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Sample records for clausius-clapeyron equation

  1. Clausius-Clapeyron Equation and Saturation Vapour Pressure: Simple Theory Reconciled with Practice

    Science.gov (United States)

    Koutsoyiannis, Demetris

    2012-01-01

    While the Clausius-Clapeyron equation is very important as it determines the saturation vapour pressure, in practice it is replaced by empirical, typically Magnus-type, equations which are more accurate. It is shown that the reduced accuracy reflects an inconsistent assumption that the latent heat of vaporization is constant. Not only is this…

  2. Heating-Cooling Asymmetry in the δ-γ Transformation in Plutonium: Clausius-Clapeyron Considerations

    Energy Technology Data Exchange (ETDEWEB)

    Schwartz, Daniel S. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Mitchell, Jeremy Neil [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

    2015-07-27

    Slides discuss the subject under the following topics: Pu phase transformations and features of the γ-δ transformation (heating-cooling asymmetry, cooling rate, effect of impurities); pressure effects in γ-δ transformations; Clausius-Clapeyron analysis; and discussion of heating-cooling asymmetry in the γ-δ transformation. The following conclusions are reached: burst behavior and extended transformation range due to pressure arrest; low slope of P-T curve for γ-δ favors this transformation for pressure arrest; asymmetry w.r.t. direction of transformation likely due to defects.

  3. Clausius-Clapeyron Scaling of Convective Available Potential Energy (CAPE) in Cloud-Resolving Simulations

    Science.gov (United States)

    Seeley, J.; Romps, D. M.

    2015-12-01

    Recent work by Singh and O'Gorman has produced a theory for convective available potential energy (CAPE) in radiative-convective equilibrium. In this model, the atmosphere deviates from a moist adiabat—and, therefore, has positive CAPE—because entrainment causes evaporative cooling in cloud updrafts, thereby steepening their lapse rate. This has led to the proposal that CAPE increases with global warming because the strength of evaporative cooling scales according to the Clausius-Clapeyron (CC) relation. However, CAPE could also change due to changes in cloud buoyancy and changes in the entrainment rate, both of which could vary with global warming. To test the relative importance of changes in CAPE due to CC scaling of evaporative cooling, changes in cloud buoyancy, and changes in the entrainment rate, we subject a cloud-resolving model to a suite of natural (and unnatural) forcings. We find that CAPE changes are primarily driven by changes in the strength of evaporative cooling; the effect of changes in the entrainment rate and cloud buoyancy are comparatively small. This builds support for CC scaling of CAPE.

  4. How closely do changes in surface and column water vapor follow Clausius-Clapeyron scaling in climate change simulations?

    International Nuclear Information System (INIS)

    The factors governing the rate of change in the amount of atmospheric water vapor are analyzed in simulations of climate change. The global-mean amount of water vapor is estimated to increase at a differential rate of 7.3% K-1 with respect to global-mean surface air temperature in the multi-model mean. Larger rates of change result if the fractional change is evaluated over a finite change in temperature (e.g., 8.2% K-1 for a 3 K warming), and rates of change of zonal-mean column water vapor range from 6 to 12% K-1 depending on latitude. Clausius-Clapeyron scaling is directly evaluated using an invariant distribution of monthly-mean relative humidity, giving a rate of 7.4% K-1 for global-mean water vapor. There are deviations from Clausius-Clapeyron scaling of zonal-mean column water vapor in the tropics and mid-latitudes, but they largely cancel in the global mean. A purely thermodynamic scaling based on a saturated troposphere gives a higher global rate of 7.9% K-1. Surface specific humidity increases at a rate of 5.7% K-1, considerably lower than the rate for global-mean water vapor. Surface specific humidity closely follows Clausius-Clapeyron scaling over ocean. But there are widespread decreases in surface relative humidity over land (by more than 1% K-1 in many regions), and it is argued that decreases of this magnitude could result from the land/ocean contrast in surface warming.

  5. Departure from Clausius-Clapeyron scaling of water entering the stratosphere in response to changes in tropical upwelling

    Science.gov (United States)

    Fueglistaler, S.; Liu, Y. S.; Flannaghan, T. J.; Ploeger, F.; Haynes, P. H.

    2014-02-01

    Water entering the stratosphere ([H2O]entry) is strongly constrained by temperatures in the tropical tropopause layer (TTL). Temperatures at tropical tropopause levels are 15-20 K below radiative equilibrium. A strengthening of the residual circulation as suggested by general circulation models in response to increasing greenhouse gases is, based on radiative transfer calculations, estimated to lead to a temperature decrease of about 2 K per 10% change in upwelling (with some sensitivity to vertical scale length). For a uniform temperature change in the inner tropics, [H2O]entry may be expected to change as predicted by the temperature dependence of the vapor pressure, referred here as "Clausius-Clapeyron (CC) scaling." Under CC scaling, this corresponds to ˜1 ppmv change in [H2O]entry per 10% change in upwelling. However, the change in upwelling also changes the residence time of air in the TTL. We show with trajectory calculations that this affects [H2O]entry, such that [H2O]entry changes ˜10 % less than expected from CC scaling. This residence time effect for water vapor is a consequence of the spatiotemporal variance in the temperature field. We show that for the present-day TTL, a little more than half of the effect is due to the systematic relation between flow and temperature field. The remainder can be understood from the perspective of a random walk problem, with slower ascent (longer path) increasing each air parcel's probability to encounter anomalously low temperatures. Our results show that atmospheric water vapor may depart from CC scaling with mean temperatures even when all physical processes of dehydration remain unchanged.

  6. Changes in intensity of precipitation extremes in Romania on very hight temporal scale and implications on the validity of the Clausius-Clapeyron relation

    Science.gov (United States)

    Busuioc, Aristita; Baciu, Madalina; Breza, Traian; Dumitrescu, Alexandru; Stoica, Cerasela; Baghina, Nina

    2016-04-01

    Many observational, theoretical and based on climate model simulation studies suggested that warmer climates lead to more intense precipitation events, even when the total annual precipitation is slightly reduced. In this way, it was suggested that extreme precipitation events may increase at Clausius-Clapeyron (CC) rate under global warming and constraint of constant relative humidity. However, recent studies show that the relationship between extreme rainfall intensity and atmospheric temperature is much more complex than would be suggested by the CC relationship and is mainly dependent on precipitation temporal resolution, region, storm type and whether the analysis is conducted on storm events rather than fixed data. The present study presents the dependence between the very hight temporal scale extreme rainfall intensity and daily temperatures, with respect to the verification of the CC relation. To solve this objective, the analysis is conducted on rainfall event rather than fixed interval using the rainfall data based on graphic records including intensities (mm/min.) calculated over each interval with permanent intensity per minute. The annual interval with available a such data (April to October) is considered at 5 stations over the interval 1950-2007. For Bucuresti-Filaret station the analysis is extended over the longer interval (1898-2007). For each rainfall event, the maximum intensity (mm/min.) is retained and these time series are considered for the further analysis (abbreviated in the following as IMAX). The IMAX data were divided based on the daily mean temperature into bins 2oC - wide. The bins with less than 100 values were excluded. The 90th, 99th and 99.9th percentiles were computed from the binned data using the empirical distribution and their variability has been compared to the CC scaling (e.g. exponential relation given by a 7% increase per temperature degree rise). The results show a dependence close to double the CC relation for

  7. HIGH ENERGY RATE EXTRUSION OF URANIUM

    Science.gov (United States)

    Lewis, L.

    1963-07-23

    A method of extruding uranium at a high energy rate is described. Conditions during the extrusion are such that the temperature of the metal during extrusion reaches a point above the normal alpha to beta transition, but the metal nevertheless remains in the alpha phase in accordance with the Clausius- Clapeyron equation. Upon exiting from the die, the metal automatically enters the beta phase, after which the metal is permitted to cool. (AEC)

  8. Estabelecimento do Conceito de Temperatura como uma grandeza derivada da Energia e da Entropia

    CERN Document Server

    De Abreu, R

    2002-01-01

    Temperature is introduced as a derived concept from energy and entropy. We consider two sub-systems in equilibrium for several configurations. The equality of temperature of the sub-systems is obtained from the equilibrium condition. The isothermal transformation is defined for several configurations and from this definition we obtain the Clausius-Clapeyron equation. We apply the analysis to the ideal gas. The classical ideal gas appears as a limit and the problem of the measurement of temperature is analysed.

  9. THERMODYNAMIC STUDY OF HIGH-PRESSURE ADSORPTION OF METHANE AND HEATS OF METHANE ADSORPTION ON MICROPOROUS CARBONS

    Institute of Scientific and Technical Information of China (English)

    杨晓东; 林文胜; 郑青榕; 顾安忠; 鲁雪生; 宋燕

    2002-01-01

    The study was done for high-pressure adsorption of methane on microporous carbons, which has an ANG vehicular application background. Adsorption isotherm of methane on super activated carbon up to 6 MPa was measured and isosteric heats of methane adsorption on a number of microporous carbons were determined from adsorption isosteres by the Clausius-Clapeyron equation. The variation of the isosteric heats of adsorption with the amount of methane adsorbed was discussed.

  10. Determinação da entalpia de vaporização de líquidos pelo método do isoteniscópio de Smith e Menzies Determination of the enthalpy of vaporization of liquid compounds by the Smith Menzies (isoteniscope method

    Directory of Open Access Journals (Sweden)

    Adriana Passarella Gerola

    2010-01-01

    Full Text Available This article proposes an experimental procedure to determine the enthalpy (and entropy of vaporization of organic liquid compounds, by the Smith-Menzies (isoteniscope method. The values of vapor pressure at different temperatures were obtained and ΔvH (and ΔvS were graphically determined, using the Clausius-Clapeyron equation. The results for diethyl-ether, propanone, ethanol and n-hexane are in very good agreement with those from literature. A historical and thermodynamic discussion on equations that correlates vapor pressures and temperature precedes the experimental proposition.

  11. Heats of adsorption for charcoal nitrogen systems

    Energy Technology Data Exchange (ETDEWEB)

    Prasad, M.; Akkimaradi, B.S.; Rastogi, S.C. [ISRO Satellite Centre, Bangalore (India). Thermal Systems Group; Rao, R.R. [Government College for Boys, Kolar, Karnataka (India); Srinivasan, K. [Indian Institute of Science, Bangalore (India). Dept. of Mechanical Engineering

    1999-07-01

    This paper develops an empirical equation for correlation of the loading dependence of the heat of adsorption for two samples of activated charcoal-nitrogen systems. Details are given of the use of isotherm data, the evaluation of the heat of adsorption using the Clausius-Clapeyron equation, the plotting of primary adsorption data, and the plotting of the heat of adsorption as a function of the loading of the two samples. The need to consider the heat of adsorption property when designing a system in which a gaseous medium is adsorbed by a solid sorbent is discussed. (UK)

  12. Precision ozone vapor pressure measurements

    Science.gov (United States)

    Hanson, D.; Mauersberger, K.

    1985-01-01

    The vapor pressure above liquid ozone has been measured with a high accuracy over a temperature range of 85 to 95 K. At the boiling point of liquid argon (87.3 K) an ozone vapor pressure of 0.0403 Torr was obtained with an accuracy of + or - 0.7 percent. A least square fit of the data provided the Clausius-Clapeyron equation for liquid ozone; a latent heat of 82.7 cal/g was calculated. High-precision vapor pressure data are expected to aid research in atmospheric ozone measurements and in many laboratory ozone studies such as measurements of cross sections and reaction rates.

  13. Thermochemical study of the monobromonitrobenzene isomers

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Lobo Ferreira, Ana I.M.C.; Santos, Ana Filipa L.O.M.; Rocha, Ines M. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2010-02-15

    The standard (p{sup o} = 0.1 MPa) molar enthalpies of formation, of the 2-, 3-, and 4-monobromonitrobenzene isomers, in the crystalline phase, at T = 298.15 K, were derived from the standard massic energies of combustion, in oxygen, at T = 298.15 K, measured by rotating bomb combustion calorimetry. From the temperature dependence of the vapour pressures of these compounds, measured by the Knudsen effusion technique, their standard molar enthalpies of sublimation, at T = 298.15 K, were derived using the Clausius-Clapeyron equation.

  14. Experimental study of damping in civil engineering structures using smart materials (CuAlBe - SMA): an application to a steel portico

    OpenAIRE

    Torra Ferré, Vicenç; Isalgue Buxeda, Antonio; Lovey, Francisco Carlos; Carreras, Guillem; Casciati, Fabio; Soul, H.

    2010-01-01

    The target of the paper focuses in the required properties for successful behavior of the CuAlBe Shape Memory Alloy (SMA) in damping of steel structures under the action of earthquakes. The appropriate fracture – life, the long time of aging, the minor creep effects on cycling, the Clausius-Clapeyron equation and the self-heating effects are also, evaluated. Analysis via simulation using a proprietary model of the SMA behavior furnishes satisfactory results. Our main interest is focused in th...

  15. The influence of substitution of Mn by Fe and Co on magnetocaloric effect and magnetoresistance properties of SmMn{sub 2}Ge{sub 2}

    Energy Technology Data Exchange (ETDEWEB)

    Dincer, I., E-mail: idincer@eng.ankara.edu.tr [Department of Engineering Physics, Faculty of Engineering, Ankara University, 06100 Besevler, Ankara (Turkey); Elerman, Y. [Department of Engineering Physics, Faculty of Engineering, Ankara University, 06100 Besevler, Ankara (Turkey)

    2013-01-15

    Magnetocaloric and magnetoresistance properties of SmMn{sub 2-x}Fe{sub x}Ge{sub 2} (x=0.05 and 0.10) and SmMn{sub 2-x}Co{sub x}Ge{sub 2} (x=0.05 and 0.15) compounds have been studied by magnetic and resistance measurements in the temperature range between 30 and 350 K. All compounds exhibit metamagnetic transition from antiferromagnetism to ferromagnetism around the Sm moments ferromagnetic ordering temperature-T{sub Sm} and Mn moments antiferromagnetic ordering temperature-T{sub N1}. The magnetic entropy changes of these compounds are estimated from the Maxwell equation, Maxwell Clausius Clapeyron equation, Landau theory and mean-field theory. The maximum magnetic entropy change values of SmMn{sub 1.90}Fe{sub 0.10}Ge{sub 2} and SmMn{sub 1.85}Co{sub 0.15}Ge{sub 2} compounds are -8.1 J kg{sup -1} K{sup -1} and -5.1 J kg{sup -1} K{sup -1} in a magnetic field change of 5 T, respectively. These compounds show negative magnetoresistance around the magnetic phase transition temperatures. The magnetoresistance value of SmMn{sub 1.95}Fe{sub 0.05}Ge{sub 2} is -23% at T{sub Sm} which is bigger than the magnetoresistance value of SmMn{sub 2}Ge{sub 2} (-15%). - Highlights: Black-Right-Pointing-Pointer Magnetic entropy changes of these compounds are estimated from the Maxwell equation, Maxwell Clausius Clapeyron equation, Landau theory and mean-field theory. Black-Right-Pointing-Pointer Our results exhibit that the Maxwell Clausius Clapeyron equation should be used for compounds that have a first-order phase transition instead of using the Maxwell equation. Black-Right-Pointing-Pointer As a result, in the antiferromagnetic state, the resistance is higher than that in the ferromagnetic state, suggesting that the MR in compounds from the RMn{sub 2}X{sub 2} family may arise on the basis of a spin-valve mechanism.

  16. Thermodynamics analysis of aluminum plasma transition induced by hypervelocity impact

    Science.gov (United States)

    Liu, Zhixiang; Zhang, Qingming; Ju, Yuanyuan

    2016-02-01

    The production of plasmas during hypervelocity meteoroid and space debris impact has been proposed to explain the presence of paleomagnetic fields on the lunar surface, and also the electromagnetic damage to spacecraft electronic devices. Based on Gibbs' ensemble theory, we deduce Saha equation of state and figure out the ionization degree; further, by using the derivation of Clausius-Clapeyron equation, we obtain the entropy increase and latent heat of plasma transition after vaporization; finally, we analyze the conversion efficiency of kinetic energy into internal energy, present two key contradictions, and revise them with the entropy increase, latent heat, and conversion efficiency. We analyze the aluminum plasma transition from multiple perspectives of the equation of state, latent heat of phase transition, and conversion efficiency and propose the internal energy and impact velocity criterion, based on the laws of thermodynamics.

  17. Adsorption properties of biomass-based activated carbon prepared with spent coffee grounds and pomelo skin by phosphoric acid activation

    Science.gov (United States)

    Ma, Xiaodong; Ouyang, Feng

    2013-03-01

    Activated carbon prepared from spent coffee grounds and pomelo skin by phosphoric acid activation had been employed as the adsorbent for ethylene and n-butane at room temperature. Prepared activated carbon was characterized by means of nitrogen adsorption-desorption, X-ray powder diffraction, scanning electron microscope and Fourier transform infrared spectroscope. It was confirmed that pore structure played an important role during the adsorption testes. Adsorption isotherms of ethylene and n-butane fitted well with Langmuir equation. The prepared samples owned better adsorption capacity for n-butane than commercial activated carbon. Isosteric heats of adsorptions at different coverage were calculated through Clausius-Clapeyron equation. Micropore filling effect was explained in a thermodynamic way.

  18. Methane hydrate formation and dissociation in synthetic seawater

    Institute of Scientific and Technical Information of China (English)

    Vikash Kumar Saw; Iqbal Ahmad; Ajay Mandal; G.Udayabhanu; Sukumar Laik

    2012-01-01

    The formation and dissociation of methane gas hydrate at an interface between synthetic seawater (SSW) and methane gas have been experimentally investigated in the present work.The amount of gas consumed during hydrate formation has been calculated using the real gas equation.Induction time for the formation of hydrate is found to depend on the degree of subcooling.All the experiments were conducted in quiescent system with initial cell pressure of 11.14 MPa.Salinity effects on the onset pressure and temperature of hydrate formation are also observed.The dissociation enthalpies of methane hydrate in synthetic seawater were determined by Clausius-Clapeyron equation based on the measured phase equilibrium data.The dissociation data have been analyzed by existing models and compared with the reported data.

  19. Methane Hydrate Formation and Dissociation in the Presence of Silica Sand and Bentonite Clay

    Directory of Open Access Journals (Sweden)

    Kumar Saw V.

    2015-11-01

    Full Text Available The formation and dissociation of methane hydrates in a porous media containing silica sand of different sizes and bentonite clay were studied in the presence of synthetic seawater with 3.55 wt% salinity. The phase equilibrium of methane hydrate under different experimental conditions was investigated. The effects of the particle size of silica sand as well as a mixture of bentonite clay and silica sand on methane hydrate formation and its dissociation were studied. The kinetics of hydrate formation was studied under different subcooling conditions to observe its effects on the induction time of hydrate formation. The amount of methane gas encapsulated in hydrate was computed using a real gas equation. The Clausius-Clapeyron equation is used to estimate the enthalpy of hydrate dissociation with measured phase equilibrium data.

  20. Entropy, Order Parameters, and Complexity: Incorporating the last 50 years into the statistical mechanics curriculum

    Science.gov (United States)

    Sethna, James

    2007-03-01

    The purview of statistical mechanics has grown rapidly in the past decades, with nonequilibrium extensions and applications to dynamical systems, molecular biology and bioinformatics, complex systems and networks, digital communication and information theory, and econophysics and other social sciences. It is our responsibility to join these new insights to the old wisdom in the field, and to distill the key ideas for the next generation. We should include (a) Shannon entropy, data compression, and reversible computation, (b) chaotic motion, ergodicity and the KAM theorem, and renormalization-group treatments of the onset of chaos, (c) molecular motors and hidden Markov models for analyzing genomic data. We should make statistical mechanics useful and comprehensible to those outside of physics, eschewing applications (Clausius-Clapeyron equations, cp vs. cv) and methods (quantum mechanics) accessible and interesting only to condensed-matter physicists and physical chemists. See Entropy, Order Parameters, and Complexity (http://www.physics.cornell.edu/sethna/StatMech/), OUP, 2006.

  1. Experimental Investigation on the Mechanical Instability of Superelastic NiTi Shape Memory Alloy

    Science.gov (United States)

    Xiao, Yao; Zeng, Pan; Lei, Liping

    2016-09-01

    In this paper, primary attention is paid to the mechanical instability of superelastic NiTi shape memory alloy (SMA) during localized forward transformation at different temperatures. By inhibiting the localized phase transformation, we can obtain the up-down-up mechanical response of NiTi SMA, which is closely related to the intrinsic material softening during localized martensitic transformation. Furthermore, the material parameters of the up-down-up stress-strain curve are extracted, in such a way that this database can be utilized for simulation and validation of the theoretical analysis. It is found that during forward transformation, the upper yield stress, lower yield stress, Maxwell stress, and nucleation stress of NiTi SMA exhibit linear dependence on temperature. The relation between nucleation stress and temperature can be explained by the famous Clausius-Clapeyron equation, while the relation between upper/lower yield stress and temperature lacks theoretical study, which needs further investigation.

  2. Pressure Effects on Thermodynamics of Phase Behavior in Polystyrene/Methylcyclohexane Solutions

    Institute of Scientific and Technical Information of China (English)

    LI Hong-fei; JIANG Shi-chun; AN Li-jia; YU Dong-hong

    2007-01-01

    The calculations presented in this paper are based on the Sanchez-Lacombe(SL) lattice fluid theory. The interaction energy parameter, g12*/k, required in this approach was obtained by fitting the cloud points of polystyrene(PS)/methylcyclohexane(MCH) polymer solutions under pressure. The SL lattice fluid theory was used to calculate the spinodals, the binodals, and the Flory-Huggins(FH) interaction parameter of the solutions. The calculated results show that the SL lattice fluid theory can describe the dependences of thermodynamics of PS/MCH solutions on temperature and pressure very well. However, the calculated enthalpy and the excess volume changes indicate that the Clausius-Clapeyron equation cannot be suitable to describe pressure effect on PS/MCH solutions. Further analysis on the thermodynamics of this system under pressure shows that the role of entropy is more important than the excess volume in the present case.

