. Morgan’s bailout of the failed Bear Stearns
investment bank in the first quarter of 2008 with the help of the
central bank. Of course the thought of dismantling his employer,
the government leaving the counterfeiting business, and a return
to using the market’s money—gold—didn’t occur to him. More
government regulation in which “the basic rules of the game
establish stronger incentives for building more robust shock
absorbers,” is what he prescribed.
Surely Murray is somewhere laughing.
My introduction to The Mystery of Banking came in 1992 as
I was finishing my thesis at UNLV under Murray’s direction. I
found the book in the university library and couldn’t put it down.
The book was long out of print by that time and being prior to
the start of Amazon.com and other online used book searches, I
was unable to find a copy of the book for purchase. Thus, I fed
dimes into the library copier one Saturday afternoon and made
myself a copy. When the online searches became available I
waited patiently and bought two copies when they surfaced, paying
many times the original \$19.95 retail price (as I write this
AbeBooks.com has three copies for sale ranging from \$199 to
\$225, and Bauman Rare Books recently sold a signed first edition for \$650).
When I discovered Rothbard’s great work I had been a banker
for six years, but like most people working in banking, I had no
clear understanding of the industry. It is not knowledge that is
taught on the job. Murray may have referred to me as “the efficient
banker,” but he was the one who knew the evil implications
of the modern fractionalized banking system: “the pernicious and
inflationary domination of the State.”
DOUGLAS E. FRENCH
JUNE 2008

Preface by Douglas E. French . . . . . . . . . . . . . . . . . . . . . . . . xi
Foreword by Joseph T. Salerno . . . . . . . . . . . . . . . . . . . . . . . xv
I. Money: Its Importance and Origins. . . . . . . . . . . . . . . . . . . 1
1. The Importance of Money . . . . . . . . . . . . . . . . . . . . . . . 1
2. How Money Begins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. The Proper Qualities of Money . . . . . . . . . . . . . . . . . . . 6
4. The Money Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II. What Determines Prices: Supply and Demand . . . . . . . . 15
III. Money and Overall Prices . . . . . . . . . . . . . . . . . . . . . . . . 29
1. The Supply and Demand for Money and Overall
Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2. Why Overall Prices Change . . . . . . . . . . . . . . . . . . . . 35
IV. The Supply of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1. What Should the Supply of Money Be? . . . . . . . . . . . . 44
2. The Supply of Gold and the Counterfeiting Process . . 47
3. Government Paper Money . . . . . . . . . . . . . . . . . . . . . 51
4. The Origins of Government Paper Money . . . . . . . . . 55
XIII. Central Banking in the United States I: The Origins. . 191
1. The Bank of North America and the First
Bank of the United States. . . . . . . . . . . . . . . . . . . . . 191
2. The Second Bank of the United States . . . . . . . . . . . . 198
XIV. Central Banking in the United States II:
The 1820s to the Civil War . . . . . . . . . . . . . . . . . . . . . . 207
1. The Jacksonian Movement and the Bank War . . . . . . 207
2. Decentralized Banking from the 1830s to the
Civil War . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
XV. Central Banking in the United States III: The National
Banking System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
1. The Civil War and the National Banking System . . . . 219
2. The National Banking Era and the Origins
of the Federal Reserve System . . . . . . . . . . . . . . . . . 229
XVI. Central Banking in the United States IV:
The Federal Reserve System . . . . . . . . . . . . . . . . . . . . . . . 235
1. The Inflationary Structure of the Fed . . . . . . . . . . . . 235
2. The Inflationary Policies of the Fed . . . . . . . . . . . . . 241
XVII. Conclusion: The Present Banking
Situation and What to Do About It . . . . . . . . . . . . . . . . 247
1. The Road to the Present . . . . . . . . . . . . . . . . . . . . . . 247
2. The Present Money Supply . . . . . . . . . . . . . . . . . . . . 252
3. How to Return to Sound Money . . . . . . . . . . . . . . . 261
Appendix: The Myth of Free Banking in Scotland . . . . . . . 269
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

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Lemma 84.8.5. In Situation 84.3.3. Let $K$ be an object of $D(\mathcal{C}_{total})$.

