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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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Transient Flow Between Aquifers and Surface Water: Analytically De...

By: G. H. De Rooij

Description: Helmholtz Centre for Environmental Research – UFZ, Theodor-Lieser-Strasse 4, 06120 Halle (Saale), Germany. The increasing importance of catchment-scale and basin-scale models of the hydrological cycle makes it desirable to have a simple, yet physically realistic model for lateral subsurface water flow. As a first building block towards such a model, analytical solutions are presented for horizontal groundwater flow to surface waters held at prescribed wa...

Abramowitz, M. and Stegun, I. A.: Handbook of mathematical functions. With formulas, graphs, and mathematical tables, Dover Publications, Inc., New York., 1046 pp., 1965.; Berggren, L., Borwein, J., and Borwein, P.: Pi: A source book, 3rd Edn., Springer, Berlin, Germany, 797 pp., 2004.; Bouwer, H.: Groundwater hydrology, International student edition, McGraw-Hill Kogakusha, Ltd., Tokyo, 480 pp., 1978.; Bruggeman, G. A.: Analytical solutions of geohydrological problems, D...

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An Eco-hydrologic Model of Malaria Outbreaks : Volume 9, Issue 3 (...

By: S. Manzoni; A. Porporato; A. Montanari; E. Montosi

Description: Faculty of Engineering, Bologna University, Bologna, Italy. Malaria is a geographically widespread infectious disease that is well known to be affected by climate variability at both seasonal and interannual timescales. In an effort to identify climatic factors that impact malaria dynamics, there has been considerable research focused on the development of appropriate disease models for malaria transmission and their consideration alongside climatic dat...

Bailey, N. T. J.: The biomathematics of malaria., Charles Griffin and Company Ltd., London, 1982.; Bayoh, M. N. Lindsay, S. W.: Effect of temperature on the development of the aquatic stages of Anopheles gambiae sensu stricto (Diptera:Culicidae), B. Entomol. Res., 93, 375–381, 2003.; Bertuzzo, E., Azaele, S., Maritan, A., Gatto, M., Rodriguez-Iturbe, I., and Rinaldo, A.: On the space-time evolution of a cholera epidemic, Water Resour. Res., 4, W01424,

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Modelling Catchment-scale Shallow Landslide Occurrence by Means of...

By: P. Tarolli; M. Borga; R. Rigon; C. Lanni

Description: Department of Civil and Environmental Engineering, University of Trento, Trento, Italy. Topographic index-based hydrological models have gained wide use to describe the hydrological control on the triggering of rainfall-induced shallow landslides at the catchment scale. A common assumption in these models is that a spatially continuous water table occurs simultaneously at any point across the catchment. However, during a rainfall event is...

Barling, R. D., Moore, I. D., and Grayson, R. B.: A quasi-dynamic wetness index for characterizing the spatial distribution of zones of surface saturation and soil water content, Water Resour. Res., 30, 1029–1044, 1994.; Baum, R. L., Savage, W. Z., and Godt, J. W.: TRIGRS– A FORTRAN program for transient rainfall infiltration and grid based regional slope stability analysis, version 2.0, US Geol. Surv. Open File Rep. 2008, 1159, 74 pp. 2008.; Beven, K. J. and Kirkby, M. ...

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Analysis of Servo-constraint Problems for Underactuated Multibody ...

By: W. Blajer; R. Seifried

Description: Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany. Underactuated multibody systems have fewer control inputs than degrees of freedom. In trajectory tracking control of such systems an accurate and efficient feedforward control is often necessary. For multibody systems feedforward control by model inversion can be designed using servo-constraints. So far servo-constraints have been m...

Blajer, W.: A Geometric Interpretation and Uniform Matrix Formulation for Multibody System Dynamics, ZAMM, 81, 247–259, 2001.; Blajer, W. and Kolodziejczyk, K.: A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework, Multibody Syst. Dyn., 11, 343–364, 2004.; Bajodah, A., Hodges, H., and Chen, Y.-H.: Inverse Dynamics of Servo-Constraints Based on the Generalized Inverse, Nonlinear Dynam., 39, 179–196, 2005.; Bastos, G., Seifried, R., a...

