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Calculus


Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

 
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Synchronicity in Predictive Modelling: a New View of Data Assimila...

By: J. J. Tribbia; J. B. Weiss; G. S. Duane

Description: National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA. The problem of data assimilation can be viewed as one of synchronizing two dynamical systems, one representing truth and the other representing model, with a unidirectional flow of information between the two. Synchronization of truth and model defines a general view of data assimilation, as machine perception, that is reminiscent of the Jung-Pauli notion of synchronicity be...

Afraimovich, V. S., Verichev, N. N., and Rabinovich, M. I.: Stochastic synchronization of oscillation in dissipative systems, Radiophys. Quantum Electron., 29, 795–803, 1986.; Anderson, J. L.: An ensemble adjustment Kalman filter for data assimilation, Mon. Wea. Rev., 129, 2884–2903, 2001.; Anderson, J. L.: A local least-squares framework for ensemble filtering, Mon. Wea. Rev., 131, 634–642, 2003.; Corazza, M., Kalnay, E., Patil, D. J., Yang, S.-C., Morss, R., Cai, M., S...

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Synchronicity in Predictive Modelling: a New View of Data Assimila...

By: J. J. Tribbia; J. B. Weiss; G. S. Duane

Description: National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA. The problem of data assimilation can be viewed as one of synchronizing two dynamical systems, one representing truth and the other representing model, with a unidirectional flow of information between the two. Synchronization of truth and model defines a general view of data assimilation, as machine perception, that is reminiscent of the Jung-Pauli notion of synchronicity be...

Afraimovich, V. S., Verichev, N. N., and Rabinovich, M. I.: Stochastic synchronization of oscillation in dissipative systems, Radiophys. Quantum Electron., 29, 795–803, 1986.; Anderson, J. L.: An ensemble adjustment Kalman filter for data assimilation, Mon. Wea. Rev., 129, 2884–2903, 2001.; Anderson, J. L.: A local least-squares framework for ensemble filtering, Mon. Wea. Rev., 131, 634–642, 2003.; Corazza, M., Kalnay, E., Patil, D. J., Yang, S.-C., Morss, R., Cai, M., S...

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A Model for Aperiodicity in Earthquakes : Volume 15, Issue 1 (07/0...

By: D. Lavallée; B. Birnir; B. Erickson

Description: Department of Mathematics, University of California, Santa Barbara, USA. Conditions under which a single oscillator model coupled with Dieterich-Ruina's rate and state dependent friction exhibits chaotic dynamics is studied. Properties of spring-block models are discussed. The parameter values of the system are explored and the corresponding numerical solutions presented. Bifurcation analysis is performed to determine the bifurcations and stability of s...

Andrews, D. J.: Rupture velocity of plane strain shear cracks, J. Geophys. Res., 81, 5679–5687, 1976.; Ascher, U. M. and Petzold, L. R.: Computer methods for ordinary differential equations and differential-algebraic equations, SIAM, Philadelphia, 1st edition, 1998.; Archuleta, R. J.: A faulting model for the 1979 Imperial Valley earthquake, J. Geophys. Res., 89, 4559–4586, 1984. %; %Archuleta, R. J.: %Personal communication, ralph@crustal.ucsb.edu, %2007.; Bak, P., Tang...

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Non-diffusive, Non-local Transport in Fluids and Plasmas : Volume ...

By: D. Del-castillo-negrete

Description: Oak Ridge National Laboratory, Oak Ridge, TN, USA. A review of non-diffusive transport in fluids and plasmas is presented. In the fluid context, non-diffusive chaotic transport by Rossby waves in zonal flows is studied following a Lagrangian approach. In the plasma physics context the problem of interest is test particle transport in pressure-gradient-driven plasma turbulence. In both systems the probability density function (PDF) of particle displaceme...

Bouchaud, J. P. and Georges, A.: , Phys. Rep., 195, p. 127, doi:10.1016/0370-1573(90)90099-N, 1990.; Callen, J. D. and Kissick, M. W.: , Plasma Phys. Contr. F., 39, B173, doi:10.1088/0741-3335/39/12B/014, 1997.; Carreras, B. A., Garcia, L., and Diamond, P. H.: , Phys. Fluids, 30, 1388, doi:10.1063/1.866518, 1987.; Cartea, A. and del-Castillo-Negrete, D.: , Phys. Rev. E, 76, 041105, doi:10.1103/PhysRevE.76.041105, 2007.; del-Castillo-Negrete, D.: , Phys. Fluids, 10, p. 57...