1. If $H^{-p}(\mathcal{C}_ p, K_ p) = 0$ for all $p \geq 0$, then $H^0(\mathcal{C}_{total}, K) = 0$.

2. If $R\Gamma (\mathcal{C}_ p, K_ p) = 0$ for all $p \geq 0$, then $R\Gamma (\mathcal{C}_{total}, K) = 0$.

Proof. With notation as in Remark 84.8.4 we see that $R\Gamma (\mathcal{C}_{total}, K)$ is represented by $\text{Tot}_\pi (A^{\bullet , \bullet })$. The assumption in (1) tells us that $H^{-p}(A^{p, \bullet }) = 0$. Thus the vanishing in (1) follows from More on Algebra, Lemma 15.103.1. Part (2) follows from part (1) and taking shifts. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


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# Lars Eighner's Homepage

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### Integration by Substitution Exercises

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Sources:

1. Elements of the Differential and Integral Calculus Wikisource
2. Calculus Made Easy by Silvanus P. Thompson Project Gutenberg
3. Davis,Ellery Williams, William Charles Brenke, Earle Raymond Hedrick. The Calculus (Macmillan Company, 1922) Google Books
4. Love, Clyde E., Earl David Rainville Differential and Integral Calculus (Macmillian, 1916) Google Books

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483 views

### Mathematical Physics Book Recommendation [duplicate]

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### Books that every physicist should read

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Math Expressions Grade 4 Unit 8 Lesson 8 Homework

Using the Vocabulary box at the right, write the name of the quadrilateral that best describes each figure. Use each word once. Describe how it is different from other quadrilaterals.

VOCABULARY
square
trapezoid
rhombus
rectangle
parallelogram

Question 1.

Square
Explanation:
It has four equal lengths called sides with four right angles.

Question 2.

Explanation:
It has four un equal lengths called sides with four different angles.

Question 3.

Rhombus
Explanation:
It has four equal lengths called sides with two angles of same measure.

Question 4.

Rectangle
Explanation:
It has two opposite sides of same lengths with four right angles and four corners.

Question 5.

Parallelogram
Explanation:
It has two opposite sides of same lengths with two angles of same measure.

Question 6.

Trapezoid
Explanation:
It has two opposite sides are parallel of different  lengths with different angle.

Math Expressions Grade 4 Unit 8 Lesson 8 Remembering

Write these amounts as decimal numbers.

Question 1.
3 tenths _________
Explanation:
$$\frac{3}{10}$$=0.3

Question 2.
7 hundredths __________
Explanation:
$$\frac{7}{100}$$=0.07

Question 3.
56 hundredths __________
Explanation:
$$\frac{56}{100}$$=0.56

Question 4.
$$\frac{6}{100}$$ ___________
Explanation:
$$\frac{6}{100}$$ = 0.06

Question 5.
$$\frac{42}{100}$$ __________
Explanation:
$$\frac{42{100}$$=0.42

Question 6.
$$\frac{9}{10}$$ ___________
Explanation:
$$\frac{9}{10}$$ = 0.9

Tell whether each pair of lines is parallel, perpendicular, or neither.

Question 7.

Neither parallel nor perpendicular
Explanation:
Lines are neither parallel nor perpendicular, the line will meet at one point, and the angle will increase at opposite end.

Question 8.

Perpendicular lines
Explanation:
The Lines are perpendicular to each other as the angle between the two lines is 90

Question 9.

Neither parallel nor perpendicular
Explanation:
Lines are neither parallel nor perpendicular, the line will meet at one point, and the angle will increase at opposite end.

Question 10.

Parallel lines
Explanation:
Lines are parallel line, the line never meet at any point.

Question 11.
First draw a line segment 4cm long. Then draw a line segment 3 cm long that is not parallel nor perpendicular to the first line.


text:
## What is square root 96 simplified?

4√6
The square root of 96 in simplified form is 4√6. We can see that the square root of 96 is four times the square root of 6. So, the square root of 96 = 4 × square root of 6.

## Can √ 10 be simplified?

√10 is already in simplest form.