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Nonlinear Evolution of the Modulational Instability Under Weak For...

By: C. Kharif; J. Touboul

Description: Laboratoire de Sondages Éléctromagnétiques de l'Environnement Terrestre, UMR 6017, Institut des Sciences de l'Ingénieur Toulon – Var, Av. G. Pompidou, BP 56, 83162, La Valette Cedex, France. The evolution of modulational instability, or Benjamin-Feir instability is investigated within the framework of the two-dimensional fully nonlinear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles' ...

Banner, M. L. and and Tian, X.: On the determination of the onset of breaking for modulating surface gravity water waves, J. Fluid Mech., 367, 107–137, 1998.; Benjamin, T. B. and Feir, J. E.: The disintegration of wave trains on deep water, J. Fluid Mech., 27, 417–430, 1967.; Conte, S. D. and Miles, J. W.: On the numerical integration of the Orr-Sommerfeld equation, J. Soc. Ind. Appl. Maths., 7, 361–366, 1959.; Dias, F., Dyachenko, A. I., and Zakharov, V. E.: Theory of w...

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A Shallow-flow Model for the Propagation of Tsunamis Over Complex ...

By: R. M. L. Ferreira; C. Sousa Oliveira; D. A. S. Conde; M. A. V. Baptista

Description: CEHIDRO, Instituto Superior Técnico, TU Lisbon, Lisbon, Portugal. A distinguishable feature of overland tsunami propagation is the incorporation of solids within the flow column, either sediment from the natural environment or remains from built infrastructure. This article describes a 2DH (two-dimensional horizontal) mathematical model particularly suited for tsunami propagation over complex and dynamic geometries, such as river and estuarine mobile be...

Andrade, C., Freitas, M. C., Miranda, J. M., and Baptista, M. A.: Recognizing Possible Tsunami Sediments In The Ultradissipative Environment Of The Tagus Estuary, in: Proceedings of the International Conference on Coastal Sediments 2003, Florida, USA, 18–23, 2003.; Bagnold, R. A.: An approach to the sediment transport problem from general physics, United States Department of Interior, Geological Survey, Professional Paper 442-I, 1966.; Baptista, M. A. and Miranda, J. M.:...

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Climate-induced Landslide Reactivation at the Edge of the Most Bas...

By: V. Smítka; J. Burda; F. Hartvich; J. Rybář; J. Valenta

Description: Brown Coal Research Institute j.s.c., Most, Budovatelů 2830, Czech Republic. The catastrophic landslide at Eisenberg in North Bohemia was reactivated during January 2011. This study integrates a range of geoscientific evidence in order to constrain the spatial and temporal development of this reactivation. It has investigated long-term geodetic measurements to assess the morphological development of the site over the last two decades. There is evidence ...

Bláha, J., Blín, J., and Stanislav, P.: Nov\'{y} system sledování bočních svahů lomu ČSA, Zpravodaj Hnědé uhlí, 1, 20–27, 2006 (in Czech).; Brown, N., Kaloustian, S., and Roeckle, M.: Monitoring of open pit mines using combined GNSS satellite receivers and robotic total stations, available at: www.leica-geosystems.com/download...

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A Perturbation of Dc Electric Field Caused by Light Ion Adhesion t...

By: M. Hayakawa; A. K. Yaschenko; V. M. Sorokin

Description: Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (IZMIRAN), Troitsk, Moscow Region, Russia. The influence of variations in conductivity and external electric current variations in the lower atmosphere on DC electric field over a seismic region is investigated. The external current is formed with the occurrence of convective upward transport of charged aerosols and their gravitational sedimentation in the atmosphere. This effect i...

Alekseev, V. A. and Alekseeva, N. G.: Investigation of metal transfer in the biosphere during gaseous emission in zones of tectonic activity using methods of nuclear physics, Nucl. Geophys., 6, 99–104, 1992.; Angel, E. and Bellman, R.: Dynamic Programming and Partial Differential Equations, Academic Press., 1972.; Barth, C. A.: Chemical Reactions in the Lower and Upper Atmosphere, J. Willey, N. Y., 1961.; Clement, C. F. and Harrison, R. G.: The charging of radioactive ae...