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A Possible Theory for the Interaction Between Convective Activitie...

By: M. Takahashi; Y. H. Ding; X. Y. Shen; N. Zhao

Description: State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, 100081, China. Theoretical studies usually attribute convections to the developments of instabilities such as the static or symmetric instabilities of the basic flows. However, the following three facts make the validities of these basic theories unconvincing. First, it seems that in most cases the basic flow with balance property cannot exist as the exact soluti...

Arakawa, A. and Schubert, W. H.: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I., J. Atmos. Sci., 31, 674–701, 1974.; Charney, J. G.: The use of the primitive equation of motion in numerical prediction, Tellus, {{7}} 22–26, 1955.; Drazin, P. G.: Hydrodynamical stability, Cambridge University Press, Cambridge, 1981.; Emanuel, K. A., Neelin, J. D., and Bretherton, C. S.:  On large-scale circulations in convecting atmospheres, Quart. J. Roy...

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Evolutionary Modeling-based Approach for Model Errors Correction :...

By: W. Jiang; L. Wang; W. Zhang; W. P. He; S. Q. Wan

Description: Yangzhou Meteorological Office, Yangzhou, China. The inverse problem of using the information of historical data to estimate model errors is one of the science frontier research topics. In this study, we investigate such a problem using the classic Lorenz (1963) equation as a prediction model and the Lorenz equation with a periodic evolutionary function as an accurate representation of reality to generate observational data.

On the basis of t...

Bäck, T., Hammel, U., and Schwefel, H. P.: Evolutionary computation: Comments on the history and current state, Evolutionary Computation, IEEE Trans. Evolut. Comput., 1, 3–17, 1997.; Cao, H. X.: Self-memorization Equation in Atmospheric Motion, Sci. China, 23, 104–112, 1993.; Cao, H. X., Kang, L., Chen, Y., and Yu, J.: EM of systems of ordinary differential equations with genetic programming, Genet. Program. Evol. M., 1, 309–337, 2000.; Chou, J. F.: Short Term Climatic ...

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Dynamics of a Two-fault System with Viscoelastic Coupling : Volume...

By: M. Dragoni; A. Amendola

Description: Dipartimento di Fisica, Università di Bologna, Viale Carlo Berti Pichat 8, 40127 Bologna, Italy. A fault system made of two segments or asperities subject to a constant strain rate is considered. The fault is modelled as a discrete dynamical system made of two blocks coupled by a Maxwell spring dashpot element and pulled at constant velocity on a rough plane. The long-term behaviour of the fault is studied by calculating the orbits of the system in the p...

Belardinelli, M. E. and Bonafede, M.: Post-seismic stress evolution for a strike-slip fault in the presence of a viscoelastic asthenosphere, Geophys. J. Int., 123, 744–756, 1995.; Filippov, A. F.: Differential equations with discontinuous righthand sides, Kluwer Academic Publishers, 307 pp., 1988.; Bürgmann, R. and Dresen, G.: Rheology of the lower crust and upper mantle: evidence from rock mechanics, geodesy, and field observation, Ann. Rev. Earth Planet. Sci., 36, 531–...

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Dynamics of Simple Earthquake Model with Time Delay and Variation ...

By: N. Vasović; I. Franović; K. Todorović; S. Kostić

Description: University of Belgrade Faculty of Mining and Geology, Ðušina 7, Belgrade, Serbia. We examine the dynamical behaviour of the phenomenological Burridge–Knopoff-like model with one and two blocks, where the friction term is supplemented by the time delay Τ and the variable friction strength c. Time delay is assumed to reflect the initial quiescent period of the fault healing, considered to be a function of history of sliding. Friction strength param...

Aronson, D., Ermentout, G., and Kopell, N.: Amplitude response of coupled oscillators, Physica D, 41, 403–449, 1990.; Bak, P. and Tang, C.: Earthquakes as a self-organized critical phenomenon, J. Geophys. Res., 94, 15635–15637, 1989.; Behnsen, J. and Faulkner, D. R.: The effect of mineralogy and effective normal stress on frictional strength of sheet silicates, J. Struct. Geol., 30, 1–13, 2012.; Beltrami, H. and Mareshal, J.: Strange seismic attractor?, Pure Appl. Geophy...