## What is the meaning of √ 10?

The square root of 10 is expressed as √10 in the radical form and as (10)½ or (10)0.5 in the exponent form. The square root of 10 rounded up to 10 decimal places is 3.1622776602.

## Does 96 have a perfect square factor?

96 is not a perfect square.

## Is 10 a perfect square?

A perfect square is a number that is generated by multiplying two equal integers by each other. For example, the number 9 is a perfect square because it can be expressed as a product of two equal integers: 9 = 3 x 3….Example 1.

Integer Perfect square
8 x 8 64
9 x 9 81
10 x 10 100
11 x 11 121

Expressing in simplest radical form just means simplifying a radical so that there are no more square roots, cube roots, 4th roots, etc left to find. It also means removing any radicals in the denominator of a fraction.

## What is root 12 simplified?

The square root of 12 is represented in the radical form as √12, which is equal to 2√3. Since 2√3 cannot be further simplified, hence such roots are called surds.

## What does ten mean in texting?

Summary of Key Points

TEN
Definition: Perfect body (10 out of 10)
Type: Slang Word (Jargon)
Guessability: 2: Quite easy to guess

## What is 10 to the power called?

Positive powers

Name Power Number
one 0 1
ten 1 10
hundred 2 100
thousand 3 1,000

## What is the square root of 96 in radical form?

Step by step simplification process to get square roots radical form: First we will find all factors under the square root: 96 has the square factor of 16. Let’s check this with √16*6=√96. As you can see the radicals are not in their simplest form. Now extract and take out the square root √16 * √6.

## What is the best way to simplify 96 into a fraction?

The process of simplifying usually makes use of the fact that . The best way to start is to find a square number which is a factor of 96. Any will do, since, just like reducing a fraction, you can keep going till you can’t simplify any more and get the same result in one step or several.

## How many possible roots are there for a number x?

There are 2 possible roots for any positive real number. A positive root and a negative root. Given a number x, the square root of x is a number a such that a 2 = x. Square roots is a specialized form of our common roots calculator. “Note that any positive real number has two square roots, one positive and one negative.

## What is the 2nd root of 100 and 10?

The 2nd root of 100, or 100 radical 2, or the square root of 100 is written as \$$\\sqrt[2]{100} = \\sqrt[]{100} = \\pm 10 \$$. The 2nd root of 10, or 10 radical 2, or the square root of 10 is written as \$$\\sqrt[2]{10} = \\sqrt[]{10} = \\pm 3.162278 \$$.


text:
# An algebra problem by Oleg Silkin

Algebra Level pending

Let $$p$$ and $$q$$ be real numbers such that $$p\ne0, p^3\ne q, p^3 \ne -q$$. If $$\alpha$$ and $$\beta$$ are non-zero complex numbers satisfying $$\alpha + \beta = -p$$ and $$\alpha^3 + \beta^3 = q$$, then a quadratic equation having $$\dfrac{ \alpha}{\beta} + \dfrac{\beta}{\alpha}$$ as its roots is

(A): $$\quad \quad x^2 (p^3 + q) - x(p^3 + 2q) + (p^3 + q) = 0$$.
(B): $$\quad \quad x^2 (p^3 + q) - x(p^3 - 2q) + (p^3 + q) = 0$$.
(C): $$\quad \quad x^2 (p^3 - q) - x(5p^3 - 2q) + (p^3 - q) = 0$$.
(D): $$\quad \quad x^2 (p^3 - q) - x(5p^3 + 2q) + (p^3 - q) = 0$$.

×


text:
# Reference request: maximal ratio of different norms of polynomials

Let us consider polynomials as functions on $[0,1]$, and so define \begin{align*} \|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\ \|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}. \end{align*} I am interested in the ratio of these norms. It is easy to see that $\|f\|_2\leq\|f\|_\infty$, with equality only for constant polynomials. In the opposite direction, put $$f_d(x) = \sum_{i=0}^d \frac{(d+1+i)!}{(d-i)!i!(i+1)!}(-x)^i.$$ Experiments make it clear that $\|f_d\|_2=1$ and $\|f_d\|_\infty=(d+1)$ and that $f_d$ maximises the ratio $\|f\|_\infty/\|f\|_2$ among polynomials of degree $d$. These facts must surely be known. Can anyone point me to a reference? Do the polynomials $f_d(x)$ have a standard name?