  3. Thermochemical study of 2,5-dimethyl-3-furancarboxylic acid, 4,5-dimethyl-2-furaldehyde, and 3-acetyl-2,5-dimethylfuran

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Amaral, Luisa M.P.F. [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2011-01-15

    The standard (p{sup o} = 0.1 MPa) molar enthalpies of formation, in the gaseous state, at T = 298.15 K, for 2,5-dimethyl-3-furancarboxylic acid, 3-acetyl-2,5-dimethylfuran, and 4,5-dimethyl-2-furaldehyde were derived from the values of the standard molar enthalpies of formation, in the condensed phase, and the standard molar enthalpies of phase transition from the condensed to the gaseous state. The values of the standard molar enthalpies of formation of the compounds in the condensed phases were calculated from the measurements of the standard massic energies of combustion obtained by static bomb combustion calorimetry. The enthalpies of vaporization/sublimation were measured by Calvet high temperature microcalorimetry. For 2,5-dimethyl-3-furancarboxylic acid the standard enthalpy of sublimation was also calculated, by the application of the Clausius-Clapeyron equation, to the temperature dependence of the vapor pressures measured by the Knudsen effusion technique. (table)

  4. SORPTION OF PHENOL ONTO GEL—TYPE CROSSLINKED POLYSTYRENEISOCYANURIC ACID RESIN

    Institute of Scientific and Technical Information of China (English)

    XUMancai; SHIZuoqing; 等

    2000-01-01

    Spherical crosslinked polystyrene-isocyanuric acid resin was synthesized by reaction of chloromethylated polystryene with isocyanuric acid.The sorption isotherms of phenol from aqueous solution and cyclohexane solution onto the resin were measured.It is of interest to notice that the resin sorbed phenol efficiently though its specific surface area was 0 and did not swell in water,and the sorption capactity from aqueous solution was close to that of phenol onto XAD-4 at the same equilibrium concentration.Sorption enthalpies calculated from the isotherms according to the Clausius-Clapeyron equation were -21-25kJ/mol and -39-41kJ/mol respectively.These values impled that the sorption processes were based on hydrogen bonding.In addition.the details of the hydrogen bonding between the active sites of the resin and phenol were suggested.

  5. Three years (2008-2010) of measurements of atmospheric concentrations of organochlorine pesticides (OCPs) at Station Nord, North East Greenland

    DEFF Research Database (Denmark)

    Bossi, Rossana; Skjøth, Carsten Ambelas; Skov, Henrik

    2013-01-01

    Atmospheric concentrations of organochlorine pesticides (OCPs) have been measured for the first time at Station Nord, North-East Greenland, from 2008 to 2010. The data obtained are reported here. Hexachlorobenzene (HCB), endosulfan I and hexachlorocyclohexanes (HCHs) were the predominant compounds...... detected in the atmosphere, followed by p,p'-DDE and dieldrin. Chlordane isomers and related compounds (trans- and cis-chlordanes, heptachlor and heptachlor epoxide, trans-and cis-nonachlor) were also detected. Atmospheric concentrations of the investigated compounds were correlated with temperature using...... the Clausius-Clapeyron equation in order to obtain information about their transport properties. The correlation between atmospheric concentrations and temperature was not significant for endosulfan I, gamma-HCH and p,p'-DDT, which indicates that direct transport from direct sources is the dominating...

  6. Ageneral approach to first order phase transitions and the anomalous behavior of coexisting phases in the magnetic case.

    Energy Technology Data Exchange (ETDEWEB)

    Gama, S.; de Campos, A.; Coelho, A. A.; Alves, C. S.; Ren, Y.; Garcia, F.; Brown, D. E.; da Silva, L. M.; Magnus, A.; Carvalho, G.; Gandra, G. C.; dos Santos, A. O.; Cardoso, L. P.; von Ranke, P. J.; X-Ray Science Division; Univ. Federal de Sao Paulo; Unv. Estadual de Champinas; Univ. Estadual de Maringa Lab. Nacional de Luz Sincrotron; Northern Univ.; Univ. de Estado do Rio de Janerio

    2009-01-01

    First order phase transitions for materials with exotic properties are usually believed to happen at fixed values of the intensive parameters (such as pressure, temperature, etc.) characterizing their properties. It is also considered that the extensive properties of the phases (such as entropy, volume, etc.) have discontinuities at the transition point, but that for each phase the intensive parameters remain constant during the transition. These features are a hallmark for systems described by two thermodynamic degrees of freedom. In this work it is shown that first order phase transitions must be understood in the broader framework of thermodynamic systems described by three or more degrees of freedom. This means that the transitions occur along intervals of the intensive parameters, that the properties of the phases coexisting during the transition may show peculiar behaviors characteristic of each system, and that a generalized Clausius-Clapeyron equation must be obeyed. These features for the magnetic case are confirmed, and it is shown that experimental calorimetric data agree well with the magnetic Clausius-Clapeyron equation for MnAs. An estimate for the point in the temperature-field plane where the first order magnetic transition turns to a second order one is obtained (the critical parameters) for MnAs and Gd{sub 5}Ge{sub 2}Si{sub 2} compounds. Anomalous behavior of the volumes of the coexisting phases during the magnetic first order transition is measured, and it is shown that the anomalies for the individual phases are hidden in the behavior of the global properties as the volume.

  7. Scaling of precipitation extremes with temperature in the French Mediterranean region: What explains the hook shape?

    OpenAIRE

    Drobinski, Philippe; Alonzo, Bastien; Bastin, Sophie; Da Silva, Nicolas; Muller, Caroline

    2016-01-01

    International audience Expected changes to future extreme precipitation remain a key uncertainty associated with anthropogenic climate change. Extreme precipitation has been proposed to scale with the precipitable water content in the atmosphere. Assuming constant relative humidity, this implies an increase of precipitation extremes at a rate of about 7% °C−1 globally as indicated by the Clausius-Clapeyron relationship. Increases faster and slower than Clausius-Clapeyron have also been rep...

  8. Cubic Equation

    Institute of Scientific and Technical Information of China (English)

    2004-01-01

    At the beginning of 16th century, mathematicians found it easy to solve equations of the first degree(linear equations, involving x) and of the second degree(quadratic equatiorts, involving x2). Equations of the third degree(cubic equations, involving x3)defeated them.

  9. Integral equations

    CERN Document Server

    Moiseiwitsch, B L

    2005-01-01

    Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco

  10. Riccati equations

    Directory of Open Access Journals (Sweden)

    Lloyd K. Williams

    1987-01-01

    Full Text Available In this paper we find closed form solutions of some Riccati equations. Attention is restricted to the scalar as opposed to the matrix case. However, the ones considered have important applications to mathematics and the sciences, mostly in the form of the linear second-order ordinary differential equations which are solved herewith.

  11. Scaling of precipitation extremes with temperature in the French Mediterranean region: What explains the hook shape?

    Science.gov (United States)

    Drobinski, P.; Alonzo, B.; Bastin, S.; Silva, N. Da; Muller, C.

    2016-04-01

    Expected changes to future extreme precipitation remain a key uncertainty associated with anthropogenic climate change. Extreme precipitation has been proposed to scale with the precipitable water content in the atmosphere. Assuming constant relative humidity, this implies an increase of precipitation extremes at a rate of about 7% °C-1 globally as indicated by the Clausius-Clapeyron relationship. Increases faster and slower than Clausius-Clapeyron have also been reported. In this work, we examine the scaling between precipitation extremes and temperature in the present climate using simulations and measurements from surface weather stations collected in the frame of the HyMeX and MED-CORDEX programs in Southern France. Of particular interest are departures from the Clausius-Clapeyron thermodynamic expectation, their spatial and temporal distribution, and their origin. Looking at the scaling of precipitation extreme with temperature, two regimes emerge which form a hook shape: one at low temperatures (cooler than around 15°C) with rates of increase close to the Clausius-Clapeyron rate and one at high temperatures (warmer than about 15°C) with sub-Clausius-Clapeyron rates and most often negative rates. On average, the region of focus does not seem to exhibit super Clausius-Clapeyron behavior except at some stations, in contrast to earlier studies. Many factors can contribute to departure from Clausius-Clapeyron scaling: time and spatial averaging, choice of scaling temperature (surface versus condensation level), and precipitation efficiency and vertical velocity in updrafts that are not necessarily constant with temperature. But most importantly, the dynamical contribution of orography to precipitation in the fall over this area during the so-called "Cevenoles" events, explains the hook shape of the scaling of precipitation extremes.

  12. Beautiful equations

    Science.gov (United States)

    Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul

    2014-07-01

    In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).

  13. Vapor Pressures of Di-n-Butyl Phthalate and Di-iso-Butyl Hexahydrophthalate at Reduced Pressures

    Institute of Scientific and Technical Information of China (English)

    齐欣; 徐立勇; 高正红; 刘志华

    2004-01-01

    In this paper the measured values of the vapor pressures by ebulliometer method of two important maleic anhydride recovery solvents, di-n-butyl phthalate (DBP) and di-iso-butyl hexahydrophthalate (DIBE), between 0.63-17.79 kPa and 0.49-30.95 kPa,are reported respectively.A comparison of the data of DBP with the published data has been made, which shows good consistency. For the convenient use of these vapor pressures, Cragoe equation, Antoine equation and Kirchhoff equation are selected to correlate them. The correlating results show that Antoine equation is the best one of the three equations to fit for the vapor pressures of the two solvents. According to Clausius-Clapeyron equation, the linear relationship between natural logarithm of pressure and reciprocal of temperature is used to calculate the molar latent heats of evaporation of the two organic solvents. The molar latent heats of evaporation of DBP and DIBE are 75.1 kJ/mol and 67.7 kJ/mol, respectively.

  14. Study of the enthalpy-entropy mechanism from water sorption of orange seeds (C. sinensis cv. Brazilian for the use of agro-industrial residues as a possible source of vegetable oil production Estudo do mecanismo entálpico-entrópico de sorção da água de sementes de laranja (C. sinensis cv. Brasileira, para a utilização de resíduos agroindustriais como uma possível fonte de produção de óleo vegetal

    Directory of Open Access Journals (Sweden)

    Daniele Penteado Rosa

    2013-02-01

    Full Text Available Orange seeds are a promising agroindustry-waste which can be implemented in the extraction and production of vegetable oil. The relationship between moisture content and water activity provides useful information for the processing and storage of this waste item. The aim of this study was to determine the mechanism of water sorption enthalpy-entropy of orange seeds (C. sinensis cv. Brazilians according to the moisture content. Therefore, desorption isotherms were determined at five different temperature (30, 40, 50, 60, and 70 ºC under a wide range of moisture content (0.005-0.057 kg kg-1 d.b. and water activity (0.02-0.756. Theoretical and empirical models were used for modeling the desorption isotherms. An analytical solution of the Clausius-Clapeyron equation was proposed to compute the isosteric heat of sorption, the differential entropy, and Gibbs free energy using the Oswin model when the effect of temperature on the hygroscopic equilibrium was considered.As sementes de laranja são resíduos promissores da agroindústria com um alto potencial de aplicação na produção de óleo vegetal. A relação entre o conteúdo de umidade de equilíbrio e a atividade de água fornece informações úteis para seu processamento e armazenamento. O objetivo deste trabalho foi determinar o mecanismo entalpia-entropia de sorção da água de sementes de laranja (C. sinensis cv. Brasileiros em função do teor de umidade. Para isso, isotermas de dessorção das sementes de laranja foram determinados em cinco níveis de temperaturas (30, 40, 50, 60 e 70 ºC em um intervalo de umidade de equilíbrio (0.005-0.057 kg kg-1 d.b. e atividade de água de 0,02-0,756. Modelos teóricos e empíricos foram usados para a modelagem das isotermas de dessorção. A solução analítica da equação de Clausius-Clapeyron foi proposta para calcular o calor isostérico de sorção, a entropia diferencial e a energia livre de Gibbs através do modelo de Oswin quando o

  15. Melt Flow and Heat Transfer in Laser Drilling

    CERN Document Server

    Yang, Youqing; Zhang, Yuwen

    2016-01-01

    During the laser drilling process the recoil pressure drives melt flow and affects the heat transfer and material removal rate. To get a more realistic picture of the melt flow, a series of differential equations are formulated here that govern the process from pre-heating to melting and evaporation. In particular, the Navier-Stokes equation governing the melt flow is solved with the use of the boundary layer theory and integral methods. Heat conduction in solid is investigated by using the classical method with the corrections that reflect the change in boundary condition from the constant heat flux to Stefan condition. The dependence of saturation temperature on the vapor pressure is taken into account by using the Clausius-Clapeyron equation. Both constantly rising radial velocity profiles and rising-fall velocity profiles are considered. The proposed approach is compared with existing ones. In spite of the assumed varying velocity profiles, the proposed model predicts that the drilling hole profiles are v...

  16. Diffuse-interface modeling of liquid-vapor coexistence in equilibrium drops using smoothed particle hydrodynamics.

    Science.gov (United States)

    Sigalotti, Leonardo Di G; Troconis, Jorge; Sira, Eloy; Peña-Polo, Franklin; Klapp, Jaime

    2014-07-01

    We study numerically liquid-vapor phase separation in two-dimensional, nonisothermal, van der Waals (vdW) liquid drops using the method of smoothed particle hydrodynamics (SPH). In contrast to previous SPH simulations of drop formation, our approach is fully adaptive and follows the diffuse-interface model for a single-component fluid, where a reversible, capillary (Korteweg) force is added to the equations of motion to model the rapid but smooth transition of physical quantities through the interface separating the bulk phases. Surface tension arises naturally from the cohesive part of the vdW equation of state and the capillary forces. The drop models all start from a square-shaped liquid and spinodal decomposition is investigated for a range of initial densities and temperatures. The simulations predict the formation of stable, subcritical liquid drops with a vapor atmosphere, with the densities and temperatures of coexisting liquid and vapor in the vdW phase diagram closely matching the binodal curve. We find that the values of surface tension, as determined from the Young-Laplace equation, are in good agreement with the results of independent numerical simulations and experimental data. The models also predict the increase of the vapor pressure with temperature and the fitting to the numerical data reproduces very well the Clausius-Clapeyron relation, thus allowing for the calculation of the vaporization pressure for this vdW fluid.

  17. Soil temperature effect in calculating attenuation and retardation factors.

    Science.gov (United States)

    Paraiba, Lourival Costa; Spadotto, Claudio Aparecido

    2002-09-01

    The effect of annual variation of daily average soil temperature, at different depths, in calculating pesticides ranking indexes retardation factor and attenuation factor is presented. The retardation factor and attenuation factor are two site-specific pesticide numbers, frequently used as screening indicator indexes for pesticide groundwater contamination potential. Generally, in the calculation of these two factors are not included the soil temperature effect on the parameters involved in its calculation. It is well known that the soil temperature affects the pesticide degradation rate, water-air partition coefficient and water-soil partition coefficient. These three parameters are components of the retardation factor and attenuation factor and contribute to determine the pesticide behavior in the environment. The Arrhenius equation, van't Hoff equation and Clausius-Clapeyron equation are used in this work for estimating the soil temperature effect on the pesticide degradation rate, water-air partition coefficient and soil-water partition coefficient, respectively. These dependence relationships, between results of calculating attenuation and retardation factors and the soil temperature at different depths, can aid to understand the potential pesticide groundwater contamination on different weather conditions. Numerical results will be presented with pesticides atrazine and lindane in a soil profile with 20 degrees C constant temperature, minimum and maximum surface temperatures varying and spreading in the soil profile between -5 and 30 degrees C and between 15 and 45 degrees C. PMID:12222785

  18. Martensitic phase transformation in shape-memory alloys

    International Nuclear Information System (INIS)

    Isothermal studies are described of the shape-recovery phenomenon, stress-strain behavior, electrical resistivity and thermo-electric power associated with the martensite-parent phase reaction in the Ni-Ti shape-memory alloys. The energy-balance equation that links the reaction kinetics with the strain energy change during the cooling-deforming and heating cycle is analyzed. The strain range in which the Clausius-Clapeyron equation satisfactorily describes this reaction is determined. A large change in the Young's modulus of the specimen is found to be associated with the M → P reaction. A hysteresis loop in the resistivity-temperature plot is found and related to the anomaly in the athermal resistivity changes during cyclic M → P → M transformation. An explanation for the resistivity anomaly is offered. The M structure is found to be electrically negative relative to the P structure. A thermal emf of greater than or equal to 0.12 mV is found at the M-P interface

  19. Structural and thermodynamic signatures of marine microlayer surfactant films

    Science.gov (United States)

    Pogorzelski, Stanislaw J.; Kogut, Anna D.

    2003-06-01

    Natural surface film experiments in inland waters and shallow offshore regions of the Baltic and Mediterranean Seas were carried out in the time period 1990-1999 under calm sea conditions using a novel device for sampling and force-area studies. The sampler-Langmuir trough-Wilhelmy filter paper plate system 'cuts out' an undisturbed film-covered sea area to perform π-A studies without any initial physico-chemical sample processing. The limiting specific area A lim (2.68-31.57 nm 2/molecule) and mean molecular mass M w (0.65-9.7 kDa) of microlayer surfactants were determined from the 2D virial equation of state applied to the isotherms. Enthalpy ΔH and entropy ΔS t of the 2D first-order phase transitions were evaluated using the Clausius-Clapeyron equation applied to the isotherms. Miscibility of film components and film structure evolution is expressed by the scaling exponent y adopting the 2D polymer film scaling theory. The stress-relaxation measurements revealed a two-step relaxation process at the interface with characteristic times τ 1=1.1-2.8 and τ 2=5.6-25.6 seconds suggesting the presence of diffusion-controlled and structural organisation relaxation phenomena. The obtained results suggest that natural films are a complex mixture of biomolecules covering a wide range of solubilities, surface activity and molecular masses with an apparent structural organisation exhibiting a spatial and temporal variability.

  20. Sorption isotherms, thermodynamic properties and glass transition temperature of mucilage extracted from chia seeds (Salvia hispanica L.).

    Science.gov (United States)

    Velázquez-Gutiérrez, Sandra Karina; Figueira, Ana Cristina; Rodríguez-Huezo, María Eva; Román-Guerrero, Angélica; Carrillo-Navas, Hector; Pérez-Alonso, César

    2015-05-01

    Freeze-dried chia mucilage adsorption isotherms were determined at 25, 35 and 40°C and fitted with the Guggenheim-Anderson-de Boer model. The integral thermodynamic properties (enthalpy and entropy) were estimated with the Clausius-Clapeyron equation. Pore radius of the mucilage, calculated with the Kelvin equation, varied from 0.87 to 6.44 nm in the temperature range studied. The point of maximum stability (minimum integral entropy) ranged between 7.56 and 7.63kg H2O per 100 kg of dry solids (d.s.) (water activity of 0.34-0.53). Enthalpy-entropy compensation for the mucilage showed two isokinetic temperatures: (i) one occurring at low moisture contents (0-7.56 kg H2O per 100 kg d.s.), controlled by changes in water entropy; and (ii) another happening in the moisture interval of 7.56-24 kg H2O per 100 kg d.s. and was enthalpy driven. The glass transition temperature Tg of the mucilage fluctuated between 42.93 and 57.93°C.

  1. Biosorption kinetics, thermodynamics and isosteric heat of sorption of Cu(II) onto Tamarindus indica seed powder.

    Science.gov (United States)

    Chowdhury, Shamik; Saha, Papita Das

    2011-12-01

    Biosorption of Cu(II) by Tamarindus indica seed powder (TSP) was investigated as a function of temperature in a batch system. The Cu(II) biosorption potential of TSP increased with increasing temperature. The rate of the biosorption process followed pseudo second-order kinetics while the sorption equilibrium data well fitted to the Langmuir and Freundlich isotherm models. The maximum monolayer Cu(II) biosorption capacity increased from 82.97 mg g(-1) at 303 K to 133.24 mg g(-1) at 333 K. Thermodynamic study showed spontaneous and endothermic nature of the sorption process. Isosteric heat of sorption, determined using the Clausius-Clapeyron equation increased with increase in surface loading showing its strong dependence on surface coverage. The biosorbent was characterized by scanning electron microscopy (SEM), surface area and porosity analyzer, X-ray diffraction (XRD) spectrum and Fourier transform infrared (FTIR) spectroscopy. The results of FTIR analysis of unloaded and Cu(II)-loaded TSP revealed that -NH(2), -OH, -C=O and C-O functional groups on the biosorbent surface were involved in the biosorption process. The present study suggests that TSP can be used as a potential, alternative, low-cost biosorbent for removal of Cu(II) ions from aqueous media.

  2. Study of the enthalpy-entropy mechanism from water sorption of orange seeds (C. sinensis cv. Brazilian for the use of agro-industrial residues as a possible source of vegetable oil production

    Directory of Open Access Journals (Sweden)

    Daniele Penteado Rosa

    2013-02-01

    Full Text Available Orange seeds are a promising agroindustry-waste which can be implemented in the extraction and production of vegetable oil. The relationship between moisture content and water activity provides useful information for the processing and storage of this waste item. The aim of this study was to determine the mechanism of water sorption enthalpy-entropy of orange seeds (C. sinensis cv. Brazilians according to the moisture content. Therefore, desorption isotherms were determined at five different temperature (30, 40, 50, 60, and 70 ºC under a wide range of moisture content (0.005-0.057 kg kg-1 d.b. and water activity (0.02-0.756. Theoretical and empirical models were used for modeling the desorption isotherms. An analytical solution of the Clausius-Clapeyron equation was proposed to compute the isosteric heat of sorption, the differential entropy, and Gibbs free energy using the Oswin model when the effect of temperature on the hygroscopic equilibrium was considered.

  3. Carbon dioxide captured by multi-walled carbon nanotube and activated charcoal: A comparative study

    Directory of Open Access Journals (Sweden)

    Khalili Soodabeh

    2013-01-01

    Full Text Available this study, the equilibrium adsorption of CO2 on activated charcoal (AC and multi-walled carbon nanotube (MWCNT were investigated. Experiments were performed at temperature range of 298-318 K and pressures up to 40 bars. The obtained results indicated that the equilibrium uptakes of CO2 by both adsorbents increased with increasing pressure and decreasing temperature. In spite of lower specific surface area, the maximum amount of CO2 uptake achieved by MWCNT at 298K and 40 bars were twice of CO2 capture by AC (15 mmol.g-1 compared to 7.93 mmol.g-1. The higher CO2 captured by MWCNT can be attributed to its higher pore volume and specific structure of MWCN T such as hollowness and light mass which had greater influence than specific surface area. The experimental data were analyzed by means of Freundlich and Langmuir adsorption isotherm models. Following a simple acidic treatment procedure increased marginally CO2 capture by MWCNT over entire range of pressure, while for AC this effect appeared at higher pressures. Small values of isosteric heat of adsorption were evaluated based on Clausius-Clapeyron equation showed the physical nature of adsorption mechanism. The high amount of CO2 capture by MWCNT renders it as a promising carrier for practical applications such as gas separation.

  4. Strong Glacial Cooling In The Middle Tropical Troposphere Due To Non-linear Effects

    Science.gov (United States)

    Lorenz, S. J.; Lohmann, G.