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Barriers to Transport in Aperiodically Time-dependent Two-dimensio...

By: S. Wiggins; A. M. Mancho

Description: School of Mathematics, University of Bristol, Bristol BS8 1TW, UK. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems p...

Acrivos, A., Aref, H., and Ottino, J. M. (Eds.): Symposium on Fluid Mechanics of Stirring and Mixing, Phys. Fluids A, Part 2, 723–1469, 1991.; Aref, H.: Stirring by chaotic advection, J. Fluid Mech., 143, 1–21, 1984.; Aref, H.: The development of chaotic advection, Phys. Fluids, 14, 1315–1325, 2002.; Aref, H. and El Naschie, M. S. (Eds.): Chaos Applied to Fluid Mixing, Chaos, Solitons, and Fractals, 4, 745–1116 , 1994.; Bower, A. S.: A simple kinematic mechanism for mixi...

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Improving the Ensemble Transform Kalman Filter Using a Second-orde...

By: X. Zheng; G. Wu; X. Zhang; L. Wang; S. Zhang; X. Liang; X. Yi

Description: State Key Laboratory of Remote Sensing Science, College of Global Change and Earth System Science, Beijing Normal University, Beijing, China. The ensemble transform Kalman filter (ETKF) assimilation scheme has recently seen rapid development and wide application. As a specific implementation of the ensemble Kalman filter (EnKF), the ETKF is computationally more efficient than the conventional EnKF. However, the current implementation of the ETKF still ha...

Anderson, J. L.: An adaptive covariance inflation error correction algorithm for ensemble filters, Tellus A, 59, 210–224, 2007.; Anderson, J. L.: Spatially and temporally varying adaptive covariance inflation for ensemble filters, Tellus A, 61, 72–83, 2009.; Anderson, J. L. and Anderson, S. L.: A Monte Carlo implementation of the non-linear filtering problem to produce ensemble assimilations and forecasts, Mon. Weather Rev., 127, 2741–2758, 1999.; Bishop, C. H. and Toth,...

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Incidence and Reflection of Internal Waves and Wave-induced Curren...

By: J. P. McHugh

Description: University of New Hampshire, Durham, NH, USA. Weakly nonlinear internal gravity waves are treated in a two-layer fluid with a set of nonlinear Schrodinger equations. The layers have a sharp interface with a jump in buoyancy frequency approximately modeling the tropopause. The waves are periodic in the horizontal but modulated in the vertical and Boussinesq flow is assumed. The equation governing the incident wave packet is directly coupled to the equati...

McHugh, J. P., Jumper, G. Y., and Chun, M.: Balloon thermosonde measurements over Mauna Kea, and comparison with seeing measurements, Pub. Astro. Soc. Pac., 120, 1318–1324, 2008b.; Shrira, V. I.: On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves, Int. J. Non-Lin. Mech., 16, 129–138, 1981.; Smith, R. B., Woods, B. K., Jensen, J., Cooper, W. A., Doyle, J. D., Jiang, Q., and Grubisic, V.: Mountain waves entering the stratosphere, J...

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Improving the Ensemble Transform Kalman Filter Using a Second-orde...

By: S. Zhang; L. Wang; G. Wu; X. Zhang; X. Liang; X. Zheng

Description: State Key Laboratory of Remote Sensing Science, College of Global Change and Earth System Science, Beijing Normal University, Beijing, China. The Ensemble Transform Kalman Filter (ETKF) assimilation scheme has recently seen rapid development and wide application. As a specific implementation of the Ensemble Kalman Filter (EnKF), the ETKF is computationally more efficient than the conventional EnKF. However, the current implementation of the ETKF stil...

Anderson, J. L.: An adaptive covariance inflation error correction algorithm for ensemble filters, Tellus A, 59, 210–224, 2007.; Anderson, J. L.: Spatially and temporally varying adaptive covariance inflation for ensemble filters, Tellus A, 61, 72–83, 2009.; Anderson, J. L. and Anderson, S. L.: A Monte Carlo implementation of the non-linear filtering problem to produce ensemble assimilations and forecasts, Mon. Weather Rev., 127, 2741–2758, 1999.; Bishop, C. H. and Toth,...

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On the Indonesian Throughflow in the Occam 1/4 Degree Ocean Model ...