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Detection of Metallic and Plastic Landmines Using the Gpr and 2-d ...

By: M. Metwaly

Description: National Research Institute of Astronomy and Geophysics, Cairo, Egypt. Low and non-metallic landmines are one of the most difficult subsurface targets to be detected using several geophysical techniques. Ground penetrating radar (GPR) performance at different field sites shows great success in detecting metallic landmines. However significant limitations are taking place in the case of low and non-metallic landmines. Electrical resistivity imaging (ERI)...

El-Qady, G. and Ushijima, K.: Detection of UXO and landmines using 2-D modeling of geoelectrical resistivity data, Proceedings of SAGEEP Meeting, 1176–1182, 2005.; Gerard, R. and Tabbagh, A.: A mobile four electrodes array and its application to electrical survey of planetary grounds at shallow depths, J. Geophys. Res., 96(B3), 4117–4123, 1991.; Lopera, O. and Milisavljevic, N.: Prediction of the effects of soil and target properties on the antipersonnel landmine detecti...

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Tsunami Hazard in La Réunion Island from Numerical Modeling of His...

By: A. Loevenbruck; H. Hébert; S. Allgeyer; E. Quentel

Description: CEA, DAM, DIF, 91297 Arpajon, France. Whereas major tsunamis have recently affected the southwest Indian Ocean, tsunami hazard in this basin has never been thoroughly examined. Our study contributes to fill in this lack and focuses on La Réunion island for which tsunami hazard related to great earthquakes is evaluated by modeling the scenarios of major historical events. Then, our numerical modeling allow us to compare the tsunami impact at regional sca...

Ammon, C. J., Kanamori, H., Lay, T., and Velasco, A. A.: The 17 July 2006 Java tsunami earthquake, Geophys. Res. Lett., 33, L24308, doi:10.1029/2006GL028005, 2006.; Byrne, D., Sykes, L. R., and Davis, D. M.: Great thrust earthquakes and aseismic slip along the plate boundary of the Makran Subduction Zone, J. Geophys. Res., 97, 449–478, doi:10.1029/91JB02165, 1992.; Choi, B....

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Three-dimensional Simulations of Turbulent Spectra in the Local In...

By: G. P. Zank; D. Shaikh

Description: Institute of Geophysics and Planetary Physics, University of California, Riverside CA 92521, USA. Three-dimensional time dependent numerical simulations of compressible magnetohydrodynamic fluids describing super-Alfvénic, supersonic and strongly magnetized space and laboratory plasmas show a nonlinear relaxation towards a state of near incompressibility. The latter is characterized essentially by a subsonic turbulent Mach number. This transition is med...

Armstrong, J. W., Cordes, J. M., and Rickett, B. J.: Density power spectrum in the local interstellar medium, Nature, 291, 561–564, 1981.; Armstrong, J. W., Rickett, B. J, and Spangler, S.: Electron density power spectrum in the local interstellar medium, ApJ, 443, 209–221, 1995.; Bayly, B. J., Levermore, C. D., and Passot, T.: Density variations in weakly compressible flows, Phys. Fluids, A4, 945–954, 1992.; Bhattacharjee, A., Ng, C. S., and Spangler, S. R.: ApJ, 494, 4...

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Numerical Detection of Unstable Periodic Orbits in Continuous-time...

By: Y. Saiki

Description: Research Institute for Mathematical Sciences, Kyoto University, Japan. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical comput...

Biham, O. and Wenzel, W.: Characterization of unstable periodic orbits in chaotic attractors and repellers, Phys. Rev. Lett., 63, 819–822, 1989.; Bowen, R.: Topological entropy and AxiomA, Global Analysis (Berkeley, CA, 1968) Proc. Sympos. Pure Math., 14, Amererican Mathathematical Society, Providence, RI, 23–41, 1970.; Chian, A. C. L., Kamide, Y., Rempel, E. L., and Santana, W. M.: On the chaotic nature of solar-terrestrial environmen...