    Numerical experiments with an atmospheric general circulation model for glacial and interglacial climates have been performed. Our model experiments reveal that slightly cooler tropical sea surface temperatures (SST) relative to the ones previously recon- structed by the CLIMAP project (1981) are sufficient to exhibit a strong glacial cool- ing reconstructed by tropical snow lines. The increased cooling in our experiments can be attributed to two non-linear effects: Firstly, there is an increased environmental lapse rate in the free atmosphere. Slightly cooler glacial SSTs provide for less abso- lute moisture content and the Clausius-Clapeyron equation of moisture is accountable for an increased lapse rate. In our LGM simulation we find an additional two degrees cooling in the tropical middle troposphere. Secondly, the surface air temperature near tropical glaciers is further cooled by a longer duration of snow cover. Our model result provides a consistent view of the last glacial maximum climate with much colder tem- peratures than today in the tropical mountains in concordance with moderate lowering of tropical SSTs. We propose that these non-linearities in the climate system are also important when detecting global warming from tropical snow lines.

  5. Temperature and Humidity Control in Multi-Layered Garments

    Science.gov (United States)

    Lee, Duck Weon

    2011-12-01

    The purpose of this research is to measure a property of a multilayered fabric system by using heat energy and vapor flow in terms of thermodynamics. By observing change in the heat energy and vapor flow passing through the multilayered fabric system, this research is able to provide precise information about a property of individual fabric layer composing the multilayered fabric system. This new research idea originates from a concept that, when heat energy and vapor flow pass through the layer or membrane, the amount of the heat energy and vapor flow is changed in accordance with a function of the layer or membrane. In particular, the amount of the vapor flow is apparently changed according to the fabric or membranes' structure and material property in a given environmental condition. The research conducts an experiment by using 'the energy source,' which is newly and innovatively developed, measuring temperature and relative humidity in the multilayered system. Through experimental data, the research calculates the amount of heat energy flow in the microclimates and fabric by using Stefan Boltzmann equation, Newton's law of cooling, Fourier's law, and Clausius- Clapeyron Relation. The research explains what properties of the fabric layers influence the energy flow attributable to conduction in the multilayered system consisting individual layers. In addition, the research shows that it is possible to build an optimized multilayered system under a variety of environmental conditions.

  6. Thermodynamic properties of water sorption of jackfruit (Artocarpus heterophyllus Lam. as a function of moisture content

    Directory of Open Access Journals (Sweden)

    Ana Paula Prette

    2013-03-01

    Full Text Available The Jackfruit tree is one of the most significant trees in tropical home gardens and perhaps the most widespread and useful tree in the important genus Artocarpus. The fruit is susceptible to mechanical and biological damage in the mature state, and some people find the aroma of the fruit objectionable, particularly in confined spaces. The dehydration process could be an alternative for the exploitation of this product, and the relationship between moisture content and water activity provides useful information for its processing and storage. The aim of this study was to determine the thermodynamic properties of the water sorption of jackfruit (Artocarpus heterophyllus Lam. as a function of moisture content. Desorption isotherms of the different parts of the jackfruit (pulp, peduncle, mesocarp, peel, and seed were determined at four different temperatures (313.15, 323.15, 333.15, and 343.15 K in a water activity range of 0.02-0.753 using the static gravimetric method. Theoretical and empirical models were used to model the desorption isotherms. An analytical solution of the Clausius-Clapeyron equation was proposed to calculate the isosteric heat of sorption, the differential entropy, and Gibbs' free energy using the Guggenhein-Anderson-de Boer and Oswin models considering the effect of temperature on the hygroscopic equilibrium.

  7. Reversible Storage of Hydrogen and Natural Gas in Nanospace-Engineered Activated Carbons

    Science.gov (United States)

    Romanos, Jimmy; Beckner, Matt; Rash, Tyler; Yu, Ping; Suppes, Galen; Pfeifer, Peter

    2012-02-01

    An overview is given of the development of advanced nanoporous carbons as storage materials for natural gas (methane) and molecular hydrogen in on-board fuel tanks for next-generation clean automobiles. High specific surface areas, porosities, and sub-nm/supra-nm pore volumes are quantitatively selected by controlling the degree of carbon consumption and metallic potassium intercalation into the carbon lattice during the activation process. Tunable bimodal pore-size distributions of sub-nm and supra-nm pores are established by subcritical nitrogen adsorption. Optimal pore structures for gravimetric and volumetric gas storage, respectively, are presented. Methane and hydrogen adsorption isotherms up to 250 bar on monolithic and powdered activated carbons are reported and validated, using several gravimetric and volumetric instruments. Current best gravimetric and volumetric storage capacities are: 256 g CH4/kg carbon and 132 g CH4/liter carbon at 293 K and 35 bar; 26, 44, and 107 g H2/kg carbon at 303, 194, and 77 K respectively and 100 bar. Adsorbed film density, specific surface area, and binding energy are analyzed separately using the Clausius-Clapeyron equation, Langmuir model, and lattice gas models.

  8. Experimental thermochemical study of 3-acetyl-2-methyl-5-phenylthiophene

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Santos, Ana Filipa L.O.M. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2010-01-15

    The standard (p{sup 0}=0.1MPa) massic energy of combustion, in oxygen, of the crystalline 3-acetyl-2-methyl-5-phenylthiophene was measured, at T = 298.15 K, by rotating-bomb combustion calorimetry, from which the standard molar enthalpy of formation, in the condensed phase, was calculated as DELTA{sub f}H{sub m}{sup 0}(cr)=-(104.3+-3.1)kJ.mol{sup -1}. The corresponding standard molar enthalpy of sublimation, at T = 298.15 K, DELTA{sub cr}{sup g}H{sub m}{sup 0}=(108.9+-0.4)kJ.mol{sup -1}, was derived by the Clausius-Clapeyron equation, from the temperature dependence of the vapour pressures of this compound, measured by the Knudsen effusion mass-loss technique. From the results presented above, the standard molar enthalpy of formation, in the gaseous phase, at T = 298.15 K, was derived, DELTA{sub f}H{sub m}{sup 0}(g)=(4.6+-3.1)kJ.mol{sup -1}. This value, in conjunction with the literature values of the experimental enthalpies of formation of thiophene, 2-methylthiophene, and 3-acetylthiophene, was used to predict the enthalpic increment due to the introduction of a phenyl group in the position 2- of the thiophene ring. The calculated increment was compared with the corresponding ones in benzene and pyridine derivatives.

  9. Experimental thermochemical study of two chlorodinitroaniline isomers

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687 P-4169-007 (Portugal); Ribeiro da Silva, Maria D.M.C.; Santos, Ana Filipa L.O.M.; Ferreira, Ana I.M.C. Lobo; Galvao, Tiago L.P. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687 P-4169-007 (Portugal)

    2010-04-15

    The standard (p{sup 0}=0.1MPa) molar enthalpies of formation of 2-chloro-4,6-dinitroaniline and 4-chloro-2,6-dinitroaniline, in the gaseous phase, at T = 298.15 K, were derived from the combination of the values of the standard molar enthalpies of formation, in the crystalline phase, and of the standard molar enthalpies of sublimation, at the same temperature. The standard molar enthalpies of formation, in the crystalline phase, were derived from the standard massic energies of combustion, in oxygen, measured by rotating-bomb combustion calorimetry. The standard molar enthalpies of sublimation were calculated, by the application of the Clausius-Clapeyron equation, to the vapour pressures at several temperatures, measured by Knudsen effusion technique. The values of the standard molar enthalpies of formation of 2-chloro-4,6-dinitroaniline and 4-chloro-2,6-dinitroaniline, in the gaseous phase, at T = 298.15 K, are discussed in terms of enthalpic increments, and the enthalpy of isomerization between the two compounds is compared with the same parameter for two isomers of chloronitroaniline, studied in previous works.

  10. Enthalpies of combustion, vapour pressures, and enthalpies of sublimation of the 1,5- and 1,8-diaminonaphthalenes

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 (Portugal); Lobo Ferreira, Ana I.M.C.; Santos, Ana Filipa L.O.M.; Ferreira, Cristiana M.A.; Barros, Delfina C.B.; Reis, Joana A.C.; Costa, Jose C.S.; Calvinho, Maria Miguel G.; Rocha, Sonia I.A.; Pinto, Sonia P.; Freire, Sonia S.L.; Almeida, Suzete M.; Guimaraes, Vanessa S.; Almeida, Vasco N.M. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 (Portugal)

    2010-03-15

    The standard (p{sup 0} = 0.1 MPa) molar enthalpies of formation, in the crystalline state, of 1,5-diaminonaphthalene and 1,8-diaminonaphthalene were derived from the standard molar energies of combustion, in oxygen, at T = 298.15 K, measured by static-bomb combustion calorimetry. The Knudsen mass-loss effusion technique was used to measure the dependence of the vapour pressure of the solid isomers of diaminonaphthalene with the temperature, from which the standard molar enthalpies of sublimation were derived using the Clausius-Clapeyron equation. Combining these two experimental values, the gas-phase standard molar enthalpies of formation, at T = 298.15 K, were derived and compared with those estimated using two different empirical methods of DELTA{sub f}H{sub m}{sup 0}(g) estimation: the Cox scheme and the Benson's Group Method. Moreover, the standard (p{sup 0} = 0.1 MPa) molar entropies and Gibbs energies of sublimation, at T = 298.15 K, were derived for the two diaminonaphthalene isomers.

  11. Thermochemical study of the 2,5-dibromonitrobenzene isomer: An approach of the energetic study for the other dibromonitrobenzene isomers

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 (Portugal)], E-mail: risilva@fc.up.pt; Lobo Ferreira, Ana I.M.C.; Santos, Ana Filipa L.O.M.; Rocha, Ines M. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 (Portugal)

    2009-11-15

    The standard (p{sup 0}=0.1MPa) molar enthalpy of formation, of the 2,5-dibromonitrobenzene, in the crystalline phase, at T = 298.15 K, was derived from the standard massic energy of combustion, in oxygen, at T = 298.15 K, measured by rotating bomb combustion calorimetry. The Knudsen mass-loss effusion technique was used to measure the vapour pressures of the crystal as a function of the temperature and applying the Clausius-Clapeyron equation, the standard molar enthalpy of sublimation of the compound, at T = 298.15 K, was calculated. The combination of the values of the standard molar enthalpy of formation, in the crystalline phase, and the standard molar enthalpy of sublimation of the dibromonitrobenzene isomer, allowed the calculation of the standard (p{sup 0}=0.1MPa) molar enthalpy of formation, in the gaseous phase, at T = 298.15 K. Additionally, this value was estimated by employing two different methodologies. One based on the conventional Cox Scheme and another one, much more accurate, based on the values of the standard molar enthalpies of formation of 2- and 3-bromonitrobenzene already determined experimentally. Once the best approach was found, it was applied in the estimation of the standard molar enthalpies of formation of the other five isomers.

  12. Experimental thermochemical study of 2,5- and 2,6-dichloro-4-nitroanilines

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)], E-mail: risilva@fc.up.pt; Ribeiro da Silva, Maria D.M.C.; Lobo Ferreira, Ana I.M.C.; Santos, Ana Filipa L.O.M.; Galvao, Tiago L.P. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2009-10-15

    The standard (p{sup o} = 0.1 MPa) molar enthalpies of formation of 2,5- and 2,6-dichloro-4-nitroanilines, in the gaseous phase, at T = 298.15 K, were derived from the combination of the values of the standard molar enthalpies of formation in the crystalline phase, at T = 298.15 K, and the standard molar enthalpies of sublimation, of each compound, at the same temperature. The standard molar enthalpies of formation, in the crystalline phase, at T = 298.15 K, were derived from the standard massic energies of combustion of the two isomers, in oxygen, at T = 298.15 K, measured by rotating-bomb combustion calorimetry. The standard molar enthalpies of sublimation were calculated, by application of the Clausius-Clapeyron equation, to the vapour pressures at several temperatures measured by Knudsen effusion technique. The values of the standard (p = 0.1 MPa) molar enthalpies of formation of 2,5- and 2,6-dichloro-4-nitroanilines, in the gaseous phase, at T = 298.15 K, were compared with those estimated by the Cox scheme.

  13. Experimental thermochemical study of 4,5-dichloro-2-nitroaniline

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Ribeiro da Silva, Maria D.M.C. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)], E-mail: risilva@fc.up.pt; Lobo Ferreira, Ana I.M.C.; Santos, Ana Filipa L.O.M.; Galvao, Tiago L.P. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2009-11-15

    The standard (p{sup 0}=0.1MPa) molar enthalpy of formation of 4,5-dichloro-2-nitroaniline, in the gaseous phase, at T = 298.15 K, was derived from the combination of the values of the standard molar enthalpy of formation, in the crystalline phase, at T = 298.15 K, and the standard molar enthalpy of sublimation, at the same temperature. The standard molar enthalpy of formation, in the crystalline phase, at T = 298.15 K, was derived as -(99.7 {+-} 1.6) kJ . mol{sup -1} from the standard massic energy of combustion, in oxygen, measured by rotating-bomb combustion calorimetry. The standard molar enthalpy of sublimation was calculated, (109.4 {+-} 0.9) kJ . mol{sup -1} by the application of the Clausius-Clapeyron equation, to the vapour pressures measured at several temperatures by Knudsen effusion technique. The standard molar enthalpies of formation, in the gaseous phase, of the six dichloro-2-nitroaniline isomers and of the four dichloro-4-nitroaniline isomers were estimated by the Cox Scheme and by the Domalski and Hearing group additivity method and compared with the available experimental values. For the Domalski and Hearing group additivity method four new correction terms were derived.

  14. Non-isothermal two-phase flow in low-permeable porous media

    Science.gov (United States)

    Kolditz, O.; De Jonge, J.

    In this paper, we consider non-isothermal two-phase flow of two components (air and water) in gaseous and liquid phases in extremely low-permeable porous media through the use of the finite element method (FEM). Interphase mass transfer of the components between any of the phases is evaluated by assuming local thermodynamic equilibrium between the phases. Heat transfer occurs by conduction and multiphase advection. General equations of state for phase changes (Clausius-Clapeyron and Henry law) as well as multiphase properties for the low-permeable bentonites are implemented in the code. Additionally we consider the impact of swelling/shrinking processes on porosity and permeability changes. The numerical model is implemented in the context of the simulator RockFlow/RockMech (RF/RM), which is based on object-oriented programming techniques. The finite element formulations are written in terms of dimensionless quantities. This has proved to be advantageous for preconditioning composite system matrices of coupled multi-field problems. Three application examples are presented. The first one examines differences between the Richards' approximation and the multicomponent/multiphase approach, and between two numerical coupling schemes. The second example serves as partial verification against experimental results and to demonstrate coherence between different element types. The last example shows simultaneous desaturation and resaturation in one system.

  15. Trends of surface humidity and temperature during 1951-2012 in Beijing, China

    Science.gov (United States)

    Chu, Q.; Xu, Z.; Peng, D.; Yang, X.; Yang, G.

    2015-05-01

    In this paper, two datasets, a long time series (1951-2012) of daily surface observations at one meteorological station and a shorter time series (1979-2012) of three-hourly data with 0.1°×0.1° spatial resolution were analysed by using non-parametric methods to identify annual and seasonal variations in surface humidity and temperature. The results reveal that: (1) saturation water vapour pressure increased exponentially with temperature. Actual daily values at Beijing Meteorological Station are very close to the theoretical values estimated by using the simplified Clausius-Clapeyron equation, but with seasonal variations. (2) For both long- and short-term data, clear increasing tendencies of annual saturation specific humidity and temperature are found. Decreasing and drying trends were detected for winter. (3) The annual relative humidity showed a decreasing trend except for some suburban areas, somehow related to the lower temperature and increased specific humidity in those areas. (4) Regional changes in topography and elevation likely influenced trends in surface humidity, while local land use showed little effect on it.

  16. Correlation vs. Causation: The Effects of Ultrasonic Melt Treatment on Cast Metal Grain Size

    Directory of Open Access Journals (Sweden)

    J. B. Ferguson

    2014-10-01

    Full Text Available Interest in ultrasonic treatment of liquid metal has waxed and waned for nearly 80 years. A review of several experiments representative of ultrasonic cavitation treatment of Al and Mg alloys shows that the theoretical mechanisms thought to be responsible for grain refinement are (1 cavitation-induced increase in melting temperature predicted by the Clausius-Clapeyron equation and (2 cavitation-induced wetting of otherwise unwetted insoluble particles. Neither of these theoretical mechanisms can be directly confirmed by experiment, and though they remain speculative, the available literature generally assumes that one or the other or both mechanisms are active. However, grain size is known to depend on temperature of the liquid, temperature of the mold, and cooling rate of the entire system. From the reviewed experiments, it is difficult to isolate temperature and cooling rate effects on grain size from the theoretical effects. Ultrasonic treatments of Al-A356 were carried out to isolate such effects, and though it was found that ultrasound produced significant grain refinement, the treatments also significantly chilled the liquid and thereby reduced the pouring temperature. The grain sizes attained closely correlated with pouring temperature suggesting that ultrasonic grain refinement is predominantly a result of heat removal by the horn and ultrasonic stirring.

  17. Difference equations by differential equation methods

    CERN Document Server

    Hydon, Peter E

    2014-01-01

    Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.

  18. Random diophantine equations, I

    OpenAIRE

    Brüdern, Jörg; Dietmann, Rainer

    2012-01-01

    We consider additive diophantine equations of degree $k$ in $s$ variables and establish that whenever $s\\ge 3k+2$ then almost all such equations satisfy the Hasse principle. The equations that are soluble form a set of positive density, and among the soluble ones almost all equations admit a small solution. Our bound for the smallest solution is nearly best possible.

  19. Kinetic energy equations for the average-passage equation system

    Science.gov (United States)

    Johnson, Richard W.; Adamczyk, John J.

    1989-01-01

    Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.

  20. Strong increase in convective precipitation in response to higher temperatures

    DEFF Research Database (Denmark)

    Berg, P.; Moseley, C.; Härter, Jan Olaf Mirko

    2013-01-01

    at higher temperature, faster than the rate of increase in the atmosphere's water-holding capacity, termed the Clausius-Clapeyron rate. Invigoration of convective precipitation (such as thunderstorms) has been favoured over a rise in stratiform precipitation (such as large-scale frontal precipitation...

  1. The Modified Magnetohydrodynamical Equations

    Institute of Scientific and Technical Information of China (English)

    EvangelosChaliasos

    2003-01-01

    After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similar fashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is done by replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vector potential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vector analysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHD equations.

  2. On the Raychaudhuri equation

    Indian Academy of Sciences (India)

    George F R Ellis

    2007-07-01

    The Raychaudhuri equation is central to the understanding of gravitational attraction in astrophysics and cosmology, and in particular underlies the famous singularity theorems of general relativity theory. This paper reviews the derivation of the equation, and its significance in cosmology.

  3. Ordinary differential equations

    CERN Document Server

    Greenberg, Michael D

    2014-01-01

    Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps

  4. Beginning partial differential equations

    CERN Document Server

    O'Neil, Peter V

    2014-01-01

    A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible,combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is or

  5. Reducible functional differential equations

    Directory of Open Access Journals (Sweden)

    S. M. Shah

    1985-01-01

    Full Text Available This is the first part of a survey on analytic solutions of functional differential equations (FDE. Some classes of FDE that can be reduced to ordinary differential equations are considered since they often provide an insight into the structure of analytic solutions to equations with more general argument deviations. Reducible FDE also find important applications in the study of stability of differential-difference equations and arise in a number of biological models.

  6. New unified evolution equation

    OpenAIRE

    Lim, Jyh-Liong; Li, Hsiang-nan

    1998-01-01

    We propose a new unified evolution equation for parton distribution functions appropriate for both large and small Bjorken variables $x$, which is an improved version of the Ciafaloni-Catani-Fiorani-Marchesini equation. In this new equation the cancellation of soft divergences between virtual and real gluon emissions is explicit without introducing infrared cutoffs, next-to-leading contributions to the Sudakov resummation can be included systematically. It is shown that the new equation reduc...

  7. Diophantine equations and identities

    Directory of Open Access Journals (Sweden)

    Malvina Baica

    1985-01-01

    Full Text Available The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are    i  x2−my2=±1 ii  x3+my3+m2z3−3mxyz=1iii  Some fifth degree diopantine equations

  8. The Modified Magnetohydrodynamical Equations

    Institute of Scientific and Technical Information of China (English)

    Evangelos Chaliasos

    2003-01-01

    After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similarfashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is doneby replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vectorpotential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vectoranalysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHDequations.

  9. Fractional Differential Equations

    Directory of Open Access Journals (Sweden)

    Jianping Zhao

    2012-01-01

    Full Text Available An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.

  10. Singular stochastic differential equations

    CERN Document Server

    Cherny, Alexander S

    2005-01-01

    The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) arise often in theory and in applications. However, known conditions for the existence and uniqueness of a solution typically fail for such equations. The book concentrates on the study of the existence, the uniqueness, and, what is most important, on the qualitative behaviour of solutions of singular stochastic differential equations. This is done by providing a qualitative classification of isolated singular points, into 48 possible types.

  11. Lanczos's equation to replace Dirac's equation ?

    CERN Document Server

    Gsponer, A; Gsponer, Andre; Hurni, Jean-Pierre

    1994-01-01

    Lanczos's quaternionic interpretation of Dirac's equation provides a unified description for all elementary particles of spin 0, 1/2, 1, and 3/2. The Lagrangian formulation given by Einstein and Mayer in 1933 predicts two main classes of solutions. (1) Point like partons which come in two families, quarks and leptons. The correct fractional or integral electric and baryonic charges, and zero mass for the neutrino and the u-quark, are set by eigenvalue equations. The electro-weak interaction of the partons is the same as with the Standard model, with the same two free parameters: e and sin^2 theta. There is no need for a Higgs symmetry breaking mechanism. (2) Extended hadrons for which there is no simple eigenvalue equation for the mass. The strong interaction is essentially non-local. The pion mass and pion-nucleon coupling constant determine to first order the nucleon size, mass and anomalous magnetic moment.

  12. On separable Pauli equations

    International Nuclear Information System (INIS)

    We classify (1+3)-dimensional Pauli equations for a spin-(1/2) particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the 11 classes of vector-potentials of the electro-magnetic field A(t,x(vector sign))=(A0(t,x(vector sign)), A(vector sign)(t,x(vector sign))) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x(vector sign)) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field

  13. Differential equations for dummies

    CERN Document Server

    Holzner, Steven

    2008-01-01

    The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.