By: U. W. Humphries; D. J. Webb

Description: King Monghut's Institute of Technology, Thailand. The Indonesian Throughflow is analysed in two runs of the OCCAM 1/4 degree global ocean model, one using monthly climatological winds and one using ECMWF analysed six-hourly winds for the period 1993 to 1998. The long-term model throughflow agrees with observations and the value predicted by Godfrey's Island Rule. The Island Rule has some skill in predicting the annual signal each year but is poor at pre...

Arief, D. and Murray, S. P: Low-frequency fluctuations in the Indonesian Throughflow through Lombok Strait, J. Geophys. Res., 101(C5), 12 455–12 464, 1996.; Arakawa, A.: Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow, J. Comput. Phys., 1, 119–143, 1966.; Armi, L. and Williams, R.: The hydraulics of a stratified fluid flowing through a contraction, J. Fluid Mech., 251, 355–375, 1993.; Bryan, K...

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On the Time to Tracer Equilibrium in the Global Ocean : Volume 5, ...

By: E. Deleersnijder; F. Primeau

Description: Department of Earth System Science, University of California at Irvine, Rowland Hall, Irvine, CA 92697, USA. An important issue for the interpretation of data from deep-sea cores is the time for tracers to be transported from the sea surface to the deep ocean. Global ocean circulation models can help shed light on the timescales over which a tracer comes to equilibrium in different regions of the ocean. In this note, we discuss how the most slowly ...

Broecker, W. S. and Peng, T.-H.: Tracers in the sea, Lamont-Doherty Geological Observatory, Palisades, NY., 690 pp., 1982.; Campin, J.-M., Fichefet, T., and Duplessy, J.-C.: Problems with using radiocarbon to infer ocean ventilation rates for past and present climates, Earth Planet. Sci. Lett., 165, 17–24, 1999.; Cox, M. D.: An Idealized Model of the World Ocean. Part I: The Global-Scale Water Masses., J. Phys. Oceanogr. 19, 1730–1752, 1989.; Delhez, E. J. M., Campin, J....

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The North American Review : Volume 0085, Issue 176 July 1857: Nort...

By: University of Northern Iowa

Description:The North American Review is a is a journal of literature and culture, published starting in the 19th century.

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The United States Democratic Review : Volume 0028, Issue 154 April...

By: J. and H. G. Langley

Description:The United States Magazine and Democratic Review was also (perhaps even primarily) a literary magazine, promoting the development of American literature

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Workbook in Higher Algebra

By: David Surowski

Mathematics document containing theorems and formulas.

Excerpt: The present set of notes was developed as a result of Higher Algebra courses that I taught during the academic years 1987-88, 1989-90 and 1991-92. The distinctive feature of these notes is that proofs are not supplied. There are two reasons for this. First, I would hope that the serious student who really intends to master the material will actually try to supply many of the missing proofs. Indeed, I have tried to break down the exposition in such a way that by ...

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Quantum Mechanics

By: Professor John W. Norbury

Physics Literature

Excerpt: 1 Wave Function 7 // 1.1 Probability Theory 8 // 1.1.1 Mean, Average, Expectation Value. 8 // 1.1.2 Average of a Function 10 // 1.1.3 Mean, Median, Mode 10 // 1.1.4 Standard Deviation and Uncertainty 11 // 1.1.5 Probability Density. 14 // 1.2 Postulates of Quantum Mechanics. 14 // 1.3 Conservation of Probability (Continuity Equation) 19 // 1.3.1 Conservation of Charge. 19 // 1.3.2 Conservation of Probability. 22 // 1.4 Interpretation of the Wave Function 23 // 1...

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Combinatorial Geometry with Applications to Field Theory : Second ...

By: Linfan Mao

In The 2nd Conference on Combinatorics and Graph Theory of China (Aug. 16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathematical sciences (abbreviated to CC Conjecture), i.e., a mathematical science can be reconstructed from or made by combinatorialization, implicated in the foreword of Chapter 5 of my book Automorphism groups of Maps, Surfaces and Smarandache Geometries (USA, 2005). This conjecture is essentially a philosophic notion for de...

1.5 ENUMERATION TECHNIQUES 1.5.1 Enumeration Principle. The enumeration problem on a finite set is to count and find closed formula for elements in this set. A fundamental principle for solving this problem in general is on account of the enumeration principle: For finite sets X and Y , the equality |X| = |Y | holds if and only if there is a bijection f : X → Y . Certainly, if the set Y can be easily countable, then we can find a closed formula for elements in X.

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