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On Stochastic Stability of Regional Ocean Models with Uncertainty ...

By: L. M. Ivanov; P. C. Chu

Description: Naval Postgraduate School, Monterey, CA, USA. A shallow-water model was used to understand model error induced by non-Gaussian wind uncertainty. Although the model was simple, it described a generic system with many degrees of freedom randomized by external noise. The study focused on the nontrivial collective behavior of finite-amplitude perturbations on different scales and their influence on model predictability. The error growth strongly depended on...

Auclair, F., Marsaleix, P., and De Mey, P.: Space-time structure and dynamics of the forecast error in a coastal circulation model of the Gulf of Lions, Dyn. Atmos. Oceans, 36, 309–346, 2003.; Benzi, R., Sutra, A., and Vulpiani, A.: The mechanism of stochastic resonance, J. Phys. A: Math. Gen., 14, L453–L457, 1981.; Berloff, P. S.: Random-forcing model of the mesoscale oceanic eddies, J. Fluid Mech., 529, 71–95, 2005.; Aurell, E., Boffetta, G., Crisanti, A., Paladin, G.,...

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Another Look at Climate Sensitivity : Volume 17, Issue 2 (17/03/2010)

By: M. Ghil; I. Zaliapin

Description: Department of Mathematics and Statistics, University of Nevada, Reno, USA. We revisit a recent claim that the Earth's climate system is characterized by sensitive dependence to parameters; in particular, that the system exhibits an asymmetric, large-amplitude response to normally distributed feedback forcing. Such a response would imply irreducible uncertainty in climate change predictions and thus have notable implications for climate science and clim...

Allen, M. R. and Frame, D. J.: Call off the quest, Science, 318(5850), 582–583, 2007.; Andronov, A. A. and Pontryagin, L. S.: Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14(5), 247–250, 1937.; Arfken, G.: Taylor's Expansion, Mathematical Methods for Physicists, 3rd ed., Orlando, FL, Academic Press, pp. 303–313, 1985.; Bryan, F.: High-latitude salinity effects and interhemispheric thermohaline circulations, Nature, 323, 301–304, 1986.; Budyko, M. I.: The effect of solar r...

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Nonlinear Chaotic Model for Predicting Storm Surges : Volume 17, I...

By: M. Siek; D. P. Solomatine

Description: Department of Hydroinformatics and Knowledge Management, UNESCO-IHE Institute for Water Education, Delft, The Netherlands. This paper addresses the use of the methods of nonlinear dynamics and chaos theory for building a predictive chaotic model from time series. The chaotic model predictions are made by the adaptive local models based on the dynamical neighbors found in the reconstructed phase space of the observables. We implemented the univariate and ...

Alexandersson, H., Schmith, T., Iden, K., and Tuomenvirta, H.: Long-term variations of the storm climate over NW Europe, Global Atmos. Ocean System, 6, 97–120, 1998.; Battjes, J. A. and Gerritsen, H.: Coastal modelling for flood defence, Philos. Transact. A Math. Phys. Eng. Sci., 360(1796), 1461–1475, 2002.; Bode, L. and Hardy, T. A.: Progress and recent developments in storm surge modeling, J. Hydraul. Eng.-ASCE, 123(4), 315–331, 1997.; Box, G., Jenkins, G., and Reinsel...

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Inversion of 2-d Dc Resistivity Data Using Rapid Optimization and ...

By: U. K. Singh; S. B. Singh; R. K. Tiwari

Description: Department of Applied Geophysics, Indian School of Mines, Dhanbad-826 004, India. The backpropagation (BP) artificial neural network (ANN) technique of optimization based on steepest descent algorithm is known to be inept for its poor performance and does not ensure global convergence. Nonlinear and complex DC resistivity data require efficient ANN model and more intensive optimization procedures for better results and interpretations. Improvements in t...