• Have a look here – user111 Dec 12 '17 at 18:31
• @user111 If you want to promote your comment to an answer, then I will accept it. – Neil Strickland Dec 12 '17 at 18:40

This is problem VI.103 in volume 2 of Polya and Szego. For the interval $[-1,1]$, the extremal polynomial is $$\frac{P_{n}(x)-P_{n+1}(x)}{1-x},$$ where $P_{n}$ is the Legendre polynomials of degree $n$ (with normalization $\int P_{n}^{2}(x)dx=2/(2n+1)$). Making use of the explicit formula $$P_{n}(x)= \sum _ { k = 0} ^ { n } \binom{ n } { k }\binom{ n + k }{ k } \left( \frac { x - 1} { 2} \right) ^ { k },$$ and the change of variables $x=1-2y$, it is straightforward to check that it gives indeed your formula on the segment $[0,1]$.


text:
# Manually calculate Arccosine and Arctangent

1. Oct 1, 2013

### Philosophaie

How do you calculate Arccosine and Arctangent if you do not have a scientific calculator.
$$\theta = atan(\frac{y}{x})$$
$$\theta2 = acos(\frac{z}{\sqrt{x^2+y^2+z^2}})$$

2. Oct 1, 2013

### arildno

Make a finite Taylor series expansion, for example.

3. Oct 1, 2013

### arildno

Or, if you recognize the value inside the acos, for example, as the cosine value to some specific angle, then that specific angle is your desired answer.

4. Oct 1, 2013

### UltrafastPED

The series expansions (Taylor series) can be used though most computational methods first transform this into a Chebyshev expansion. You will study these if you take a course in numerical analysis.

For an introduction see: http://en.wikipedia.org/wiki/Chebyshev_polynomials It takes a while to get to the application of the polynomials to approximation theory.

5. Oct 1, 2013

### arildno

Essentially, what you want is a finite algorithm of a) minimal length and retaining b) optimal accuracy for your answer.
These two competing issues generate in general, a tricky balancing act, although much research has uncovered a lot of techniques that justifiably have become "favoured".

Last edited: Oct 1, 2013
6. Nov 23, 2013

### Zeda

Though it is nearly 8 weeks late, if you need only about 4 digits of accuracy, you could use $tan^{-1}\approx \frac{x(240+115x^{2})}{240+195x^{2}+17x^{4}}, x\in [-1,1]$ and for x outside that range, use the identity $tan^{-1}(x)=\frac{\pi}{2}-tan^{-1}(\frac{1}{x})$. It might be a pain by hand, but it is doable.

Using Chebyshev polynomials or Taylor Series will take you a looong time :P

So, as an example, take x=.1
$tan^{-1}(.1)\approx \frac{.1(240+115*.01)}{240+195*.01+17*.0001}$
$= \frac{.1(241.15)}{241.9517}$
$= \frac{24.115}{241.9517}$
$\approx .0996686529$

Compare to my calculator which returns .0996686525 ;)

See this post for the derivation of that formula.

For arccos, you can use the identity:
$cos^{-1}(x)=2tan^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right), x\in (-1,1]$

To compute square roots by hand, there are a number of methods, but the method I am most familiar with is in binary, so I don't know how useful that will be :/ (Note that the square root will be squared in two places plugging it into the arctangent approximation, so you don't technically need to compute the square root, there.)


text:
# Getting your ALT data in the right format¶

Because the ALT probability models failures and right censored data from many stress levels, it was not practical to make an input for each stress level. Instead, the failure times are combined in a single input and the failure_stress input provides a list of the corresponding stresses at which each failure occurred. The same is true of the right_censored and right_censored_stress inputs.