  14. Elliptic partial differential equations

    CERN Document Server

    Volpert, Vitaly

    If we had to formulate in one sentence what this book is about it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Mathematical anaylsis of reaction-diffusion equations will be based on the theory of Fredholm operators presented in the first volume. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equ...

  15. Nonlinear Dirac Equations

    Directory of Open Access Journals (Sweden)

    Wei Khim Ng

    2009-02-01

    Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.

  16. Fractional Chemotaxis Diffusion Equations

    CERN Document Server

    Langlands, T A M

    2010-01-01

    We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.

  17. Drift-Diffusion Equation

    Directory of Open Access Journals (Sweden)

    K. Banoo

    1998-01-01

    equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.

  18. Solving Ordinary Differential Equations

    Science.gov (United States)

    Krogh, F. T.

    1987-01-01

    Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

  19. Fundamental Equation of Economics

    OpenAIRE

    Wayne, James J.

    2013-01-01

    Recent experience of the great recession of 2008 has renewed one of the oldest debates in economics: whether economics could ever become a scientific discipline like physics. This paper proves that economics is truly a branch of physics by establishing for the first time a fundamental equation of economics (FEOE), which is similar to many fundamental equations governing other subfields of physics, for example, Maxwell’s Equations for electromagnetism. From recently established physics laws of...

  20. Differential equations I essentials

    CERN Document Server

    REA, Editors of

    2012-01-01

    REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.

  1. On separable Pauli equations

    OpenAIRE

    Zhalij, Alexander

    2002-01-01

    We classify (1+3)-dimensional Pauli equations for a spin-1/2 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the eleven classes of vector-potentials of the electro-magnetic field A(t,x) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is...

  2. A new evolution equation

    International Nuclear Information System (INIS)

    A new evolution equation is proposed for the gluon density relevant (GLR) for the region of small xB. It generalizes the GLR equation and allows deeper penetration in dense parton systems than the GLR equation does. This generalization consists of taking shadowing effects more comprehensively into account by including multi gluon correlations, and allowing for an arbitrary initial gluon distribution in a hadron. We solve the new equation for fixed αs. It is found that the effects of multi gluon correlations on the deep-inelastic structure function are small. (author) 15 refs, 5 figs, 2 tabs

  3. Analysis on flow characteristic of nuclear heating reactor

    International Nuclear Information System (INIS)

    The experiment was carried out on the test loop HRTL-5, which simulates the geometry and system design of a 5 MW Nuclear heating reactor. The analysis was based on a one-dimensional two-phase flow drift model with conservation equations for mass, steam mass, energy and momentum. Clausius-Clapeyron equation was used for the calculation of flashing front in the riser. A set of ordinary equation, which describes the behavior of two-phase flow in the natural circulation system, was derived through integration of the above conservation equations in subcooled boiling region, bulk boiling region in the heated section and in the riser. The method of time-domain was used for the calculation. Both static and dynamic results are presented. System pressure, inlet subcooling and heat flux are varied as input parameters. The results show that, firstly, subcooled boiling in the heated section and void flashing in the riser have significant influence on the distribution of the void fraction, mass flow rate and stability of the system, especially at lower pressure, secondly, in a wide range of two-phase flow conditions, only subcooled boiling occurs in the heated section. For the designed two-phase regime operation of the 5 MW nuclear heating reactor, the temperature at the core exit has not reaches its saturation value. Thirdly, the mechanism of two-phase flow oscillation, namely, 'zero-pressure-drop', is described. In the wide range of inlet subcooling (0 K<ΔT<28 K) there exists three regions for system flow condition, namely, (1) stable two-phase flow, (2) bulk and subcooled boiling unstable flow, (3) subcooled boiling and single phase stable flow. The response of mass flow rate, after a small disturbance in the heat flux, is showed in the above inlet subcooling range, and based on it the instability map of the system is given through experiment and calculation. (3 refs., 9 figs.)

  4. Thermal hydraulic modeling of a natural circulation loop

    International Nuclear Information System (INIS)

    The experiment was carried out on the test loop HRTL-5, which simulates the geometry and system design of a 5 MW nuclear heating reactor. The analysis was based on a one-dimensional two-phase flow drift model with conservation equations for mass, steam, energy and momentum. Clausius-Clapeyron equation was used for the calculation of flashing front in the riser. A set of ordinary equations, which describes the behavior of two-phase flow in the natural circulation system, was derived through integration of the above conservation equations for the subcooled boiling region, bulk boiling region in the heated section and for the riser. The method of time-domain was used for the calculation. Both static and dynamic results are presented. System pressure, inlet subcooling and heat flux are varied as input parameters. The results show that subcooled boiling in the heated section and void flashing in the riser have significant influence on the distribution of the void fraction, mass flow rate and flow instability of the system, especially at low pressure. The response of mass flow rate, after a small disturbance in the heat flux is shown, and based on it the instability map of the system is given through experiment and calculation. There exists three regions in the instability map of the investigated natural circulation system, namely, the stable two-phase flow region, the unstable bulk and subcooled boiling flow region and the stable subcooled boiling and single phase flow region. The mechanism of two-phase flow oscillation is interpreted. (orig.)

  5. Gauge invariant flow equation

    CERN Document Server

    Wetterich, C

    2016-01-01

    We propose a gauge invariant flow equation for Yang-Mills theories and quantum gravity that only involves one macroscopic gauge field or metric. It is based on a projection on physical and gauge fluctuations, corresponding to a particular gauge fixing. The freedom in the precise choice of the macroscopic field can be exploited in order to keep the flow equation simple.

  6. On the Diophantine equation

    Science.gov (United States)

    Zahari, N. M.; Sapar, S. H.; Mohd Atan, K. A.

    2013-04-01

    This paper discusses an integral solution (a, b, c) of the Diophantine equations x3n+y3n = 2z2n for n ≥ 2 and it is found that the integral solution of these equation are of the form a = b = t2, c = t3 for any integers t.

  7. Some classical Diophantine equations

    Directory of Open Access Journals (Sweden)

    Nikita Bokarev

    2014-09-01

    Full Text Available An attempt to find common solutions complete some Diophantine equations of the second degree with three variables, traced some patterns, suggest a common approach, which being elementary, however, lead to a solution of such equations. Using arithmetic functions allowed to write down the solutions in a single formula with no restrictions on the parameters used.

  8. Braneworld flow equations

    OpenAIRE

    Ramirez, Erandy; Liddle, Andrew

    2004-01-01

    We generalize the flow equations approach to inflationary model building to the Randall–Sundrum Type II braneworld scenario. As the flow equations are quite insensitive to the expansion dynamics, we find results similar to, though not identical to, those found in the standard cosmology.

  9. The Wouthuysen equation

    NARCIS (Netherlands)

    Hazewinkel, M.

    1995-01-01

    Dedication: I dedicate this paper to Prof. P.C. Baayen, at the occasion of his retirement on 20 December 1994. The beautiful equation which forms the subject matter of this paper was invented by Wouthuysen after he retired. The four complex variable Wouthuysen equation arises from an original space-

  10. Linear Equations: Equivalence = Success

    Science.gov (United States)

    Baratta, Wendy

    2011-01-01

    The ability to solve linear equations sets students up for success in many areas of mathematics and other disciplines requiring formula manipulations. There are many reasons why solving linear equations is a challenging skill for students to master. One major barrier for students is the inability to interpret the equals sign as anything other than…

  11. Navier-Stokes equation

    Directory of Open Access Journals (Sweden)

    Hannelore Breckner

    2000-01-01

    Full Text Available We consider a stochastic equation of Navier-Stokes type containing a noise part given by a stochastic integral with respect to a Wiener process. The purpose of this paper is to approximate the solution of this nonlinear equation by the Galerkin method. We prove the convergence in mean square.

  12. The relativistic Pauli equation

    CERN Document Server

    Delphenich, David

    2012-01-01

    After discussing the way that C2 and the algebra of complex 2x2 matrices can be used for the representation of both non-relativistic rotations and Lorentz transformations, we show that Dirac bispinors can be more advantageously represented as 2x2 complex matrices. One can then give the Dirac equation a form for such matrix-valued wave functions that no longer necessitates the introduction of gamma matrices or a choice for their representation. The minimally-coupled Dirac equation for a charged spinning particle in an external electromagnetic field then implies a second order equation in the matrix-valued wave functions that is of Klein-Gordon type and represents the relativistic analogue of the Pauli equation. We conclude by presenting the Lagrangian form for the relativistic Pauli equation.

  13. Applied partial differential equations

    CERN Document Server

    Logan, J David

    2004-01-01

    This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...

  14. Moisture sorption isotherms and thermodynamic properties of Oak wood (Quercus robur and Quercus canariensis): optimization of the processing parameters

    Science.gov (United States)

    Bahar, Rim; Azzouz, Soufien; Remond, Romain; Ouertani, Sahbi; Elaieb, Mohamed Taher; El Cafci, Mohamed Afif

    2016-09-01

    The aim of this paper was to determine the moisture desorption isotherms and essentials thermodynamic properties of two Oak wood varieties. Desorption isotherms were measured using a static gravimetric method at 50, 60, 70 and 80 °C within the range of 5-90 % relative humidity. The equilibrium moisture content decreased with increasing temperature and decreased with decreasing relative humidity at a constant temperature. The `Thermodynamic' sorption equation was found to be the best for describing the experimental moisture sorption isotherms of woods within the range of temperature and water activity investigated. The Fiber saturation point, deduced from the `Thermodynamic' model parameters, depends on the temperature and varying from 22.6 to 54.4 (% kg water/kg dry matter). Isosteric heat of desorption and differential entropy were calculated by applying Clausius-Clapeyron equation to the desorption data fitted by the `Thermodynamic' model. The isosteric heat of desorption and the differential entropy decreased with increasing moisture content according to an exponential law equation and varying from 2.03 to 31.14 kJ/mol and from 73.98 to 4.34 J/(mol K), respectively. The linear relationship between differential enthalpy and entropy satisfied the enthalpy-entropy compensation theory. The sign of Gibbs free energy was found to be positive (+283 J/mol) and (+97 J/mol) for Quercus robur and Quercus canariensis, respectively. The isokinetic temperature was found to be greater than the harmonic temperature. Based on the enthalpy-entropy compensation theory, it could be concluded that the moisture desorption isotherm of Oak wood is a non-spontaneous and enthalpy-controlled process.

  15. Differential equations problem solver

    CERN Document Server

    Arterburn, David R

    2012-01-01

    REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and

  16. Ordinary differential equations

    CERN Document Server

    Miller, Richard K

    1982-01-01

    Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,

  17. Stochastic Gauss equations

    Science.gov (United States)

    Pierret, Frédéric

    2016-02-01

    We derived the equations of Celestial Mechanics governing the variation of the orbital elements under a stochastic perturbation, thereby generalizing the classical Gauss equations. Explicit formulas are given for the semimajor axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle, and the mean anomaly, which are expressed in term of the angular momentum vector H per unit of mass and the energy E per unit of mass. Together, these formulas are called the stochastic Gauss equations, and they are illustrated numerically on an example from satellite dynamics.

  18. Beginning partial differential equations

    CERN Document Server

    O'Neil, Peter V

    2011-01-01

    A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres

  19. Hyperbolic partial differential equations

    CERN Document Server

    Witten, Matthew

    1986-01-01

    Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M

  20. Uncertain differential equations

    CERN Document Server

    Yao, Kai

    2016-01-01

    This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.

  1. Modern introduction to differential equations

    CERN Document Server

    Ricardo, Henry J

    2009-01-01

    A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equat

  2. A Comparison of IRT Equating and Beta 4 Equating.

    Science.gov (United States)

    Kim, Dong-In; Brennan, Robert; Kolen, Michael

    Four equating methods were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional mean squared error (CMSE) difference, and the equipercentile equating property. The four methods were: (1) three parameter logistic (3PL) model true score equating; (2) 3PL observed score equating; (3) beta 4 true…

  3. Kinetic equations: computation

    CERN Document Server

    Pareschi, Lorenzo

    2013-01-01

    Kinetic equations bridge the gap between a microscopic description and a macroscopic description of the physical reality. Due to the high dimensionality the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity.

  4. Applied partial differential equations

    CERN Document Server

    Logan, J David

    2015-01-01

    This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs.  Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...

  5. Diophantine Equations and Computation

    Science.gov (United States)

    Davis, Martin

    Unless otherwise stated, we’ll work with the natural numbers: N = \\{0,1,2,3, dots\\}. Consider a Diophantine equation F(a1,a2,...,an,x1,x2,...,xm) = 0 with parameters a1,a2,...,an and unknowns x1,x2,...,xm For such a given equation, it is usual to ask: For which values of the parameters does the equation have a solution in the unknowns? In other words, find the set: \\{ mid exists x_1,ldots,x_m [F(a_1,ldots,x_1,ldots)=0] \\} Inverting this, we think of the equation F = 0 furnishing a definition of this set, and we distinguish three classes: a set is called Diophantine if it has such a definition in which F is a polynomial with integer coefficients. We write \\cal D for the class of Diophantine sets.

  6. Nonlinear differential equations

    Energy Technology Data Exchange (ETDEWEB)

    Dresner, L.

    1988-01-01

    This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

  7. Stochastic Gauss Equations

    CERN Document Server

    Frédéric, Pierret

    2014-01-01

    The equations of celestial mechanics that govern the variation of the orbital elements are completely derived for stochastic perturbation which generalized the classic perturbation equations which are used since Gauss, starting from Newton's equation and it's solution. The six most understandable orbital element, the semi-major axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle and the mean motion are express in term of the angular momentum vector $\\textbf{H}$ per unit of mass and the energy $E$ per unit of mass. We differentiate those expressions using It\\^o's theory of differential equations due to the stochastic nature of the perturbing force. The result is applied to the two-body problem perturbed by a stochastic dust cloud and also perturbed by a stochastic dynamical oblateness of the central body.

  8. Nonlinear differential equations

    International Nuclear Information System (INIS)

    This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics

  9. Exciton laser rate equations

    OpenAIRE

    Garkavenko A. S.

    2011-01-01

    The rate equations of the exciton laser in the system of interacting excitons have been obtained and the inverted population conditions and generation have been derived. The possibility of creating radically new gamma-ray laser has been shown.

  10. Complex Maxwell's equations

    Institute of Scientific and Technical Information of China (English)

    A.I.Arbab

    2013-01-01

    A unified complex model of Maxwell's equations is presented.The wave nature of the electromagnetic field vector is related to the temporal and spatial distributions and the circulation of charge and current densities.A new vacuum solution is obtained,and a new transformation under which Maxwell's equations are invariant is proposed.This transformation extends ordinary gauge transformation to include charge-current as well as scalar-vector potential.An electric dipole moment is found to be related to the magnetic charges,and Dirac's quantization is found to determine an uncertainty relation expressing the indeterminacy of electric and magnetic charges.We generalize Maxwell's equations to include longitudinal waves.A formal analogy between this formulation and Dirac's equation is also discussed.

  11. On Dust Charging Equation

    OpenAIRE

    Tsintsadze, Nodar L.; Tsintsadze, Levan N.

    2008-01-01

    A general derivation of the charging equation of a dust grain is presented, and indicated where and when it can be used. A problem of linear fluctuations of charges on the surface of the dust grain is discussed.

  12. Relativistic Guiding Center Equations

    Energy Technology Data Exchange (ETDEWEB)

    White, R. B. [PPPL; Gobbin, M. [Euratom-ENEA Association

    2014-10-01

    In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.

  13. SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER

    Science.gov (United States)

    Collier, D.M.; Meeks, L.A.; Palmer, J.P.

    1960-05-10

    A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.

  14. Hedin Equations for Superconductors

    OpenAIRE

    Linscheid, A.; Essenberger, F.

    2015-01-01

    We generalize Hedin equations to a system of superconducting electrons coupled with a system of phonons. The electrons are described by an electronic Pauli Hamiltonian which includes the Coulomb interaction among electrons and an external vector and scalar potential. We derive the continuity equation in the presence of the superconducting condensate and point out how to cast vertex corrections in the form of a non-local effective interaction that can be used to describe both fluctuations of s...

  15. The effect of chemical treatment on adsorption of natural gas by multi-walled carbon nanotubes: Sorption equilibria and thermodynamic studies

    Directory of Open Access Journals (Sweden)

    Delavar M.

    2012-01-01

    Full Text Available In this study, adsorption of methane as the main constituent of natural gas was firstly studied on the pristine multi-walled carbon nanotubes (MWCNTs and then purification and chemical treatments of MWCNTs was performed to enhance the natural gas adsorption capacity. MWCNTs were chemically treated using different methods in this research. The results revealed that chemical treatment of the MWCNTs in presence of H2SO4/HNO3 acidic mixture in 3:1 volume ratio, enhanced considerably natural gas adsorption capacity (an optimal up to 45 mmol/g at temperature of 298.15 K and the pressure of 50 bar compared to the pristine MWCNTs (about 27 mmol/g at the same operating conditions. This effect can be attributed to the opening of the nanotubes caps with a major alteration in its structural properties due to chemical treatment. The experimental data of adsorption were almost equally well described by Langmuir, Freundlich and Sips equations to determine the model isotherms. The best fit was obtained by the Sips model isotherm with the r-squared value near to unity. Furthermore, using the experimental data obtained in different temperatures the isosteric heat of natural gas adsorption onto pristine MWCNTs was also calculated in the interested range of pressures and temperatures using the thermodynamic-based Clausius-Clapeyron equation from the Sips isotherm model. The results revealed an energetically heterogeneous surface of MWCNTs in natural gas adsorption. Also the natural gas adsorption process was kinetically studied through pseudo-second order and intra-particle diffusion models which indicated the intra-particular diffusion is rate limiting step in adsorption of methane on MWCNTs.

  16. Experimental study of water desorption isotherms and thin-layer convective drying kinetics of bay laurel leaves

    Science.gov (United States)

    Ghnimi, Thouraya; Hassini, Lamine; Bagane, Mohamed

    2016-02-01

    The aim of this work is to determine the desorption isotherms and the drying kinetics of bay laurel leaves (Laurus Nobilis L.). The desorption isotherms were performed at three temperature levels: 50, 60 and 70 °C and at water activity ranging from 0.057 to 0.88 using the statistic gravimetric method. Five sorption models were used to fit desorption experimental isotherm data. It was found that Kuhn model offers the best fitting of experimental moisture isotherms in the mentioned investigated ranges of temperature and water activity. The Net isosteric heat of water desorption was evaluated using The Clausius-Clapeyron equation and was then best correlated to equilibrium moisture content by the empirical Tsami's equation. Thin layer convective drying curves of bay laurel leaves were obtained for temperatures of 45, 50, 60 and 70 °C, relative humidity of 5, 15, 30 and 45 % and air velocities of 1, 1.5 and 2 m/s. A non linear regression procedure of Levenberg-Marquardt was used to fit drying curves with five semi empirical mathematical models available in the literature, The R2 and χ2 were used to evaluate the goodness of fit of models to data. Based on the experimental drying curves the drying characteristic curve (DCC) has been established and fitted with a third degree polynomial function. It was found that the Midilli Kucuk model was the best semi-empirical model describing thin layer drying kinetics of bay laurel leaves. The bay laurel leaves effective moisture diffusivity and activation energy were also identified.

  17. Functional Equations and Fourier Analysis

    OpenAIRE

    Yang, Dilian

    2010-01-01

    By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations -- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation, on compact groups.

  18. Integral equations and computation problems

    International Nuclear Information System (INIS)

    Volterra's Integral Equations and Fredholm's Integral Equations of the second kind are discussed. Computational problems are found in the derivations and the computations. The theorem of the solution of the Fredholm's Integral Equation is discussed in detail. (author)

  19. Scaling Equation for Invariant Measure

    Institute of Scientific and Technical Information of China (English)

    LIU Shi-Kuo; FU Zun-Tao; LIU Shi-Da; REN Kui

    2003-01-01

    An iterated function system (IFS) is constructed. It is shown that the invariant measure of IFS satisfies the same equation as scaling equation for wavelet transform (WT). Obviously, IFS and scaling equation of WT both have contraction mapping principle.

  20. Introduction to partial differential equations

    CERN Document Server

    Greenspan, Donald

    2000-01-01

    Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.

  1. Unified derivation of evolution equations

    OpenAIRE

    Li, Hsiang-nan

    1998-01-01

    We derive the evolution equations of parton distribution functions appropriate in different kinematic regions in a unified and simple way using the resummation technique. They include the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation for large momentum transfer $Q$, the Balitskii-Fadin-Kuraev-Lipatov equation for a small Bjorken variable $x$, and the Ciafaloni-Catani-Fiorani-Marchesini equation which embodies the above two equations. The relation among these equations is explored, and p...

  2. Quadratic Diophantine equations

    CERN Document Server

    Andreescu, Titu

    2015-01-01

    This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems, and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory. The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis.

  3. Boussinesq evolution equations

    DEFF Research Database (Denmark)

    Bredmose, Henrik; Schaffer, H.; Madsen, Per A.

    2004-01-01

    This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic...... Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over...... a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave...

  4. Stochastic porous media equations

    CERN Document Server

    Barbu, Viorel; Röckner, Michael

    2016-01-01

    Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.

  5. Generalization of Hopf Functional Equation

    Institute of Scientific and Technical Information of China (English)

    2002-01-01

    This paper generalizes the Hopf functional equation in order to apply it to a wider class of not necessarily incompressible fluid flows. We start by defining characteristic functionals of the velocity field, the density field and the temperature field of a compressible field. Using the continuity equation, the Navier-Stokes equations and the equation of energy we derive a functional equation governing the motion of an ideal gas flow and a van der Waals gas flow, and then give some general methods of deriving a functional equation governing the motion of any compressible fluid flow. These functional equations can be considered as the generalization of the Hopf functional equation.

  6. Symplectic Dirac Equation

    CERN Document Server

    Amorim, R G G; Silva, Edilberto O

    2015-01-01

    Symplectic unitary representations for the Poincar\\'{e} group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Klein-Gordon and Dirac equations are derived in phase space. As an application, we study the Dirac equation with electromagnetic interaction in phase space.

  7. Gas Dynamics Equations: Computation

    CERN Document Server

    Chen, Gui-Qiang G

    2012-01-01

    Shock waves, vorticity waves, and entropy waves are fundamental discontinuity waves in nature and arise in supersonic or transonic gas flow, or from a very sudden release (explosion) of chemical, nuclear, electrical, radiation, or mechanical energy in a limited space. Tracking these discontinuities and their interactions, especially when and where new waves arise and interact in the motion of gases, is one of the main motivations for numerical computation for the gas dynamics equations. In this paper, we discuss some historic and recent developments, as well as mathematical challenges, in designing and formulating efficient numerical methods and algorithms to compute weak entropy solutions for the Euler equations for gas dynamics.