Battiti, R.: First and second order methods for learning between steepest descent and Newton's methods, Neural Comput., 4, 141–166, 1992.; Baum, E. and Haussler, D.: What size net gives valid generalization?, Neural Comput., 1, 151–160, 1989.; Brown, M. and Poulton, M.: Locating buried objects for environmental site investigations using neural networks, J. Environ. Eng. Geoph., 1, 179–188, 1996.; Constable, S., Parker, R., and Constable, C.: Occam's inversion: a practica...

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The Effect of Stochastic Perturbations on Plankton Transport by In...

By: J. V. Mecking; C. Subich; F. J. Poulin; M. Stastna

Description: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada. Internal solitary and solitary-like waves are a commonly observed feature of density stratified natural waters, including lakes and the coastal ocean. Since such waves induce significant currents throughout the water column they can be responsible for significant transport of both passive and swimming biota. We consider simple models of moving zooplankton based...

Aref, H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows, Ann. Rev. Fluid Mech., 15, 345–389, 1983.; Bartosch, L.: Generation of Colored Noise, J. Modern Phys., 6, 851–855, 2001.; Bearon, R. N.: A model for bacterial colonization of sinking aggregates, Bull. Math. BIol., 69, 417–431, 2007.; Farmer, D. and Armi, L.: The generation and trapping of solitary waves over topography, Science, 283, 188–190, 1999.; de la Fuente, A., Shimizu, K., Imberge...

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Chaotic Behavior in the Flow Along a Wedge Modeled by the Blasius ...

By: B. Basu; A. S. Sharma; E. Foufoula-georgiou

Description: Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55414, USA. The Blasius equation describes the properties of steady-state two dimensional boundary layer forming over a semi-infinite plate parallel to a unidirectional flow field. The flow is governed by a modified Blasius equation when the surface is aligned along the flow. In this paper, we demonstrate using numerical solution, that as the wedge angle increases, bifurcation occ...

Schwartz, L. W. and Eley, R. R.: Flow of architectural coatings on complex surfaces theory and experiment, J. Eng. Math., 43, 153–171, 2002.; Adomian, G.: A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501–544, doi:10.1016/0022-247X(88)90170-9, 1988.; Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations, Comput. Math. Appl., 21(...

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Optimal Solution Error Covariance in Highly Nonlinear Problems of ...

By: I. Gejadze; G. J. M. Copeland; F.-x. Le Dimet; V. Shutyaev

Description: Institute of Numerical Mathematics, Russian Academy of Sciences, 119333 Gubkina 8, Moscow, Russia. The problem of variational data assimilation (DA) for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition, boundary conditions and/or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covariance matri...

Ratkowsky, D. A.: Nonlinear regression modelling: a unified practical approach, Marcel Dekker, New York, 1983.; Thacker, W. C.: The role of the Hessian matrix in fitting models to measurements, J. Geophys. Res., 94, 6177–6196, 1989.; Amemiya, T.: Handbook of econometrics, 1, North-Holland Publishing Company, Amsterdam, 1983.; Courtier, P., Thépaut, J. N., and Hollingsworth, A.: A strategy for operational implementation of 4D-Var, using an incremental approach, Q. J. Roy....

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Bistable Systems with Stochastic Noise: Virtues and Limits of Effe...

By: V. Lucarini; D. Faranda; M. Willeit

Description: Klimacampus, Institute of Meteorology, Hamburg Universität, Hamburg, Germany. The understanding of the statistical properties and of the dynamics of multistable systems is gaining more and more importance in a vast variety of scientific fields. This is especially relevant for the investigation of the tipping points of complex systems. Sometimes, in order to understand the time series of given observables exhibiting bimodal distributions, simple one-dime...

Benzi, R., Parisi G., Sutera A., and Vulpiani A.: Stochastic resonance in climatic change, Tellus 34, 10–16, 1982.; Benzi, R.: Stochastic resonance: from climate to biology, Nonlin. Processes Geophys., 17, 431–441, doi:10.5194/npg-17-431-2010, 2010.; Boyle, E. and Kegwin, L.: North Atlantic thermohaline circulation during the past 20,000 years linked to high-latitude surface temperature, Nature, 330, 35–40, 1987.; Ces...

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