To get your data in the correct format, ensure you have combined all your failure times into a single list or numpy array and there is a corresponding list or array of the same length that provides all of the stresses. The following example illustrates one method to do this if you do not have the list already imported from Excel or another source. This is done for failures only but if you have right_censored data then you would do the same thing, but keep it separate to the failure data. There is no need to sort the data in any particular order as this is all done automatically. The only requirement is that the length of failures matches the length of the failure_stress, and that there are no new stresses in right_censored_stress that are not present in failure_stress.

import numpy as np
#create the data
failure_times_at_stress_1 = [800,850,910,940]
failure_stress_1 = [40,40,40,40]
failure_times_at_stress_2 = [650,670,715,740]
failure_stress_2 = [50,50,50,50]
failure_times_at_stress_3 = [300,320,350,380]
failure_stress_3 = [60,60,60,60]
# combine the data
failures = np.hstack([failure_times_at_stress_1,failure_times_at_stress_2,failure_times_at_stress_3])
failure_stresses = np.hstack([failure_stress_1,failure_stress_2,failure_stress_3])
# print for inspection
print(failures)
print(failure_stresses)

'''
[800 850 910 940 650 670 715 740 300 320 350 380]
[40 40 40 40 50 50 50 50 60 60 60 60]
'''


text:
# Image Sources¶

Custom image soruces are a way to plug in custon code for loading images to Calcam, and having it integrate nicely in to the Calcam GUI. A custom image source takes the form of a python module or package. The module or package is required to have the following attributes at its top level:

• get_image_function : A function, or referece to a function, which actually loads the image data.

• display_name : A string specifying a user-friendly name for the image source. This will be used to denote the image source in the Calcam GUI.

• get_image_arguments : A list of dictionaries documenting the required inputs to the image loading function.

The return type of the image loading dunction must be a Python dictionary. The following keys are mandatory:

Key

Data Type

Description

image_data

NumPy ndarray

Array containing the image data.

source

String

Description of where the image was loaded from.

The dictionary can also contain the following optional image metadata:

Key

Data Type

Description

NumPy ndarray

For images with multiple sub-views: array of
integers the same shape as the image
specifying which sub-view each image pixel belongs to.

subview_names

List of strings

For images with multiple sub-views: list of
strings containing user-friendly names for each sub-view

transform_actions

List of strings

List of strings specifying the geometrical transforms
required to convert the image from original to display
coordinates. Actions are performed from the beginning
to the end of the list. Valid actions are:
flip_up_down
flip_left_right
rotate_clockwise_90
rotate_clockwise_180
rotate_clockwise_270

pixel_size

float

Physical height of the camera pixels, in metres

coords

String

Either Display or Original: whether the image is
being returned in display or original orientation.
If not provided, Display is assumed.

pixel_aspect

float

For cameras with non-square pixels, the pixel
aspect ratio defined as pixel height/pixel width.

image_offset

tuple

For cameras where only a sub-region of the full detector
is read out or recorded, this specifies the (x,y) pixel
coordinates of the top-left pixel of the image, relative to
the full detector.

## Structure of get_image_arguments¶

The top-level module attribute get_image_arguments must be a list of dictionaries describing the required input parameters to the image loading function. Each dictionary in the list corresponds to one input arguument. All argument dictionaries must contain the following keys:

Key

Data Type

Description

arg_name

String

Name of the argument as written in the function definition.

gui_label

String

User-friendly argument name which will be displayed in the GUI.

type

String

String specifying the type of input argument. See description below.

Additional keys are required depending on the type of variable.

### Input argument type: filename¶

If the inpit variable type is filename, the user will be presented with a text box and “Browse…” button to browse for the file. Additional keys in the dictionary describing the parameter are:

Key

Data Type

Description

filter

String

File name filter to be used in the file browsing window.
Format: “File type description (*.ext1, *.ext2)”

default

String

Default value.

### Input argument type: float¶

If the inpit variable type is float, the user will be presented with a box in which to input the value. Additional keys in the dictionary describing the parameter are:

Key

Data Type

Description

limits

array-like

2 element array-like specifying the lower and upper limits
of acceptable values for the argument.

default

float

Default value.

decimals

int

Number of decimal places to be shown in the user input box.