  8. Generalized estimating equations

    CERN Document Server

    Hardin, James W

    2002-01-01

    Although powerful and flexible, the method of generalized linear models (GLM) is limited in its ability to accurately deal with longitudinal and clustered data. Developed specifically to accommodate these data types, the method of Generalized Estimating Equations (GEE) extends the GLM algorithm to accommodate the correlated data encountered in health research, social science, biology, and other related fields.Generalized Estimating Equations provides the first complete treatment of GEE methodology in all of its variations. After introducing the subject and reviewing GLM, the authors examine th

  9. The relativistic Pauli equation

    OpenAIRE

    Delphenich, David

    2012-01-01

    After discussing the way that C2 and the algebra of complex 2x2 matrices can be used for the representation of both non-relativistic rotations and Lorentz transformations, we show that Dirac bispinors can be more advantageously represented as 2x2 complex matrices. One can then give the Dirac equation a form for such matrix-valued wave functions that no longer necessitates the introduction of gamma matrices or a choice for their representation. The minimally-coupled Dirac equation for a charge...

  10. Systematic Equation Formulation

    DEFF Research Database (Denmark)

    Lindberg, Erik

    2007-01-01

    A tutorial giving a very simple introduction to the set-up of the equations used as a model for an electrical/electronic circuit. The aim is to find a method which is as simple and general as possible with respect to implementation in a computer program. The “Modified Nodal Approach”, MNA, and th......, and the “Controlled Source Approach”, CSA, for systematic equation formulation are investigated. It is suggested that the kernel of the P Spice program based on MNA is reprogrammed....

  11. ON A FUNCTIONAL EQUATION

    Institute of Scientific and Technical Information of China (English)

    Ding Yi

    2009-01-01

    In this article, the author derives a functional equation η(s)=[(π/4)s-1/2√2/πг(1-s)sin(πs/2)]η(1-s) of the analytic function η(s) which is defined by η(s)=1-s-3-s-5-s+7-s…for complex variable s with Re s>1, and is defined by analytic continuation for other values of s. The author proves (1) by Ramanujan identity (see [1], [3]). Her method provides a new derivation of the functional equation of Riemann zeta function by using Poisson summation formula.

  12. Solving Diophantine Equations

    OpenAIRE

    Cira, Octavian; Smarandache, Florentin

    2016-01-01

    In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation {\\eta}({\\pi}(x)) = {\\pi}({\\eta}(x)), where {\\eta} is the Smarandache function and {\\pi} is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and th...

  13. Theory of differential equations

    CERN Document Server

    Gel'fand, I M

    1967-01-01

    Generalized Functions, Volume 3: Theory of Differential Equations focuses on the application of generalized functions to problems of the theory of partial differential equations.This book discusses the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant coefficients and eigenfunction expansions for self-adjoint differential operators. The topics covered include the bounded operators in spaces of type W, Cauchy problem in a topological vector space, and theorem of the Phragmén-Lindelöf type. The correctness classes for the Cau

  14. Kepler Equation solver

    Science.gov (United States)

    Markley, F. Landis

    1995-01-01

    Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.

  15. Equations of mathematical physics

    CERN Document Server

    Tikhonov, A N

    2011-01-01

    Mathematical physics plays an important role in the study of many physical processes - hydrodynamics, elasticity, and electrodynamics, to name just a few. Because of the enormous range and variety of problems dealt with by mathematical physics, this thorough advanced-undergraduate or graduate-level text considers only those problems leading to partial differential equations. The authors - two well-known Russian mathematicians - have focused on typical physical processes and the principal types of equations deailing with them. Special attention is paid throughout to mathematical formulation, ri

  16. On difference Riccati equations and second order linear difference equations

    OpenAIRE

    Ishizaki, Katsuya

    2011-01-01

    In this paper, we treat difference Riccati equations and second order linear difference equations in the complex plane. We give surveys of basic properties of these equations which are analogues in the differential case. We are concerned with the growth and value distributions of transcendental meromorphic solutions of these equations. Some examples are given.

  17. Test equating methods and practices

    CERN Document Server

    Kolen, Michael J

    1995-01-01

    In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...

  18. Standard molar enthalpies of formation of 3'- and 4'-nitroacetophenones

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Amaral, Luisa M.P.F. [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2011-06-15

    Research highlights: The standard molar enthalpies of formation, in the condensed phase, of 3'- and 4'-nitroacetophenones have been determined by combustion calorimetry. The vapor pressures of the crystalline 3'- and 4'-nitroacetophenones were measured as function of temperature by the Knudsen effusion mass loss technique. The standard molar enthalpies, entropies and Gibbs functions of sublimation, at T = 298.15 K, were calculated for both compounds. - Abstract: The standard (p{sup o} = 0.1 MPa) molar enthalpies of formation, in the condensed phase, of 3'- and 4'-nitroacetophenones, presented in this work, were obtained from measurements of their combustion energies, at T = 298.15 K, using a static bomb calorimeter. The vapor pressures of the two crystalline 3'- and 4'-nitroacetophenones were measured as a function of temperature by the Knudsen effusion mass loss technique. The standard molar enthalpies of sublimation, at T = 298.15 K, were derived from the Clausius-Clapeyron equation. The standard molar enthalpies, entropies, and Gibbs functions of sublimation, at T = 298.15 K, were calculated for the two compounds. The experimental values obtained were used to calculate the standard molar enthalpies of formation of 3'- and 4'-nitroacetophenones, in the gaseous phase, as {Delta}{sub f}H{sub m}{sup 0}(g)=-(99.4{+-}1.6)kJ{center_dot}mol{sup -1} and {Delta}{sub f}H{sub m}{sup 0}(g)=-(99.1{+-}1.7)kJ{center_dot}mol{sup -1}, respectively, and these derived values are analyzed in terms of structural enthalpic increments.

  19. Thermochemical study of some dichloroacetophenone isomers

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Amaral, Luisa M.P.F. [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2011-03-15

    The standard (p{sup 0}=0.1MPa) molar enthalpies of formation in the condensed phase, {Delta}{sub f}H{sub m}{sup 0}(cr,l), for 2',4'-, 2',5'-, and 3',4'-dichloroacetophenones were derived from the standard molar energies of combustion, {Delta}{sub c}U{sub m}{sup 0} in oxygen, to yield CO{sub 2}(g) and HCl . 600H{sub 2}O(l), at T = 298.15 K, measured by rotating bomb combustion calorimetry. The standard molar enthalpies of vapourization or sublimation, {Delta}{sub cr,l}{sup g}H{sub m}{sup 0}, of these compounds, at T = 298.15 K were determined by Calvet microcalorimetry. For the 3',4'-dichoroacetophenone, the standard molar enthalpy of sublimation, at T = 298.15 K, was derived by the Clausius-Clapeyron equation, from the temperature dependence of the vapour pressures of this compound, measured by the Knudsen effusion technique. From the values of {Delta}{sub f}H{sub m}{sup 0}(cr,l) and {Delta}{sub cr,l}{sup g}H{sub m}{sup 0} the standard molar enthalpies of formation of the three isomers, in the gaseous phase, {Delta}{sub f}H{sub m}{sup 0}(g), at T = 298.15 K were derived and compared with the same parameters estimated by the Cox Scheme. (table)

  20. Thermodynamic study of 1,2,3-triphenylbenzene and 1,3,5-triphenylbenzene

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Santos, Luis M.N.B.F.; Lima, Luis M. Spencer S. [Centro de Investigacao em Quimica, Department of Chemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2010-01-15

    The energetic study of 1,2,3-triphenylbenzene (1,2,3-TPhB) and 1,3,5-triphenylbenzene (1,3,5-TPhB) isomers was carried out by making use of the mini-bomb combustion calorimetry and Knudsen mass-loss effusion techniques. The mini-bomb combustion calorimetry technique was used to derive the standard (p{sup o} = 0.1 MPa) molar enthalpies of formation in the crystalline state from the measured standard molar energies of combustion for both isomers. The Knudsen mass-loss effusion technique was used to measure the dependence with the temperature of the vapour pressure of crystalline 1,2,3-TPhB, which allowed the derivation of the standard molar enthalpy of sublimation, by application of the Clausius-Clapeyron equation. The sublimation study of 1,3,5-TPhB had been performed previously. From the combination of data obtained by both techniques, the standard molar enthalpies of formation in the gaseous state, for both isomers, at T = 298.15 K, were calculated. The results indicate a higher stability of the 1,3,5-TPhB isomer relative to 1,2,3-TPhB, similarly to the terphenyls. Nevertheless, the 1,2,3-TPhB isomer is not as energetically destabilized as one might expect, supporting the existence of a pi-pi displacive stacking interaction between both pairs of outer phenyl rings. The volatility difference between the two isomers is ruled by the enthalpy of sublimation. The volatility of the 1,2,3-TPhB is two orders of magnitude higher than the 1,3,5-TPhB isomer, at T = 298.15 K.

  1. Experimental study on the thermochemistry of 3-nitrobenzophenone, 4-nitrobenzophenone and 3,3'-dinitrobenzophenone

    Energy Technology Data Exchange (ETDEWEB)

    Ribeiro da Silva, Manuel A.V., E-mail: risilva@fc.up.p [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal); Amaral, Luisa M.P.F.; Ortiz, Rodrigo V. [Centro de Investigacao em Quimica, Department of Chemistry and Biochemistry, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, P-4169-007 Porto (Portugal)

    2011-04-15

    Research highlights: Standard molar enthalpies of formation of 3- and 4-nitrobenzophenones and of the 3,3'-dinitrobenzophenone, in the crystalline state, were determined, at the temperature T = 298.15 K. Vapour pressures of 3- and 4- nitrobenzophenones as function of temperature were measured by the Knudsen effusion technique. Enthalpies of sublimation of 3- and 4-nitrobenzophenones and of the 3,3'-dinitrobenzophenone were derived. The derived standard molar enthalpies of formation in the gaseous state are analyzed in terms of structural enthalpic increments. - Abstract: The standard (p{sup o} = 0.1 MPa) molar enthalpies of combustion, {Delta}{sub c}H{sub m}{sup 0}, for the 3- and 4-nitrobenzophenones and for the 3,3'-dinitrobenzophenone, in the crystalline state, were determined, at the temperature T = 298.15 K, using a static bomb combustion calorimeter. For these compounds, the standard molar enthalpies of sublimation, {Delta}{sub cr}{sup g}H{sub m}{sup 0}, at T = 298.15 K, were determined by Calvet microcalorimetry. For the 3- and 4-nitrobenzophenones the vapour pressures as function of temperature were measured by the Knudsen effusion technique and the standard molar enthalpies of sublimation, {Delta}{sub cr}{sup g}H{sub m}{sup 0}, at T = 298.15 K, were derived by the Clausius-Clapeyron equation. The results are as follows: (table) These values were used to derive the standard molar enthalpies of formation of the compounds in their condensed and gaseous phases, respectively. For 3- and 4-nitrobenzophenones, the standard (p{sup o} = 0.1 MPa) molar enthalpies, entropies and Gibbs functions of sublimation, at T = 298.15 K, were derived. The derived standard molar enthalpies of formation in the gaseous state are analysed in terms of structural enthalpic increments.

  2. A novel zinc(ii) metal-organic framework with a diamond-like structure: synthesis, study of thermal robustness and gas adsorption properties.

    Science.gov (United States)

    Almáši, Miroslav; Zeleňák, Vladimír; Zukal, Arnošt; Kuchár, Juraj; Čejka, Jiří

    2016-01-21

    A solvothermal reaction of Zn(ii) salt with methanetetrabenzoic acid (H4MTB) and 1,4,8,11-tetraazacyclotetradecane (cyclam, CYC) created a new microporous metal-organic framework {[Zn2(μ4-MTB)(κ(4)-CYC)2]·2DMF·7H2O}n (DMF = N,N'-dimethylformamide). Single crystal X-ray diffraction showed that the complex exhibits a four-fold interpenetrated diamond-like structure topology with 1D jar-like channels with sizes about 14.1 × 14.1 and 2.4 × 2.4 Å(2). The stability of the framework and activation conditions of the compound have been studied by high-energy powder X-ray diffraction during in situ heating, thermogravimetric analysis coupled with mass spectrometry and infrared spectroscopy performed at different temperatures. The gas adsorption behaviour of {[Zn2(μ4-MTB)(κ(4)-CYC)2]·2DMF·7H2O}n was studied by adsorption of Ar, N2, CO2 and H2. Nitrogen and argon adsorption showed that the activated sample exhibits Brunauer-Emmet-Teller (BET) specific surface areas of 644 m(2) g(-1) (N2) and 562 m(2) g(-1) (Ar). The complex adsorbs carbon dioxide with a maximum storage capacity of 10.5 wt% at 273 K and 101 kPa. The observed hydrogen uptake was 1.27 wt% at 77 K and 800 Torr, which is the highest value reported for the compounds containing a MTB(4-) linker. The adsorption heats of carbon dioxide and hydrogen, calculated according to the Clausius-Clapeyron equation, were in the range 22.8-22.4 kJ mol(-1) for CO2 and 8.9-3.2 kJ mol(-1) for H2, indicating weak interactions of the gases with the framework.

  3. Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation

    Institute of Scientific and Technical Information of China (English)

    LIU Yan-hong; ZHANG Hui-ming

    2007-01-01

    Combining the symplectic variations theory, the homogeneous control equation and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isoparametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which are often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.

  4. Dunkl Hyperbolic Equations

    Directory of Open Access Journals (Sweden)

    Hatem Mejjaoli

    2008-12-01

    Full Text Available We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.

  5. Modelling by Differential Equations

    Science.gov (United States)

    Chaachoua, Hamid; Saglam, Ayse

    2006-01-01

    This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…

  6. Do Differential Equations Swing?

    Science.gov (United States)

    Maruszewski, Richard F., Jr.

    2006-01-01

    One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…

  7. Structural Equation Model Trees

    Science.gov (United States)

    Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman

    2013-01-01

    In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…

  8. On the Breit Equation

    OpenAIRE

    Kasari, Hikoya; Yamaguchi, Yoshio

    2001-01-01

    Contrary to the conventional belief, it was shown that the Breit equation has the eigenvalues for bound states of two oppositely charged Dirac particles interacting through the (static) Coulomb potential. All eigenvalues reduced to those of the Sch\\"odinger case in the non-relativistic limit.

  9. The Equation of Causality

    OpenAIRE

    Chi, Do Minh

    1999-01-01

    We research the natural causality of the Universe. We find that the equation of causality provides very good results on physics. That is our first endeavour and success in describing a quantitative expression of the law of causality. Hence, our theoretical point suggests ideas to build other laws including the law of the Universe's evolution.

  10. Exciton laser rate equations

    Directory of Open Access Journals (Sweden)

    Garkavenko A. S.

    2011-08-01

    Full Text Available The rate equations of the exciton laser in the system of interacting excitons have been obtained and the inverted population conditions and generation have been derived. The possibility of creating radically new gamma-ray laser has been shown.

  11. Nonlocal electrical diffusion equation

    Science.gov (United States)

    Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.

    2016-07-01

    In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0solar panels, electrochemical phenomena and the description of anomalous complex processes.

  12. Calculus & ordinary differential equations

    CERN Document Server

    Pearson, David

    1995-01-01

    Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.

  13. Equational binary decision diagrams

    NARCIS (Netherlands)

    Groote, J.F.; Pol, J.C. van de

    2000-01-01

    We incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and tautology checkin

  14. Thermodynamic properties of water sorption of jackfruit (Artocarpus heterophyllus Lam. as a function of moisture content Propriedades termodinâmicas de sorção de água da jaca (Artocarpus heterophyllus Lam. em função do teor de umidade

    Directory of Open Access Journals (Sweden)

    Ana Paula Prette

    2013-03-01

    Full Text Available The Jackfruit tree is one of the most significant trees in tropical home gardens and perhaps the most widespread and useful tree in the important genus Artocarpus. The fruit is susceptible to mechanical and biological damage in the mature state, and some people find the aroma of the fruit objectionable, particularly in confined spaces. The dehydration process could be an alternative for the exploitation of this product, and the relationship between moisture content and water activity provides useful information for its processing and storage. The aim of this study was to determine the thermodynamic properties of the water sorption of jackfruit (Artocarpus heterophyllus Lam. as a function of moisture content. Desorption isotherms of the different parts of the jackfruit (pulp, peduncle, mesocarp, peel, and seed were determined at four different temperatures (313.15, 323.15, 333.15, and 343.15 K in a water activity range of 0.02-0.753 using the static gravimetric method. Theoretical and empirical models were used to model the desorption isotherms. An analytical solution of the Clausius-Clapeyron equation was proposed to calculate the isosteric heat of sorption, the differential entropy, and Gibbs' free energy using the Guggenhein-Anderson-de Boer and Oswin models considering the effect of temperature on the hygroscopic equilibrium.A jaqueira é uma das árvores mais significativas nos quintais tropicais e, talvez, a árvore mais importante e útil do gênero Artocarpus. O fruto é suscetível a danos mecânicos e biológicos no estado maduro, e seu aroma é desagradável para algumas pessoas, quando em espaços fechados. O processo de desidratação pode ser uma alternativa para a exploração deste produto, e a relação entre a umidade e a atividade de água fornece informações úteis para seu processamento e armazenamento. O objetivo do trabalho foi determinar as propriedades termodinâmicas de sorção da água em frutos de jaca (Artocarpus

  15. Lie Symmetries of Ishimori Equation

    Institute of Scientific and Technical Information of China (English)

    SONG Xu-Xia

    2013-01-01

    The Ishimori equation is one of the most important (2+1)-dimensional integrable models,which is an integrable generalization of (1+1)-dimensional classical continuous Heisenberg ferromagnetic spin equations.Based on importance of Lie symmetries in analysis of differential equations,in this paper,we derive Lie symmetries for the Ishimori equation by Hirota's direct method.

  16. Anticipated backward stochastic differential equations

    OpenAIRE

    Peng, Shige; Yang, Zhe

    2009-01-01

    In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.

  17. Elements of partial differential equations

    CERN Document Server

    Sneddon, Ian N

    2006-01-01

    Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st

  18. Differential Equations with Linear Algebra

    CERN Document Server

    Boelkins, Matthew R; Potter, Merle C

    2009-01-01

    Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t

  19. Stochastic differential equations and applications

    CERN Document Server

    Friedman, Avner

    2006-01-01

    This text develops the theory of systems of stochastic differential equations, and it presents applications in probability, partial differential equations, and stochastic control problems. Originally published in two volumes, it combines a book of basic theory and selected topics with a book of applications.The first part explores Markov processes and Brownian motion; the stochastic integral and stochastic differential equations; elliptic and parabolic partial differential equations and their relations to stochastic differential equations; the Cameron-Martin-Girsanov theorem; and asymptotic es

  20. Classical Diophantine equations

    CERN Document Server

    1993-01-01

    The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, ...

  1. Differential equations with Mathematica

    CERN Document Server

    Abell, Martha L

    2004-01-01

    The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica

  2. The open boundary equation

    Directory of Open Access Journals (Sweden)

    D. Diederen

    2015-06-01

    Full Text Available We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave in a frictionless, prismatic, tidal channel with a horizontal bed. Assessment of a large number of numerical simulations, where an open boundary condition is posed at a certain distance landward, suggests that it can also be considered accurate in the more natural case of converging estuaries with nonlinear friction and a bed slope. The equation follows from the open boundary condition and is therefore a part of the problem formulation for an infinite tidal channel. This finding provides a practical tool for evaluating tidal wave dynamics, by reconstructing the temporal variation of the velocity based on local observations of the water level, providing a fully local open boundary condition and allowing for local friction calibration.

  3. Information Equation of State

    Directory of Open Access Journals (Sweden)

    M. Paul Gough

    2008-07-01

    Full Text Available Landauer’s principle is applied to information in the universe. Once stars began forming there was a constant information energy density as the increasing proportion of matter at high stellar temperatures exactly compensated for the expanding universe. The information equation of state was close to the dark energy value, w = -1, for a wide range of redshifts, 10 > z > 0.8, over one half of cosmic time. A reasonable universe information bit content of only 1087 bits is sufficient for information energy to account for all dark energy. A time varying equation of state with a direct link between dark energy and matter, and linked to star formation in particular, is clearly relevant to the cosmic coincidence problem. In answering the ‘Why now?’ question we wonder ‘What next?’ as we expect the information equation of state to tend towards w = 0 in the future.c

  4. New application to Riccati equation

    Science.gov (United States)

    Taogetusang; Sirendaoerji; Li, Shu-Min

    2010-08-01

    To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov-Kuznetsov equation, Karamoto-Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.

  5. Telegrapher's equation for light derived from the transport equation

    OpenAIRE

    Hoenders, Bernhard J.; Graaff, R.

    2005-01-01

    Shortcomings of diffusion theory when applied to turbid media such as biological tissue makes the development of more accurate equations desirable. Several authors developed telegrapher's equations in the well known P-1 approximation. The method used in this paper is different: it is based on the asymptotic evaluation of the solutions of the equation of radiative transport with respect to place and time for all values of the albedo. Various coefficients for the telegrapher's equations were de...

  6. Converting fractional differential equations into partial differential equations

    OpenAIRE

    He Ji-Huan; Li Zheng-Biao

    2012-01-01

    A transform is suggested in this paper to convert fractional differential equations with the modified Riemann-Liouville derivative into partial differential equations, and it is concluded that the fractional order in fractional differential equations is equivalent to the fractal dimension.

  7. The compressible adjoint equations in geodynamics: equations and numerical assessment

    Science.gov (United States)

    Ghelichkhan, Siavash; Bunge, Hans-Peter

    2016-04-01

    The adjoint method is a powerful means to obtain gradient information in a mantle convection model relative to past flow structure. While the adjoint equations in geodynamics have been derived for the conservation equations of mantle flow in their incompressible form, the applicability of this approximation to Earth is limited, because density increases by almost a factor of two from the surface to the Core Mantle Boundary. Here we introduce the compressible adjoint equations for the conservation equations in the anelastic-liquid approximation. Our derivation applies an operator formulation in Hilbert spaces, to connect to recent work in seismology (Fichtner et al (2006)) and geodynamics (Horbach et al (2014)), where the approach was used to derive the adjoint equations for the wave equation and incompressible mantle flow. We present numerical tests of the newly derived equations based on twin experiments, focusing on three simulations. A first, termed Compressible, assumes the compressible forward and adjoint equations, and represents the consistent means of including compressibility effects. A second, termed Mixed, applies the compressible forward equation, but ignores compressibility effects in the adjoint equations, where the incompressible equations are used instead. A third simulation, termed Incompressible, neglects compressibility effects entirely in the forward and adjoint equations relative to the reference twin. The compressible and mixed formulations successfully restore earlier mantle flow structure, while the incompressible formulation yields noticeable artifacts. Our results suggest the use of a compressible formulation, when applying the adjoint method to seismically derived mantle heterogeneity structure.

  8. Ordinary differential equations

    CERN Document Server

    Cox, William

    1995-01-01

    Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further

  9. Differential Equations as Actions

    DEFF Research Database (Denmark)

    Ronkko, Mauno; Ravn, Anders P.

    1997-01-01

    We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....

  10. Dimensional Equations of Entropy

    CERN Document Server

    Sparavigna, Amelia Carolina

    2015-01-01

    Entropy is a quantity which is of great importance in physics and chemistry. The concept comes out of thermodynamics, proposed by Rudolf Clausius in his analysis of Carnot cycle and linked by Ludwig Boltzmann to the number of specific ways in which a physical system may be arranged. Any physics classroom, in its task of learning physics, has therefore to face this crucial concept. As we will show in this paper, the lectures can be enriched by discussing dimensional equations linked to the entropy of some physical systems.

  11. Partial differential equations

    CERN Document Server

    Sloan, D; Süli, E

    2001-01-01

    /homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight in

  12. Generalized estimating equations

    CERN Document Server

    Hardin, James W

    2013-01-01

    Generalized Estimating Equations, Second Edition updates the best-selling previous edition, which has been the standard text on the subject since it was published a decade ago. Combining theory and application, the text provides readers with a comprehensive discussion of GEE and related models. Numerous examples are employed throughout the text, along with the software code used to create, run, and evaluate the models being examined. Stata is used as the primary software for running and displaying modeling output; associated R code is also given to allow R users to replicat

  13. Matlab differential equations

    CERN Document Server

    Lopez, Cesar

    2014-01-01

    MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Differential Equations introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to giving an introduct

  14. Conservational PDF Equations of Turbulence

    Science.gov (United States)

    Shih, Tsan-Hsing; Liu, Nan-Suey

    2010-01-01

    Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application

  15. On Certain Dual Integral Equations

    Directory of Open Access Journals (Sweden)

    R. S. Pathak

    1974-01-01

    Full Text Available Dual integral equations involving H-Functions have been solved by using the theory of Mellin transforms. The proof is analogous to that of Busbridge on solutions of dual integral equations involving Bessel functions.

  16. Program Transformation by Solving Equations

    Institute of Scientific and Technical Information of China (English)

    朱鸿

    1991-01-01

    Based on the theory of orthogonal program expansion[8-10],the paper proposes a method to transform programs by solving program equations.By the method,transformation goals are expressed in program equations,and achieved by solving these equations.Although such equations are usually too complicated to be solved directly,the orthogonal expansion of programs makes it possible to reduce such equations into systems of equations only containing simple constructors of programs.Then,the solutions of such equations can be derived by a system of solving and simplifying rules,and algebraic laws of programs.The paper discusses the methods to simplify and solve equations and gives some examples.

  17. Kepler's Differential Equations

    CERN Document Server

    Holder, Martin

    2011-01-01

    Although the differential calculus was invented by Newton, Kepler established his famous laws 70 years earlier by using the same idea, namely to find a path in a nonuniform field of force by small steps. It is generally not known that Kepler demonstrated the elliptic orbit to be composed of intelligeable differential pieces, in modern language, to result from a differential equation. Kepler was first to attribute planetary orbits to a force from the sun, rather than giving them a predetermined geometric shape. Even though neither the force was known nor its relation to motion, he could determine the differential equations of motion from observation. This is one of the most important achievements in the history of physics. In contrast to Newton's Principia and Galilei's Dialogo Kepler's text is not easy to read, for various reasons. Therefore, in the present article, his results -- most of them well known -- are first presented in modern language. Then, in order to justify the claim, the full text of some rele...

  18. The Dirac equation

    International Nuclear Information System (INIS)

    This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics

  19. Growth Equation with Conservation Law

    OpenAIRE

    Lauritsen, Kent Baekgaard

    1995-01-01

    A growth equation with a generalized conservation law characterized by an integral kernel is introduced. The equation contains the Kardar-Parisi-Zhang, Sun-Guo-Grant, and Molecular-Beam Epitaxy growth equations as special cases and allows for a unified investigation of growth equations. From a dynamic renormalization-group analysis critical exponents and universality classes are determined for growth models with a conservation law.

  20. ``Riemann equations'' in bidifferential calculus

    Science.gov (United States)

    Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.

    2015-10-01

    We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.

  1. Hyperbolic Methods for Einstein's Equations

    OpenAIRE

    Reula Oscar

    1998-01-01

    I review evolutionary aspects of general relativity, in particular those related to the hyperbolic character of the field equations and to the applications or consequences that this property entails. I look at several approaches to obtaining symmetric hyperbolic systems of equations out of Einstein's equations by either removing some gauge freedoms from them, or by considering certain linear combinations of a subset of them.

  2. Successfully Transitioning to Linear Equations

    Science.gov (United States)

    Colton, Connie; Smith, Wendy M.

    2014-01-01

    The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

  3. An Extented Wave Action Equation

    Institute of Scientific and Technical Information of China (English)

    左其华

    2003-01-01

    Based on the Navier-Stokes equation, an average wave energy equation and a generalized wave action conservation equation are presented in this paper. The turbulence effects on water particle velocity ui and wave surface elavation ξ as well as energy dissipation are included. Some simplified forms are also given.

  4. Solving Nonlinear Coupled Differential Equations

    Science.gov (United States)

    Mitchell, L.; David, J.

    1986-01-01

    Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.

  5. The Schroedinger equation and spin

    International Nuclear Information System (INIS)

    Galilei invariance of the Schroedinger equation requires linearization of the operator by the introduction of anticommuting matrices as coefficients of the linear form. In an external field this leads directly to the Pauli equation, the non-relativistic limit of Dirac's equation. An overview of the complete argument that defines spin as a non-relativistic concept is presented. 9 refs

  6. Solution of Finite Element Equations

    DEFF Research Database (Denmark)

    Krenk, Steen

    An important step in solving any problem by the finite element method is the solution of the global equations. Numerical solution of linear equations is a subject covered in most courses in numerical analysis. However, the equations encountered in most finite element applications have some special...

  7. The anti-Einstein equations

    OpenAIRE

    Chaliasos, Evangelos

    2006-01-01

    As we know, from the Einstein equations the vanishing of the four-divergence of the energy-momentum tensor follows. This is the case because the four-divergence of the Einstein tensor vanishes identically. Inversely, we find that from the vanishing of the four-divergence of the energy-momentum tensor not only the Einstein equations follow. Besides, the so-named anti-Einstein equations follow. These equations must be considered as complementary to the Einstein equations. And while from the Ein...

  8. A generalized advection dispersion equation

    Indian Academy of Sciences (India)

    Abdon Atangana

    2014-02-01

    This paper examines a possible effect of uncertainties, variability or heterogeneity of any dynamic system when being included in its evolution rule; the notion is illustrated with the advection dispersion equation, which describes the groundwater pollution model. An uncertain derivative is defined; some properties of the operator are presented. The operator is used to generalize the advection dispersion equation. The generalized equation differs from the standard equation in four properties. The generalized equation is solved via the variational iteration technique. Some illustrative figures are presented.

  9. Equation with the many fathers

    DEFF Research Database (Denmark)

    Kragh, Helge

    1984-01-01

    of the usual Schrödinger equation. In most of the early versions the Klein-Gordon equation was connected with the general theory of relativity. Klein and some other physicists attempted to express quantum mechanics within a five-dimensional unified theory, embracing general relativity as well......In this essay I discuss the origin and early development of the first relativistic wave equation, known as the Klein-Gordon equation. In 1926 several physicists, among them Klein, Fock, Schrödinger, and de Broglie, announced this equation as a candidate for a relativistic generalization...

  10. Discovering evolution equations with applications

    CERN Document Server

    McKibben, Mark

    2011-01-01

    Most existing books on evolution equations tend either to cover a particular class of equations in too much depth for beginners or focus on a very specific research direction. Thus, the field can be daunting for newcomers to the field who need access to preliminary material and behind-the-scenes detail. Taking an applications-oriented, conversational approach, Discovering Evolution Equations with Applications: Volume 2-Stochastic Equations provides an introductory understanding of stochastic evolution equations. The text begins with hands-on introductions to the essentials of real and stochast

  11. Generalized Klein-Kramers equations

    Science.gov (United States)

    Fa, Kwok Sau

    2012-12-01

    A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000), 10.1021/jp993491m]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.

  12. Scaling of differential equations

    CERN Document Server

    Langtangen, Hans Petter

    2016-01-01

    The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and exam...

  13. Elliptic scattering equations

    CERN Document Server

    Cardona, Carlos

    2016-01-01

    Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a $\\mathbb{C}P^2$ space. We show that for the simplest integrand, namely the ${\\rm n-gon}$, our proposal indeed reproduces the expected result. By using the recently formulated $\\Lambda-$algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.

  14. $\\Lambda$ Scattering Equations

    CERN Document Server

    Gomez, Humberto

    2016-01-01

    The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter $\\Lambda$ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting $\\Lambda$ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the $\\Lambda$ algorithm.

  15. EXTENDED MILD-SLOPE EQUATION

    Institute of Scientific and Technical Information of China (English)

    黄虎; 丁平兴; 吕秀红

    2001-01-01

    The Hamiltonian formalism for surface waves and the mild-slope approximation were empolyed in handling the case of slowly varying three-dimensional currents and an uneven bottom, thus leading to an extended mild-slope equation. The bottom topography consists of two components: the slowly varying component whose horizontal length scale is longer than the surface wave length, and the fast varying component with the amplitude being smaller than that of the surface wave. The frequency of the fast varying depth component is, however, comparable to that of the surface waves. The extended mild- slope equation is more widely applicable and contains as special cases famous mild-slope equations below: the classical mild-slope equation of Berkhoff , Kirby' s mild-slope equation with current, and Dingemans' s mild-slope equation for rippled bed. The extended shallow water equations for ambient currents and rapidly varying topography are also obtained.

  16. Entropy: From Thermodynamics to Hydrology

    OpenAIRE

    Demetris Koutsoyiannis

    2014-01-01

    Some known results from statistical thermophysics as well as from hydrology are revisited from a different perspective trying: (a) to unify the notion of entropy in thermodynamic and statistical/stochastic approaches of complex hydrological systems and (b) to show the power of entropy and the principle of maximum entropy in inference, both deductive and inductive. The capability for deductive reasoning is illustrated by deriving the law of phase change transition of water (Clausius-Clapeyron)...

  17. The Riccati Differential Equation and a Diffusion-Type Equation

    CERN Document Server

    Suazo, Erwin; Vega-Guzman, Jose M

    2008-01-01

    We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equation with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding nonhomogeneous equation is also found.

  18. Comparison between characteristics of mild slope equations and Boussinesq equations

    Institute of Scientific and Technical Information of China (English)

    2005-01-01

    Boussinesq-type equations and mild-slope equations are compared in terms of their basic forms and characteristics. It is concluded that linear mild-slope equations on dispersion relation are better than non-linear Boussinesq equations. In addition, Berkhoff experiments are computed and compared by the two models, and agreement between model results and available experimental data is found to be quite reasonable, which demonstrates the two models' capacity to simulate wave transformation. However they can deal with different physical processes respectively, and they have their own characteristics.

  19. JWL Equation of State

    Energy Technology Data Exchange (ETDEWEB)

    Menikoff, Ralph [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

    2015-12-15

    The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.

  20. Algebraic Approaches to Partial Differential Equations

    CERN Document Server

    Xu, Xiaoping

    2012-01-01

    Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations...

  1. Differential equations methods and applications

    CERN Document Server

    Said-Houari, Belkacem

    2015-01-01

    This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .

  2. Energy Conservation Equations of Motion

    CERN Document Server

    Vinokurov, Nikolay A

    2015-01-01

    A conventional derivation of motion equations in mechanics and field equations in field theory is based on the principle of least action with a proper Lagrangian. With a time-independent Lagrangian, a function of coordinates and velocities that is called energy is constant. This paper presents an alternative approach, namely derivation of a general form of equations of motion that keep the system energy, expressed as a function of generalized coordinates and corresponding velocities, constant. These are Lagrange equations with addition of gyroscopic forces. The important fact, that the energy is defined as the function on the tangent bundle of configuration manifold, is used explicitly for the derivation. The Lagrangian is derived from a known energy function. A development of generalized Hamilton and Lagrange equations without the use of variational principles is proposed. The use of new technique is applied to derivation of some equations.

  3. Quaternion Dirac Equation and Supersymmetry

    OpenAIRE

    Rawat, Seema; Negi, O. P. S.

    2007-01-01

    Quaternion Dirac equation has been analyzed and its supersymetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, non zero mass, scalar pote...

  4. Introduction to ordinary differential equations

    CERN Document Server

    Rabenstein, Albert L

    1966-01-01

    Introduction to Ordinary Differential Equations is a 12-chapter text that describes useful elementary methods of finding solutions using ordinary differential equations. This book starts with an introduction to the properties and complex variable of linear differential equations. Considerable chapters covered topics that are of particular interest in applications, including Laplace transforms, eigenvalue problems, special functions, Fourier series, and boundary-value problems of mathematical physics. Other chapters are devoted to some topics that are not directly concerned with finding solutio

  5. THE ERMAKOV EQUATION: A COMMENTARY

    OpenAIRE

    P.G.L. Leach; Andriopoulos, K.

    2008-01-01

    We present a short history of the Ermakov Equation with an emphasis on its discovery by theWest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008), ...

  6. Two-component Dirac equation

    OpenAIRE

    Luo, Da-Wei; Pyshkin, P. V.; Yu, Ting; Lin, Hai-Qing; You, J. Q.; Wu, Lian-Ao

    2016-01-01

    We provide an alternative approach to relativistic dynamics based on the Feshbach projection technique. Instead of directly studying the Dirac equation, we derive a two-component equation for the upper spinor. This approach allows one to investigate the underlying physics in a different perspective. For particles with small mass such as the neutrino, the leading order equation has a Hermitian effective Hamiltonian, implying there is no leakage between the upper and lower spinors. In the weak ...

  7. Spinor wave equation of photon

    CERN Document Server

    Wu, Xiang-Yao; Liu, Xiao-Jing; Zhang, Si-Qi; Wang, Jing; Li, Hong; Fan, Xi-Hui; Li, Jing-Wu

    2012-01-01

    In this paper, we give the spinor wave equations of free and unfree photon, which are the differential equation of space-time one order. For the free photon, the spinor wave equations are covariant, and the spinors $\\psi$ are corresponding to the the reducibility representations $D^{10}+D^{01}$ and $D^{10}+D^{01}+D^{1/2 1/2}$ of the proper Lorentz group.

  8. Correct Linearization of Einstein's Equations

    Directory of Open Access Journals (Sweden)

    Rabounski D.

    2006-06-01

    Full Text Available Regularly Einstein's equations can be reduced to a wave form (linearly dependent from the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel's symbols. As shown here, the origin of the problem is that one uses the general covariant theory of measurement. Here the wave form of Einstein's equations is obtained in the terms of Zelmanov's chronometric invariants (physically observable projections on the observer's time line and spatial section. The obtained equations depend on solely the second derivatives even if gravitation, the space rotation and Christoffel's symbols. The correct linearization proves: the Einstein equations are completely compatible with weak waves of the metric.

  9. Electronic representation of wave equation

    Science.gov (United States)

    Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav

    2016-06-01

    The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.

  10. Quaternion Dirac Equation and Supersymmetry

    Science.gov (United States)

    Rawat, Seema; Negi, O. P. S.

    2009-08-01

    Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.

  11. Quaternion Dirac Equation and Supersymmetry

    CERN Document Server

    Rawat, S; Rawat, Seema

    2007-01-01

    Quaternion Dirac equation has been analyzed and its supersymetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, non zero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.

  12. A modified electromagnetic wave equation

    International Nuclear Information System (INIS)

    The aim of this paper is to find an alternative to the usual electromagnetic wave equation: that is, we want to find a different equation with the same solutions. The final goal is to solve electromagnetic problems with iterative methods. The curl curl operator that appears in the electromagnetic wave equation is difficult to invert numerically, and this cannot be done iteratively. The addition of a higher order term that emphasizes the diagonal terms in the operator may help the solution of the problem, and the new equation should be solvable by an iterative algorithm. The additional mode is suppressed by suitable boundary conditions. (author) 5 figs., 9 refs

  13. Tippe Top Equations and Equations for the Related Mechanical Systems

    Directory of Open Access Journals (Sweden)

    Nils Rutstam

    2012-04-01

    Full Text Available The equations of motion for the rolling and gliding Tippe Top (TT are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion. To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis 3ˆ during the inversion. We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT.

  14. Solutions of relativistic radial quasipotential equations

    Energy Technology Data Exchange (ETDEWEB)

    Minh, V.X.; Kadyshevskii, V.G.; Zhidkov, E.P.

    1985-11-01

    A systematic approach to the investigation of relativistic radial quasipotential equations is developed. The quasipotential equations can be interpreted either as linear equations in finite differences of fourth and second orders, respectively, or as differential equations of infinite order.

  15. Anomalous Fractional Diffusion Equation for Transport Phenomena

    Institute of Scientific and Technical Information of China (English)

    QiuhuaZENG; HouqiangLI; 等

    1999-01-01

    We derive the standard diffusion equation from the continuity equation and by discussing the defectiveness of earlier proposed equations,we get the generalized fractional diffusion equation for anomalous diffusion.

  16. Students' Understanding of Quadratic Equations

    Science.gov (United States)

    López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael

    2016-01-01

    Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…

  17. Enclosing Solutions of Integral Equations

    DEFF Research Database (Denmark)

    Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole

    1996-01-01

    We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because...

  18. Partial Completion of Equational Theories

    Institute of Scientific and Technical Information of China (English)

    孙永强; 林凯; 陆朝俊

    2000-01-01

    In this paper, the notion of partial completion of equational theories is proposed, which is a procedure to construct a confluent term rewriting system from an equational theory without requirement of termination condition. A partial completion algorithm is presented with a brief description of its application in a program development system.

  19. Uncertainty of empirical correlation equations

    Science.gov (United States)

    Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.

    2016-08-01

    The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.

  20. Numerical Solution of Parabolic Equations

    DEFF Research Database (Denmark)

    Østerby, Ole

    These lecture notes are designed for a one-semester course on finite-difference methods for parabolic equations. These equations which traditionally are used for describing diffusion and heat-conduction problems in Geology, Physics, and Chemistry have recently found applications in Finance Theory...

  1. A Search on Dirac Equation

    Institute of Scientific and Technical Information of China (English)

    M. Ko(c)ak; B. G(o)nül

    2007-01-01

    The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schr(o)dinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.

  2. Differential equations a concise course

    CERN Document Server

    Bear, H S

    2011-01-01

    Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complex-valued solutions, linear differential operators, inverse operators and variation of parameters method, the Laplace transform, Picard's existence theorem, and an exploration of various interpretations of systems of equations. Numerous clearly stated theorems and proofs, examples, and problems followed by solutions.

  3. On asymptotics for difference equations

    NARCIS (Netherlands)

    Rafei, M.

    2012-01-01

    In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for nonlinear difference equations are constructed by using the recently developed perturbation method based on invariance vectors. The asymptotic approximations of the solutions of the

  4. A search on Dirac equation

    OpenAIRE

    Kocak, M.; Gonul, B.

    2007-01-01

    The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given.

  5. Loewner equations and dispersionless hierarchies

    Energy Technology Data Exchange (ETDEWEB)

    Takebe, Takashi [Department of Mathematics, Ochanomizu University, Otsuka 2-1-1, Bunkyo-ku, Tokyo, 112-8610 (Japan); Teo, Lee-Peng [Faculty of Information Technology, Multimedia University, Jalan Multimedia, Cyberjaya, 63100, Selangor Darul Ehsan (Malaysia); Zabrodin, Anton [Institute of Biochemical Physics, Kosygina str. 4, 119991 Moscow, Russia and ITEP, Bol. Cheremushkinskaya str. 25, 117259 Moscow (Russian Federation)

    2006-09-15

    Using the Hirota representation of dispersionless dKP and dToda hierarchies, we show that the chordal Loewner equations and radial Loewner equations respectively serve as consistency conditions for one-variable reductions of these integrable hierarchies. We also clarify the geometric meaning of this result by relating it to the eigenvalue distribution of normal random matrices in the large N limit.

  6. Singularity: Raychaudhuri equation once again

    Indian Academy of Sciences (India)

    Naresh Dadhich

    2007-07-01

    I first recount Raychaudhuri's deep involvement with the singularity problem in general relativity. I then argue that precisely the same situation has arisen today in loop quantum cosmology as obtained when Raychaudhuri discovered his celebrated equation. We thus need a new analogue of the Raychaudhuri equation in quantum gravity.

  7. Non-relativistic BUU equation

    International Nuclear Information System (INIS)

    The Boltzmann-Uhlenbeck (BUU) equation, which is the time evolution of the wigner function of the single particle Green's function, is dervied by using the closed-time Green's function approach. The quantum mechanical approximation in derving the BUU equation is discussed

  8. Conservation Laws of Differential Equations in Finance

    Institute of Scientific and Technical Information of China (English)

    QIN Mao-Chang; MEI Feng-Xiang; SHANG Mei

    2005-01-01

    Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.

  9. Transport Equations for Oscillating Neutrinos

    CERN Document Server

    Zhang, Yunfan

    2013-01-01

    We derive a suite of generalized Boltzmann equations, based on the density-matrix formalism, that incorporates the physics of neutrino oscillations for two- and three-flavor oscillations, matter refraction, and self-refraction. The resulting equations are straightforward extensions of the classical transport equations that nevertheless contain the full physics of quantum oscillation phenomena. In this way, our broadened formalism provides a bridge between the familiar neutrino transport algorithms employed by supernova modelers and the more quantum-heavy approaches frequently employed to illuminate the various neutrino oscillation effects. We also provide the corresponding angular-moment versions of this generalized equation set. Our goal is to make it easier for astrophysicists to address oscillation phenomena in a language with which they are familiar. The equations we derive are simple and practical, and are intended to facilitate progress concerning oscillation phenomena in the context of core-collapse su...

  10. A Generalized Cubic Functional Equation

    Institute of Scientific and Technical Information of China (English)

    P. K. SAHOO

    2005-01-01

    In this paper, we determine the general solution of the functional equation f1 (2x + y) +f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4,f5: R → R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) = g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszu. The solution of this functional equation can also be determined in certain type of groups using two important results due to Székelyhidi.