### Input argument type: int¶

If the inpit variable type is int, the user will be presented with a box in which to input the value. Additional keys in the dictionary describing the parameter are:

Key

Data Type

Description

limits

array-like

2 element array-like specifying the lower and upper limits
of acceptable values for the argument.

default

int

Default value.

### Input argument type: string¶

If the inpit variable type is string, the user will be presented with a single line box in which which to input the string. Additional keys in the dictionary describing the parameter are:

Key

Data Type

Description

default

string

Default value.

### Input argument type: bool¶

If the inpit variable type is bool, the user will be presented with a checkbox to set the value true or false. Additional keys in the dictionary describing the parameter are:

Key

Data Type

Description

default

bool

Default value.

### Input argument type: choice¶

If the inpit variable type is choice, the user will be presented with a dropdown box from whih they can choose a number of strings. The value passed to the image loading function is whatever string the user selects. Additional keys in the dictionary describing the parameter are:

Key

Data Type

Description

choices

list of str

List of strings from which the user should choose.

default

string

Default value.

## Example¶

Below is a minimal example: the build-in Calcam image source for loading images from standard format image files.

'''

'''

import cv2
import os

# The function which loads the image
def get_image(filename):

# Get the image data from a file
if dat is None:
raise UserWarning('Could not read specified image file "' + filename + '"')

# If it's a colour image, note OpenCV loads images in BGR.
# Here we change the channel order to RGB.
if len(dat.shape) == 3:

if dat.shape[2] == 3:
dat[:,:,:3] = dat[:,:,2::-1]

# If R, G and B channels are all the same, just return a monochrome image
if (dat[:,:,0] == dat[:,:,1]).all() and (dat[:,:,0] == dat[:,:,2]).all():
dat = dat[:,:,0]

# Minimal return dictionary.
return_dict = {
'image_data': dat,
'source': 'Loaded from image file {:s}'.format(os.path.split(filename)[-1])
}

return return_dict

# Display name
display_name = 'Image File'

# Point calcam to the get_image function
get_image_function = get_image

# Description of input parameters to get_image
get_image_arguments =  [
{
'arg_name':'filename',
'gui_label': 'File Name' ,
'type': 'filename' ,
'filter':'Image Files (*.png *.jpg *.jpeg *.bmp *.jp2 *.tiff *.tif)'
},
]


text:
## GR0568 Question 42

Forum for the GRE subject test in mathematics.
bonaluram
Posts: 7
Joined: Fri Oct 30, 2009 1:51 pm

### GR0568 Question 42

Hello,

Suppose X is a discrete random variable on the set of positive integers such that for each positive integer n, the probability that X = n is 1/(2^n). If Y is a random variable with the same probability distribution and X and Y are independent, what is the probability that the value of at least one of the variables X and Y is greater than 3?

(A) 1/64
(B) 15/64
(C) 1/4
(D) 3/8
(E) 4/9

The answer is B. Thank you all.

joey
Posts: 32
Joined: Fri Oct 16, 2009 3:53 pm

### Re: GR0568 Question 42

This is pretty straightforward when you know a few rules of probability. Write
$\mbox{Pr}\left(X > 3 \mbox{ or } Y > 3\right) = 1 - \mbox{Pr}\left(X \le 3 \mbox{ and } Y \le 3\right)$
Since X and Y are independent and identically distributed, this becomes
$1 - \mbox{Pr}\left(X \le 3\right)\cdot\mbox{Pr}\left(Y \le 3\right)$
$= 1 - \mbox{Pr}\left(X \le 3\right)^2$
$= 1 - \left[\mbox{Pr}\left(X = 1\right) + \mbox{Pr}\left(X = 2\right) + \mbox{Pr}\left(X = 3\right)\right]^2$

Then we just substitute and simplify.

bonaluram
Posts: 7
Joined: Fri Oct 30, 2009 1:51 pm

### Re: GR0568 Question 42

joey wrote:This is pretty straightforward when you know a few rules of probability. Write
$\mbox{Pr}\left(X > 3 \mbox{ or } Y > 3\right) = 1 - \mbox{Pr}\left(X \le 3 \mbox{ and } Y \le 3\right)$
Since X and Y are independent and identically distributed, this becomes
$1 - \mbox{Pr}\left(X \le 3\right)\cdot\mbox{Pr}\left(Y \le 3\right)$
$= 1 - \mbox{Pr}\left(X \le 3\right)^2$
$= 1 - \left[\mbox{Pr}\left(X = 1\right) + \mbox{Pr}\left(X = 2\right) + \mbox{Pr}\left(X = 3\right)\right]^2$

Then we just substitute and simplify.