  11. Nominal Logic with Equations Only

    CERN Document Server

    Clouston, Ranald

    2011-01-01

    Many formal systems, particularly in computer science, may be captured by equations modulated by side conditions asserting the "freshness of names"; these can be reasoned about with Nominal Equational Logic (NEL). Like most logics of this sort NEL employs this notion of freshness as a first class logical connective. However, this can become inconvenient when attempting to translate results from standard equational logic to the nominal setting. This paper presents proof rules for a logic whose only connectives are equations, which we call Nominal Equation-only Logic (NEoL). We prove that NEoL is just as expressive as NEL. We then give a simple description of equality in the empty NEoL-theory, then extend that result to describe freshness in the empty NEL-theory.

  12. Determining dynamical equations is hard

    CERN Document Server

    Cubitt, Toby S; Wolf, Michael M

    2010-01-01

    The behaviour of any physical system is governed by its underlying dynamical equations--the differential equations describing how the system evolves with time--and much of physics is ultimately concerned with discovering these dynamical equations and understanding their consequences. At the end of the day, any such dynamical law is identified by making measurements at different times, and computing the dynamical equation consistent with the acquired data. In this work, we show that, remarkably, this process is a provably computationally intractable problem (technically, it is NP-hard). That is, even for a moderately complex system, no matter how accurately we have specified the data, discovering its dynamical equations can take an infeasibly long time (unless P=NP). As such, we find a complexity-theoretic solution to both the quantum and the classical embedding problems; the classical version is a long-standing open problem, dating from 1937, which we finally lay to rest.

  13. Some Variations on Maxwell's Equations

    CERN Document Server

    Ascoli, G A; Ascoli, Giorgio A.; Goldin, Gerald A.

    2006-01-01

    In the first sections of this article, we discuss two variations on Maxwell's equations that have been introduced in earlier work---a class of nonlinear Maxwell theories with well-defined Galilean limits (and correspondingly generalized Yang-Mills equations), and a linear modification motivated by the coupling of the electromagnetic potential with a certain nonlinear Schroedinger equation. In the final section, revisiting an old idea of Lorentz, we write Maxwell's equations for a theory in which the electrostatic force of repulsion between like charges differs fundamentally in magnitude from the electrostatic force of attraction between unlike charges. We elaborate on Lorentz' description by means of electric and magnetic field strengths, whose governing equations separate into two fully relativistic Maxwell systems---one describing ordinary electromagnetism, and the other describing a universally attractive or repulsive long-range force. If such a force cannot be ruled out {\\it a priori\\/} by known physical ...

  14. Stochastic differential equations, backward SDEs, partial differential equations

    CERN Document Server

    Pardoux, Etienne

    2014-01-01

    This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has...

  15. COMPARISON BETWEEN BOUSSINESQ EQUATIONS AND MILD-SLOPE EQUATIONS MODEL

    Institute of Scientific and Technical Information of China (English)

    2006-01-01

    In this paper, the Boussinesq equations and mild-slope equation of wave transformation in near-shore shallow water were introduced and the characteristics of the two forms of equations were compared and analyzed. Meanwhile, a Boussinesq wave model which includes effects of bottom friction, wave breaking and subgrid turbulent mixing is established, slot technique dealing with moving boundary and damping layer dealing with absorbing boundary were established. By adopting empirical nonlinear dispersion relation and including nonlinear term, the mild-slope equation model was modified to take nonlinear effects into account. The two types of models were validated with the experiment results given by Berkhoff and their accuracy was analysed and compared with that of correlated methods.

  16. Higher derivative gravity: Field equation as the equation of state

    Science.gov (United States)

    Dey, Ramit; Liberati, Stefano; Mohd, Arif

    2016-08-01

    One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. The extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher-curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism-invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.

  17. Higher derivative gravity: field equation as the equation of state

    CERN Document Server

    Dey, Ramit; Mohd, Arif

    2016-01-01

    One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. Extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.

  18. Wave equations for pulse propagation

    Science.gov (United States)

    Shore, B. W.

    1987-06-01

    Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity.

  19. Integral equation methods for electromagnetics

    CERN Document Server

    Volakis, John

    2012-01-01

    This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo

  20. Galois theory of difference equations

    CERN Document Server

    Put, Marius

    1997-01-01

    This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.

  1. THE ERMAKOV EQUATION: A COMMENTARY

    Directory of Open Access Journals (Sweden)

    P. G. L. Leach

    2008-08-01

    Full Text Available We present a short history of the Ermakov Equation with an emphasis on its discovery by theWest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008, 123–145 for an English translation of Ermakov’s original paper.

  2. Reflection algebra and functional equations

    Energy Technology Data Exchange (ETDEWEB)

    Galleas, W., E-mail: w.galleas@uu.nl; Lamers, J., E-mail: j.lamers@uu.nl

    2014-09-15

    In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations satisfied by the partition function.

  3. Soliton equations and Hamiltonian systems

    CERN Document Server

    Dickey, L A

    2002-01-01

    The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also becau

  4. Equational theories of tropical sernirings

    DEFF Research Database (Denmark)

    Aceto, Luca; Esik, Zoltan; Ingolfsdottir, Anna

    2003-01-01

    of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results. Udgivelsesdato: APR 11......This paper studies the equational theories of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime...

  5. Direct 'delay' reductions of the Toda equation

    International Nuclear Information System (INIS)

    A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated with this equation is obtained, also by reduction. (fast track communication)

  6. Direct "Delay" Reductions of the Toda Equation

    OpenAIRE

    Joshi, Nalini

    2008-01-01

    A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painlev\\'e equations. The Lax pair associated to this equation is obtained, also by reduction.

  7. Integral Transform Approach to Generalized Tricomi Equations

    OpenAIRE

    Yagdjian, Karen

    2014-01-01

    We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14],[41]-[46] the particular version of this transform in order to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes.

  8. Geophysical interpretation using integral equations

    CERN Document Server

    Eskola, L

    1992-01-01

    Along with the general development of numerical methods in pure and applied to apply integral equations to geophysical modelling has sciences, the ability improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assu­ med to have a back...

  9. Solving Differential Equations in R

    Science.gov (United States)

    Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...

  10. IKT for quantum hydrodynamic equations

    Science.gov (United States)

    Tessarotto, Massimo; Ellero, Marco; Nicolini, Piero

    2007-11-01

    A striking feature of standard quantum mechanics (SQM) is its analogy with classical fluid dynamics. In fact, it is well-known that the Schr"odinger equation is equivalent to a closed set of partial differential equations for suitable real-valued functions of position and time (denoted as quantum fluid fields) [Madelung, 1928]. In particular, the corresponding quantum hydrodynamic equations (QHE) can be viewed as the equations of a classical compressible and non-viscous fluid, endowed with potential velocity and quantized velocity circulation. In this reference, an interesting theoretical problem, in its own right, is the construction of an inverse kinetic theory (IKT) for such a type of fluids. In this note we intend to investigate consequences of the IKT recently formulated for QHE [M.Tessarotto et al., Phys. Rev. A 75, 012105 (2007)]. In particular a basic issue is related to the definition of the quantum fluid fields.

  11. Spin equation and its solutions

    CERN Document Server

    Bagrov, V G; Baldiotti, M C; Levin, A D

    2005-01-01

    The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0+1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE.

  12. Diophantine approximations and Diophantine equations

    CERN Document Server

    Schmidt, Wolfgang M

    1991-01-01

    "This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum

  13. Invariant foliations for parabolic equations

    Institute of Scientific and Technical Information of China (English)

    2000-01-01

    It is proved for parabolic equations that under certain conditions the weak (un-)stable manifolds possess invariant foliations, called strongly (un-)stable foliations. The relevant results on center manifolds are generalized to weak hyperbolic manifolds.

  14. Overdetermined Systems of Linear Equations.

    Science.gov (United States)

    Williams, Gareth

    1990-01-01

    Explored is an overdetermined system of linear equations to find an appropriate least squares solution. A geometrical interpretation of this solution is given. Included is a least squares point discussion. (KR)

  15. Correct Linearization of Einstein's Equations

    Directory of Open Access Journals (Sweden)

    Rabounski D.

    2006-04-01

    Full Text Available Routinely, Einstein’s equations are be reduced to a wave form (linearly independent of the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel’s symbols. As shown herein, the origin of the problem is the use of the general covariant theory of measurement. Herein the wave form of Einstein’s equations is obtained in terms of Zelmanov’s chronometric invariants (physically observable projections on the observer’s time line and spatial section. The equations so obtained depend solely upon the second derivatives, even for gravitation, the space rotation and Christoffel’s symbols. The correct linearization proves that the Einstein equations are completely compatible with weak waves of the metric.

  16. Hidden Statistics of Schroedinger Equation

    Science.gov (United States)

    Zak, Michail

    2011-01-01

    Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.

  17. A New Unified Evolution Equation

    OpenAIRE

    Lim, Jyh-Liong

    1998-01-01

    WE propose a new unified evolution equation for parton distribution functions appropriate for both large and small Bjorken x. Compared with the Ciafaloni- Catani-Fiorani-Marchesini equation, the cancellation of soft poles between virtual and real gluon emissions is made explicitly without introducing infrared cutoffs, next-to-leading contributions to the Sudakov resummation can be included systematically, and the scales of the running coupling constants are determined unambiguously.

  18. On basic equation of statistical physics

    Institute of Scientific and Technical Information of China (English)

    邢修三

    1996-01-01

    Considering that thermodynamic irreversibility, the principle of entropy increase and hydrodynamic equations cannot be derived rigorously and in a unified way from the Liouville equations, the anomalous Langevin equation in Liouville space or its equivalent generalized Liouville equation is proposed as a basic equation of statistical physics. This equation reflects the fact that the law of motion of statistical thermodynamics is stochastic, but not deterministic. From that the nonequilibrium entropy, the principle of entropy increase, the theorem of minimum entropy production and the BBGKY diffusion equation hierarchy have been derived. The hydrodynamic equations, such as the generalized Navier-Stokes equation and the mass drift-diffusion equation, etc. have been derived from the BBGKY diffusion equation hierarchy. This equation has the same equilibrium solution as that of the Liouville equation. All these are unified and rigorous without adding any extra assumption. But it is difficult to prove that th

  19. Computational partial differential equations using Matlab

    CERN Document Server

    Li, Jichun

    2008-01-01

    Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE

  20. Equationally Compact Acts : Coproducts / Peeter Normak

    Index Scriptorium Estoniae

    Normak, Peeter

    1998-01-01

    In this article equational compactness of acts and its generalizations are discussed. As equational compactness does not carry over to coproducts a slight generalization of c-equational campactness is introduced. It is proved that a coproduct of acts is c-equationally compact if and only if all components are c-equationally campact

  1. Exact results for the Boltzmann equation and Smoluchowski's coagulation equation

    International Nuclear Information System (INIS)

    Almost no analytical solutions have been found for realistic intermolecular forces, largely due to the complicated structure of the collision term which calls for the construction of simplified models, in which as many physical properties are maintained as possible. In the first three chapters of this thesis such model Boltzmann equations are studied. Only spatially homogeneous gases with isotropic distribution functions are considered. Chapter I considers transition kernels, chapter II persistent scattering models and chapter III very hard particles. The second part of this dissertation deals with Smoluchowski's coagulation equation for the size distribution function in a coagulating system, with chapters devoted to the following topics: kinetics of gelation and universality, coagulation equations with gelation and exactly soluble models of nucleation. (Auth./C.F.)

  2. ON THE EQUIVALENCE OF THE ABEL EQUATION

    Institute of Scientific and Technical Information of China (English)

    2006-01-01

    This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations.

  3. Non-linear constitutive equations for gravitoelectromagnetism

    OpenAIRE

    Duplij, Steven; Di Grezia, Elisabetta; Esposito, Giampiero; Kotvytskiy, Albert

    2013-01-01

    This paper studies non-linear constitutive equations for gravitoelectromagnetism. Eventually, the problem is solved of finding, for a given particular solution of the gravity-Maxwell equations, the exact form of the corresponding non-linear constitutive equations.

  4. Multi-Time Equations, Classical and Quantum

    CERN Document Server

    Petrat, Sören

    2013-01-01

    Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics.

  5. First-order partial differential equations

    CERN Document Server

    Rhee, Hyun-Ku; Amundson, Neal R

    2001-01-01

    This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of mo

  6. How to Obtain the Covariant Form of Maxwell's Equations from the Continuity Equation

    Science.gov (United States)

    Heras, Jose A.

    2009-01-01

    The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.

  7. How to obtain the covariant form of Maxwell's equations from the continuity equation

    Energy Technology Data Exchange (ETDEWEB)

    Heras, Jose A [Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Av. San Pablo No. 180, Col. Reynosa, 02200, Mexico D. F. (Mexico); Departamento de Fisica y Matematicas, Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Mexico D. F. 01210 (Mexico)

    2009-07-15

    The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.

  8. Introductory course on differential equations

    CERN Document Server

    Gorain, Ganesh C

    2014-01-01

    Introductory Course on DIFFERENTIAL EQUATIONS provides an excellent exposition of the fundamentals of ordinary and partial differential equations and is ideally suited for a first course of undergraduate students of mathematics, physics and engineering. The aim of this book is to present the elementary theories of differential equations in the forms suitable for use of those students whose main interest in the subject are based on simple mathematical ideas. KEY FEATURES: Discusses the subject in a systematic manner without sacrificing mathematical rigour. A variety of exercises drill the students in problem solving in view of the mathematical theories explained in the book. Worked out examples illustrated according to the theories developed in the book with possible alternatives. Exhaustive collection of problems and the simplicity of presentation differentiate this book from several others. Material contained will help teachers as well as aspiring students of different competitive examinations.

  9. Differential Equations for Morphological Amoebas

    Science.gov (United States)

    Welk, Martin; Breuß, Michael; Vogel, Oliver

    This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We investigate this correspondence by analysing a space-continuous formulation of iterated median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates a partial differential equation related to self-snakes and the well-known (mean) curvature motion equation. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.

  10. Quantum corrections for Boltzmann equation

    Institute of Scientific and Technical Information of China (English)

    M.; Levy; PETER

    2008-01-01

    We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.

  11. The respiratory system in equations

    CERN Document Server

    Maury, Bertrand

    2013-01-01

    The book proposes an introduction to the mathematical modeling of the respiratory system. A detailed introduction on the physiological aspects makes it accessible to a large audience without any prior knowledge on the lung. Different levels of description are proposed, from the lumped models with a small number of parameters (Ordinary Differential Equations), up to infinite dimensional models based on Partial Differential Equations. Besides these two types of differential equations, two chapters are dedicated to resistive networks, and to the way they can be used to investigate the dependence of the resistance of the lung upon geometrical characteristics. The theoretical analysis of the various models is provided, together with state-of-the-art techniques to compute approximate solutions, allowing comparisons with experimental measurements. The book contains several exercises, most of which are accessible to advanced undergraduate students.

  12. Stability Analysis of Ecomorphodynamic Equations

    CERN Document Server

    Bärenbold, Fabian; Perona, Paolo

    2014-01-01

    Although riparian vegetation is present in or along many water courses of the world, its active role resulting from the interaction with flow and sediment processes has only recently become an active field of research. Especially, the role of vegetation in the process of river pattern formation has been explored and demonstrated mostly experimentally and numerically until now. In the present work, we shed light on this subject by performing a linear stability analysis on a simple model for riverbed vegetation dynamics coupled with the set of classical river morphodynamic equations. The vegetation model only accounts for logistic growth, local positive feedback through seeding and resprouting, and mortality by means of uprooting through flow shear stress. Due to the simplicity of the model, we can transform the set of equations into an eigenvalue problem and assess the stability of the linearized equations when slightly perturbated away from a spatially homogeneous solution. If we couple vegetation dynamics wi...

  13. Random equations in nilpotent groups

    CERN Document Server

    Gilman, Robert; Romankov, Vitalii

    2011-01-01

    In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. We show that the set SAT(G,k) of all equations in k > 1 variables over G which are satisfiable in G has an intermediate asymptotic density in the space of all equations in k variables over G. When G is a free abelian group of finite rank, we compute this density precisely; otherwise we give some non-trivial upper and lower bounds. For k = 1 the set SAT(G,k) is negligible. Usually the asymptotic densities of interesting sets in groups are either zero or one. The results of this paper provide new examples of algebraically significant sets of intermediate asymptotic density.

  14. Students' understanding of quadratic equations

    Science.gov (United States)

    López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael

    2016-05-01

    Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.

  15. Basic linear partial differential equations

    CERN Document Server

    Treves, Francois

    2006-01-01

    Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their

  16. Hamiltonian systems as selfdual equations

    Institute of Scientific and Technical Information of China (English)

    2008-01-01

    Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative action func-tionals obtained by a generalization of Bogomolnyi's trick of 'completing squares'. Reminiscent of the selfdual Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the correspond- ing Euler-Lagrange equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study certain Lagrangian intersections.

  17. Stability theory of differential equations

    CERN Document Server

    Bellman, Richard

    2008-01-01

    Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from

  18. Nielsen number and differential equations

    Directory of Open Access Journals (Sweden)

    Andres Jan

    2005-01-01

    Full Text Available In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations, two main approaches are presented. The first is via Poincaré's translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics are indicated, jointly with some further consequences like the nontrivial -structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.

  19. Fundamentals of equations of state

    CERN Document Server

    Eliezer, Shalom; Hora, Heinrich

    2002-01-01

    The equation of state was originally developed for ideal gases, and proved central to the development of early molecular and atomic physics. Increasingly sophisticated equations of state have been developed to take into account molecular interactions, quantization, relativistic effects, etc. Extreme conditions of matter are encountered both in nature and in the laboratory, for example in the centres of stars, in relativistic collisions of heavy nuclei, in inertial confinement fusion (where a temperature of 10 9 K and a pressure exceeding a billion atmospheres can be achieved). A sound knowledg

  20. Applied analysis and differential equations

    CERN Document Server

    Cârj, Ovidiu

    2007-01-01

    This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.

  1. Group analysis of differential equations

    CERN Document Server

    Ovsiannikov, L V

    1982-01-01

    Group Analysis of Differential Equations provides a systematic exposition of the theory of Lie groups and Lie algebras and its application to creating algorithms for solving the problems of the group analysis of differential equations.This text is organized into eight chapters. Chapters I to III describe the one-parameter group with its tangential field of vectors. The nonstandard treatment of the Banach Lie groups is reviewed in Chapter IV, including a discussion of the complete theory of Lie group transformations. Chapters V and VI cover the construction of partial solution classes for the g

  2. Differential equations and mathematical biology

    CERN Document Server

    Jones, DS; Sleeman, BD

    2009-01-01

    ""… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions.""-SIAM Review, Sept. 2010, Vol. 52, No. 3""… Where this text stands out i

  3. Partial differential equations an introduction

    CERN Document Server

    Colton, David

    2004-01-01

    Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of

  4. Integral equations on time scales

    CERN Document Server

    Georgiev, Svetlin G

    2016-01-01

    This book offers the reader an overview of recent developments of integral equations on time scales. It also contains elegant analytical and numerical methods. This book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses. The students in mathematical and physical sciences will find many sections of direct relevance. The book contains nine chapters and each chapter is pedagogically organized. This book is specially designed for those who wish to understand integral equations on time scales without having extensive mathematical background.

  5. Radar equations for modern radar

    CERN Document Server

    Barton, David K

    2012-01-01

    Based on the classic Radar Range-Performance Analysis from 1980, this practical volume extends that work to ensure applicability of radar equations to the design and analysis of modern radars. This unique book helps you identify what information on the radar and its environment is needed to predict detection range. Moreover, it provides equations and data to improve the accuracy of range calculations. You find detailed information on propagation effects, methods of range calculation in environments that include clutter, jamming and thermal noise, as well as loss factors that reduce radar perfo

  6. On a nonhomogeneous Burgers' equation

    Institute of Scientific and Technical Information of China (English)

    DING; Xiaqi(

    2001-01-01

    [1]Hopf, E., The partial differential equation ut + uux = μuxx, Comm. Pure Appl. Math., 1950, 3: 201-230.[2]Ding, X. Q. , Luo, P. Z. , Generalized expansions in Hilbert space, Acta Mathematica Scientia, 1999, 19(3): 241 250.[3]Titchmarsh, E., Introduction to the Theory of Fourier Integrals, 2nd ed., Oxford: Oxford University Press, 1948.[4]Ladyzhenskaya, O. A., Solonnikov, V. A., Ural' ceva, N. N., Linear and Quasilinear Equations of Parabolic Type,Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, 1968.

  7. The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics

    OpenAIRE

    E. M. E. Zayed; K. A. E. Alurrfi

    2014-01-01

    We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

  8. The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics

    Directory of Open Access Journals (Sweden)

    E. M. E. Zayed

    2014-01-01

    Full Text Available We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

  9. Algebraic solution of master equations

    OpenAIRE

    R. Rangel; L. Carvalho

    2003-01-01

    We present a simple analytical method to solve master equations for finite temperatures and any initial conditions, which consists in the expansion of the density operator into normal modes. These modes and the expansion coefficients are obtained algebraically by using ladder superoperators. This algebraic technique is successful in cases in which the Liouville superoperator is quadratic in the creation and annihilation operators.

  10. Lithium equation-of-state

    International Nuclear Information System (INIS)

    In 1977, Dave Young published an equation-of-state (EOS) for lithium. This EOS was used by Lew Glenn in his AFTON calculations of the HYLIFE inertial-fusion-reactor hydrodynamics. In this paper, I summarize Young's development of the EOS and demonstrate a computer program (MATHSY) that plots isotherms, isentropes and constant energy lines on a P-V diagram

  11. Equational axioms of test algebra

    NARCIS (Netherlands)

    Hollenberg, M.

    2008-01-01

    We present a complete axiomatization of test algebra ([24,18,29]), the two-sorted algebraic variant of Propositional Dynamic Logic (PDL,[21,7]). The axiomatization consists of adding a finite number of equations to any axiomatization of Kleene algebra ([15,26,17,4]) and algebraic translations of the

  12. Sonar equations for planetary exploration.

    Science.gov (United States)

    Ainslie, Michael A; Leighton, Timothy G

    2016-08-01

    The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus. PMID:27586766

  13. Stability of Functional Differential Equations

    CERN Document Server

    Lemm, Jeffrey M

    1986-01-01

    This book provides an introduction to the structure and stability properties of solutions of functional differential equations. Numerous examples of applications (such as feedback systrems with aftereffect, two-reflector antennae, nuclear reactors, mathematical models in immunology, viscoelastic bodies, aeroautoelastic phenomena and so on) are considered in detail. The development is illustrated by numerous figures and tables.