Thank you so much joey, that really helped! =)

joey
Posts: 32
Joined: Fri Oct 16, 2009 3:53 pm

### Re: GR0568 Question 42

I'm glad to help! It beats the hell out of working, which is what I probably _should_ be doing right now...

diogenes
Posts: 73
Joined: Sun Aug 31, 2008 9:31 pm

### Re: GR0568 Question 42

joey wrote:I'm glad to help! It beats the hell out of working, which is what I probably _should_ be doing right now...

Heh...off today, but I understand.

bonaluram
Posts: 7
Joined: Fri Oct 30, 2009 1:51 pm

### Re: GR0568 Question 42

joey wrote:I'm glad to help! It beats the hell out of working, which is what I probably _should_ be doing right now...

I am so bored of this stuff, now i have less than 9 hours to the test. Last temptations

thmsrhn
Posts: 17
Joined: Fri Mar 26, 2010 7:18 am

### Re: GR0568 Question 42

hey joey finish the whole problem please!!

joey
Posts: 32
Joined: Fri Oct 16, 2009 3:53 pm

### Re: GR0568 Question 42

We're told that $\mbox{Pr}(X=n)=\frac{1}{2^n}$. There's a joke about a mathematician, a physicist, and an engineer who are asked to put out a fire, but I won't get into it.


text:
# Limit definition of definite integrals

• Feb 6th 2009, 02:34 PM
fattydq
Limit definition of definite integrals
In a practice quiz I'm working on for my calc class I'm asked to evaluate the given definite integral from 3 to 7 of (x^2-3x+5)dx using the limit definition of a definite integral. I know how to express an integral as a limit using the limit definition, and I know how to evaluate definite integrals usint areas, but I don't know how to EVALUATE a definite integral using the limit definition. What does this even mean exactly and how could I go about doing it?
• Feb 6th 2009, 02:42 PM
Mush
• Feb 6th 2009, 02:45 PM
Abu-Khalil
You may define the partition $x_0, where $x_k=3+\left(\frac{7-3}{n}\right)k\Rightarrow \Delta x_k=x_k-x_{k-1}=\left(\frac{7-3}{n}\right)$ so when $n\to\infty, \Delta x_k \to 0$.

Finally, $\int_3^7x^2-3x+5dx=\lim_{n\to\infty}\frac{7-3}{n}\sum_{k=1}^nx_k^2-3x_k+5$ and calculate that sum isn't a problem knowing $\sum_{k=1}^nk=\frac{n(n+1)}{2}$ and $\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$.
• Feb 6th 2009, 02:47 PM
fattydq
thanks guys I'll check that link out for sure
• Feb 6th 2009, 02:50 PM
fattydq
Quote:

Originally Posted by Abu-Khalil
You may define the partition $x_0, where $x_k=3+\left(\frac{7-3}{n}\right)k\Rightarrow \Delta x_k=x_k-x_{k-1}=\left(\frac{7-3}{n}\right)$ so when $n\to\infty, \Delta x_k \to 0$.

Finally, $\int_3^7x^2-3x+5dx=\lim_{n\to\infty}\frac{7-3}{n}\sum_{k=1}^nx_k^2-3x_k+5$ and calculate that sum isn't a problem knowing $\sum_{k=1}^nk=\frac{n(n+1)}{2}$ and $\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$.

I'm sorry that's a bit too jumbled for me I don't really understand any of what you said. What is x subscript k even representing here?
• Feb 8th 2009, 03:49 PM
fattydq
I keep getting wrong answers. I just found the answer by using the integral function on my calculator but I can't seem to get it. Could someone walk me through this step by step?