  14. Homographic scheme for Riccati equation

    CERN Document Server

    Dubois, François

    2011-01-01

    In this paper we present a numerical scheme for the resolution of matrix Riccati equation, usualy used in control problems. The scheme is unconditionnaly stable and the solution is definite positive at each time step of the resolution. We prove the convergence in the scalar case and present several numerical experiments for classical test cases.

  15. Renaissance Learning Equating Study. Report

    Science.gov (United States)

    Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben

    2007-01-01

    An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…

  16. Quaternionic Monge-Ampere equations

    OpenAIRE

    Alesker, Semyon

    2002-01-01

    The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternionic strictly pseudoconvex bounded domains in H^n. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].

  17. Pendulum Motion and Differential Equations

    Science.gov (United States)

    Reid, Thomas F.; King, Stephen C.

    2009-01-01

    A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…

  18. Solution and transcritical bifurcation of Burgers equation

    Institute of Scientific and Technical Information of China (English)

    Tang Jia-Shi; Zhao Ming-Hua; Han Feng; Zhang Liang

    2011-01-01

    Burgers equation is reduced into a first-order ordinary differential equation by using travelling wave transformation and it has typical bifurcation characteristics. We can obtain many exact solutions of the Burgers equation, discuss its transcritical bifurcation and control dynamical behaviours by extending the stable region. The transcritical bifurcation exists in the (2 + 1)-dimensional Burgers equation.

  19. Stochastic dynamic equations on general time scales

    OpenAIRE

    Martin Bohner; Olexandr M. Stanzhytskyi; Anastasiia O. Bratochkina

    2013-01-01

    In this article, we construct stochastic integral and stochastic differential equations on general time scales. We call these equations stochastic dynamic equations. We provide the existence and uniqueness theorem for solutions of stochastic dynamic equations. The crucial tool of our construction is a result about a connection between the time scales Lebesgue integral and the Lebesgue integral in the common sense.

  20. A Bayesian Nonparametric Approach to Test Equating

    Science.gov (United States)

    Karabatsos, George; Walker, Stephen G.

    2009-01-01

    A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are…

  1. Exact Vacuum Solutions to the Einstein Equation

    Institute of Scientific and Technical Information of China (English)

    2007-01-01

    In this paper, the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation. This procedure of resolution is based on a canonical form of the metric. According to this procedure, the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations,which are much convenient for the resolution.

  2. Functional Equations and Inequalities with Applications

    CERN Document Server

    Kannappan, Palaniappan

    2009-01-01

    Presents a comprehensive study of the classical topic of functional equations. This monograph explores different aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis.

  3. On Backward Stochstic Partial Differential Equations.

    OpenAIRE

    2001-01-01

    We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations. This is a type of equations which appear as adjoint equations in the maximum principle approach to optimal control of systems described by stochastic partial differential equations.

  4. Algebraic entropy for differential-delay equations

    OpenAIRE

    Viallet, Claude M.

    2014-01-01

    We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.

  5. The AGL equation from the dipole picture

    CERN Document Server

    Gay-Ducati, M B

    1999-01-01

    The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to an unitarized gluon distribution in the small x regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this paper that the AGL equation and, consequently, the GLR equation, can also be obtained from the dipole picture in the double logarithmic limit, using an evolution equation, recently proposed, which includes all multiple pomeron exchanges in the leading logarithmic approximation. Our conclusion is that the AGL equation is a good candidate for an unitarized evolution equation at small x in the DLA limit.

  6. The AGL equation from the dipole picture

    International Nuclear Information System (INIS)

    The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to a unitarized gluon distribution in the small x regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this paper that the AGL equation and, consequently, the GLR equation, can also be obtained from the dipole picture in the double logarithmic limit, using an evolution equation, recently proposed, which includes all multiple pomeron exchanges in the leading logarithmic approximation. Our conclusion is that the AGL equation is a good candidate for a unitarized evolution equation at small x in the DLA limit

  7. Explicit Integration of Friedmann's Equation with Nonlinear Equations of State

    CERN Document Server

    Chen, Shouxin; Yang, Yisong

    2015-01-01

    This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshev's result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics ...

  8. Thermoviscous Model Equations in Nonlinear Acoustics

    DEFF Research Database (Denmark)

    Rasmussen, Anders Rønne

    Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....

  9. Trial Equation Method to Nonlinear Evolution Equations with Rank Inhomogeneous:Mathematical Discussions and Its Applications

    Institute of Scientific and Technical Information of China (English)

    2006-01-01

    A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation,generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.

  10. The AGL Equation from a Dipole Picture

    CERN Document Server

    Gay-Ducati, M B

    1999-01-01

    The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to an unitarized gluon distribution in the small $x$ regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this contribution that the AGL equation can also be obtained from the dipole picture. Our conclusion is that the AGL equation is a good candidate for an unitarized evolution equation at small $x$ in the DLA limit.

  11. Dual Isomonodromic Problems and Whitham Equations

    OpenAIRE

    Takasaki, Kanehisa

    1997-01-01

    The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter $\\epsilon$. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the Seiberg- Wit...

  12. Techniques for solving Boolean equation systems

    OpenAIRE

    Keinänen, Misa

    2006-01-01

    Boolean equation systems are ordered sequences of Boolean equations decorated with least and greatest fixpoint operators. Boolean equation systems provide a useful framework for formal verification because various specification and verification problems, for instance, μ-calculus model checking can be represented as the problem of solving Boolean equation systems. The general problem of solving a Boolean equation system is a computationally hard task, and no polynomial time solution technique ...

  13. The Pauli equation in scale relativity

    OpenAIRE

    Celerier, Marie-Noelle; Nottale, Laurent

    2006-01-01

    In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and cont...

  14. Integrable (2k)-Dimensional Hitchin Equations

    CERN Document Server

    Ward, R S

    2016-01-01

    This letter describes a completely-integrable system of Yang-Mills-Higgs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k=2 case are described.

  15. Algebrization of Nonautonomous Differential Equations

    Directory of Open Access Journals (Sweden)

    María Aracelia Alcorta-García

    2015-01-01

    Full Text Available Given a planar system of nonautonomous ordinary differential equations, dw/dt=F(t,w, conditions are given for the existence of an associative commutative unital algebra A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t,w=H(te,w and the maps H1(τ=H(τ,ξ and H2(ξ=H(τ,ξ are Lorch differentiable with respect to A for all (τ,ξ∈Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ of the differential equation dξ/dτ=H(τ,ξ over A define solutions (x(t,y(t=ξ(te of the planar system.

  16. Decoherent Histories and Hydrodynamic Equations

    CERN Document Server

    Halliwell, J J

    1998-01-01

    For a system consisting of a large collection of particles, a set of variables that will generally become effectively classical are the local densities (number, momentum, energy). That is, in the context of the decoherent histories approach to quantum theory, it is expected that histories of these variables will be approximately decoherent, and that their probabilites will be strongly peaked about hydrodynamic equations. This possibility is explored for the case of the diffusion of the number density of a dilute concentration of foreign particles in a fluid. It is shown that, for certain physically reasonable initial states, the probabilities for histories of number density are strongly peaked about evolution according to the diffusion equation. Decoherence of these histories is also shown for a class of initial states which includes non-trivial superpositions of number density. Histories of phase space densities are also discussed. The case of histories of number, momentum and energy density for more general...

  17. Power equations in endurance sports.

    Science.gov (United States)

    van Ingen Schenau, G J; Cavanagh, P R

    1990-01-01

    This paper attempts to clarify the formulation of power equations applicable to a variety of endurance activities. An accurate accounting of the relationship between the metabolic power input and the mechanical power output is still elusive, due to such issues as storage and recovery of strain energy and the differing energy costs of concentric and eccentric muscle actions. Nevertheless, an instantaneous approach is presented which is based upon the application of conventional Newtonian mechanics to a rigid segment model of the body, and does not contain assumptions regarding the exact nature of segmental interactions--such as energy transfer, etc. The application of the equation to running, cycling, speed skating, swimming and rowing is discussed and definitions of power, efficiency, and economy are presented.

  18. Differential Equations of Ideal Memristors

    Directory of Open Access Journals (Sweden)

    Z. Biolek

    2015-06-01

    Full Text Available Ideal memristor is a resistor with a memory, which adds dynamics to its behavior. The most usual characteristics describing this dynamics are the constitutive relation (i.e. the relation between flux and charge, or Parameter-vs-state- map (PSM, mostly represented by the memristance-to-charge dependence. One of the so far unheeded tools for memristor description is its differential equation (DEM, composed exclusively of instantaneous values of voltage, current, and their derivatives. The article derives a general form of DEM that holds for any ideal memristor and shows that it is always a nonlinear equation of the first order; the PSM forms are found for memristors which are governed by DEMs of the Bernoulli and the Riccati types; a classification of memristors according to the type of their dynamics with respect to voltage and current is carried out.

  19. Nielsen number and differential equations

    Directory of Open Access Journals (Sweden)

    Jan Andres

    2005-06-01

    Full Text Available In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations, two main approaches are presented. The first is via Poincaré's translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics are indicated, jointly with some further consequences like the nontrivial Rδ-structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.

  20. The equations icons of knowledge

    CERN Document Server

    Bais, Sander

    2005-01-01

    For thousands of years mankind has tried to understand nature. Exploring the world on all scales with instruments of ever more ingenuity, we have been able to unravel some of the great mysteries that surround us. While collecting an overwhelming multitude of observational facts, we discovered fundamental laws that govern the structure and evolution of physical reality. We know that nature speaks to us in the language of mathematics. In this language most of our basic understanding of the physical world can be expressed in an unambiguous and concise way. The most artificial language turns out to be the most natural of all. The laws of nature correspond to equations. These equations are the icons of knowledge that mark crucial turning points in our thinking about the world we happen to live in. They form the symbolic representation of most of what we know, and as such constitute an important and robust part of our culture.

  1. Nonlocal higher order evolution equations

    KAUST Repository

    Rossi, Julio D.

    2010-06-01

    In this article, we study the asymptotic behaviour of solutions to the nonlocal operator ut(x, t)1/4(-1)n-1 (J*Id -1)n (u(x, t)), x ∈ ℝN, which is the nonlocal analogous to the higher order local evolution equation vt(-1)n-1(Δ)nv. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. © 2010 Taylor & Francis.

  2. Sensitivity for the Smoluchowski equation

    Energy Technology Data Exchange (ETDEWEB)

    Bailleul, I F, E-mail: i.bailleul@statslab.cam.ac.uk [Statistical Laboratory, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB (United Kingdom)

    2011-06-17

    This paper investigates the question of sensitivity of the solutions {mu}{sup {lambda}}{sub t} of the Smoluchowski equation on R{sub +}* with respect to the parameters {lambda} in the interaction kernel K{sup {lambda}}. It is proved that {mu}{sup {lambda}}{sub t} is a C{sup 1} function of (t, {lambda}) with values in a good space of measures under the hypotheses K{sup {lambda}}(x, y) {<=} {psi}(x) {psi}(y), for some sub-linear function {psi}, and {integral}{psi}{sup 4+{epsilon}}(x) {mu}{sub 0}(dx) < {infinity}, and that the derivative is the unique solution of a related equation.

  3. Eigenwavelets of the Wave equation

    OpenAIRE

    Kaiser, Gerald

    2004-01-01

    We study a class of localized solutions of the wave equation, called eigenwavelets, obtained by extending its fundamental solutions to complex spacetime in the sense of hyperfunctions. The imaginary spacetime variables y, which form a timelike vector, act as scale parameters generalizing the scale variable of wavelets in one dimension. They determine the shape of the wavelets in spacetime, making them pulsed beams that can be focused as tightly as desired around a single ray by letting y appr...

  4. Handbook of structural equation modeling

    CERN Document Server

    Hoyle, Rick H

    2012-01-01

    The first comprehensive structural equation modeling (SEM) handbook, this accessible volume presents both the mechanics of SEM and specific SEM strategies and applications. The editor, contributors, and editorial advisory board are leading methodologists who have organized the book to move from simpler material to more statistically complex modeling approaches. Sections cover the foundations of SEM; statistical underpinnings, from assumptions to model modifications; steps in implementation, from data preparation through writing the SEM report; and basic and advanced applications, inclu

  5. Generalized bootstrap for estimating equations

    OpenAIRE

    Chatterjee, Snigdhansu; Bose, Arup

    2005-01-01

    We introduce a generalized bootstrap technique for estimators obtained by solving estimating equations. Some special cases of this generalized bootstrap are the classical bootstrap of Efron, the delete-d jackknife and variations of the Bayesian bootstrap. The use of the proposed technique is discussed in some examples. Distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained.

  6. Equation of State Project Overview

    Energy Technology Data Exchange (ETDEWEB)

    Crockett, Scott [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

    2015-09-11

    A general overview of the Equation of State (EOS) Project will be presented. The goal is to provide the audience with an introduction of what our more advanced methods entail (DFT, QMD, etc.. ) and how these models are being utilized to better constrain the thermodynamic models. These models substantially reduce our regions of interpolation between the various thermodynamic limits. I will also present a variety example of recent EOS work.

  7. Instantaneous Bethe-Salpeter equation

    International Nuclear Information System (INIS)

    We present a systematic algebraic and numerical investigation of the instantaneous Beth-Salpeter equation. Emphasis is placed on confining interaction kernels of the Lorentz scalar, time component vector, and full vector-types. We explore the stability of the solutions and Regge behavior for each of these interactions, and conclude that only time component vector confinement leads to normal Regge structure and stable solutions for all quark masses

  8. Operator equations and invariant subspaces

    Directory of Open Access Journals (Sweden)

    Valentin Matache

    1994-05-01

    Full Text Available Banach space operators acting on some fixed space X are considered. If two such operators A and B verify the condition A2=B2 and if A has nontrivial hyperinvariant subspaces, then B has nontrivial invariant subspaces. If A and B commute and satisfy a special type of functional equation, and if A is not a scalar multiple of the identity, the author proves that if A has nontrivial invariant subspaces, then so does B.

  9. Differential equations in airplane mechanics

    Science.gov (United States)

    Carleman, M T

    1922-01-01

    In the following report, we will first draw some conclusions of purely theoretical interest, from the general equations of motion. At the end, we will consider the motion of an airplane, with the engine dead and with the assumption that the angle of attack remains constant. Thus we arrive at a simple result, which can be rendered practically utilizable for determining the trajectory of an airplane descending at a constant steering angle.

  10. Wave equations in higher dimensions

    CERN Document Server

    Dong, Shi-Hai

    2011-01-01

    Higher dimensional theories have attracted much attention because they make it possible to reduce much of physics in a concise, elegant fashion that unifies the two great theories of the 20th century: Quantum Theory and Relativity. This book provides an elementary description of quantum wave equations in higher dimensions at an advanced level so as to put all current mathematical and physical concepts and techniques at the reader’s disposal. A comprehensive description of quantum wave equations in higher dimensions and their broad range of applications in quantum mechanics is provided, which complements the traditional coverage found in the existing quantum mechanics textbooks and gives scientists a fresh outlook on quantum systems in all branches of physics. In Parts I and II the basic properties of the SO(n) group are reviewed and basic theories and techniques related to wave equations in higher dimensions are introduced. Parts III and IV cover important quantum systems in the framework of non-relativisti...

  11. Effective Schroedinger equations on submanifolds

    Energy Technology Data Exchange (ETDEWEB)

    Wachsmuth, Jakob

    2010-02-11

    In this thesis the time dependent Schroedinger equation is considered on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When the potential is scaled in the directions normal to C by a small parameter epsilon, the solutions concentrate in an epsilon-neighborhood of the submanifold. An effective Schroedinger equation on the submanifold C is derived and it is shown that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order {epsilon}{sup 3} vertical stroke t vertical stroke at time t. Furthermore, it is proved that, under reasonable conditions, the eigenvalues of the corresponding Hamiltonians below a certain energy coincide upto errors of order {epsilon}{sup 3}. These results holds in the situation where tangential and normal energies are of the same order, and where exchange between normal and tangential energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were needed, to ensure that the separation of energies is maintained during the time evolution. The most important consequence of this thesis is that now constraining potentials that change their shape along the submanifold can be treated, which is the typical situation in applications like molecular dynamics and quantum waveguides.

  12. Torsion Effects and LLG Equation

    CERN Document Server

    Ferreira, Cristine N; Neto, J A Helayël

    2016-01-01

    Based on the non-relativistic regime of the Dirac equation coupled to a torsion pseudo-vector, we study the dynamics of magnetization and how it is affected by the presence of torsion. We consider that torsion interacting terms in Dirac equation appear in two ways one of these is thhrough the covariant derivative considering the spin connection and gauge magnetic field and the other is through a non-minimal spin torsion coupling. We show within this framework, that it is possible to obtain the most general Landau, Lifshitz and Gilbert (LLG) equation including the torsion effects, where we refer to torsion as a geometric field playing an important role in the spin coupling process. We show that the torsion terms can give us two important landscapes in the magnetization dynamics: one of them related with damping and the other related with the screw dislocation that give us a global effect like a helix damping sharped. These terms are responsible for changes in the magnetization precession dynamics.

  13. ADVANCED WAVE-EQUATION MIGRATION

    Energy Technology Data Exchange (ETDEWEB)

    L. HUANG; M. C. FEHLER

    2000-12-01

    Wave-equation migration methods can more accurately account for complex wave phenomena than ray-tracing-based Kirchhoff methods that are based on the high-frequency asymptotic approximation of waves. With steadily increasing speed of massively parallel computers, wave-equation migration methods are becoming more and more feasible and attractive for imaging complex 3D structures. We present an overview of several efficient and accurate wave-equation-based migration methods that we have recently developed. The methods are implemented in the frequency-space and frequency-wavenumber domains and hence they are called dual-domain methods. In the methods, we make use of different approximate solutions of the scalar-wave equation in heterogeneous media to recursively downward continue wavefields. The approximations used within each extrapolation interval include the Born, quasi-Born, and Rytov approximations. In one of our dual-domain methods, we use an optimized expansion of the square-root operator in the one-way wave equation to minimize the phase error for a given model. This leads to a globally optimized Fourier finite-difference method that is a hybrid split-step Fourier and finite-difference scheme. Migration examples demonstrate that our dual-domain migration methods provide more accurate images than those obtained using the split-step Fourier scheme. The Born-based, quasi-Born-based, and Rytov-based methods are suitable for imaging complex structures whose lateral variations are moderate, such as the Marmousi model. For this model, the computational cost of the Born-based method is almost the same as the split-step Fourier scheme, while other methods takes approximately 15-50% more computational time. The globally optimized Fourier finite-difference method significantly improves the accuracy of the split-step Fourier method for imaging structures having strong lateral velocity variations, such as the SEG/EAGE salt model, at an approximately 30% greater

  14. a Multiple Riccati Equations Rational-Exponent Method and its Application to Whitham-Broer Equation

    Science.gov (United States)

    Liu, Qing; Wang, Zi-Hua; Jia, Dong-Li

    2013-03-01

    According to two dependent solutions to a generalized Riccati equation together with the equation itself, a multiple Riccati equations rational-exponent method is proposed and applied to Whitham-Broer-Kaup equation. It shows that this method is a more concise and efficient approach and can uniformly derive many types of combined solutions to nonlinear partial differential equations.

  15. Gibbs adsorption and the compressibility equation

    International Nuclear Information System (INIS)

    A new approach for deriving the equation of state is developed. It is shown that the integral in the compressibility equation is identical to the isotherm for Gibbs adsorption in radial coordinates. The Henry, Langmuir, and Frumkin adsorption isotherms are converted into equations of state. It is shown that using Henry's law gives an expression for the second virial coefficient that is identical to the result from statistical mechanics. Using the Langmuir isotherm leads to a new analytic expression for the hard-sphere equation of state which can be explicit in either pressure or density. The Frumkin isotherm results in a new equation of state for the square-well potential fluid. Conversely, new adsorption isotherms can be derived from equations of state using the compressibility equation. It is shown that the van der Waals equation gives an adsorption isotherm equation that describes both polymolecular adsorption and the unusual adsorption behavior observed for supercritical fluids. copyright 1995 American Institute of Physics

  16. Kinetic equations for an unstable plasma

    International Nuclear Information System (INIS)

    In this work, we establish the plasma kinetic equations starting from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations. We demonstrate that relations existing between correlation functions may help to justify the truncation of the hierarchy. Then we obtain the kinetic equations of a stable or unstable plasma. They do not reduce to an equation for the one-body distribution function, but generally involve two coupled equations for the one-body distribution function and the spectral density of the fluctuating electric field. We study limiting cases where the Balescu-Lenard equation, the quasi-linear theory, the Pines-Schrieffer equations and the equations of weak turbulence in the random phase approximation are recovered. At last we generalise the H-theorem for the system of equations and we define conditions for irreversible behaviour. (authors)

  17. Transport Properties of the Universal Quantum Equation

    Institute of Scientific and Technical Information of China (English)

    A.I.Arbab

    2012-01-01

    The universal quantum equation (UQE) is found to describe the transport properties of the quantum particles.This equation describes a wave equation interacting with constant scalar and vector potentials propagating in spacetime.A new transformation that sends the Schr(o)dinger equation with a potential energy V =-1/2mc2 to Dirac's equation is proposed.The Cattaneo telegraph equation as well as a one-dimensional UQE are compatible with our recently proposed generalized continuity equations.Furthermore,a new wave equation resulted from the invariance of the UQE under the post-Galilean transformations is derived.This equation is found to govern a Klein Gordon's particle interacting with a photon-like vector field (ether) whose magnitude is proportional to the particle's mass.

  18. On Reducing a System of Equations to a Single Equation

    DEFF Research Database (Denmark)

    Frandsen, G.S.; Shparlinski, I.E.

    2004-01-01

    For a system of polynomial equations over Q;p; we present an efficient construction of a single polynomial of quite small degree whose zero set over Q;p; coincides with the zero set over Q;p; of the original system. We also show that the polynomial has some other attractive features such as low a...... additive and straight-line complexity.The proof is based on a link established here between the above problem and some recent number theoretic result about zeros of p-adic forms....

  19. From Newton's Equation to Fractional Diffusion and Wave Equations

    Directory of Open Access Journals (Sweden)

    Vázquez Luis

    2011-01-01

    Full Text Available Fractional calculus represents a natural instrument to model nonlocal (or long-range dependence phenomena either in space or time. The processes that involve different space and time scales appear in a wide range of contexts, from physics and chemistry to biology and engineering. In many of these problems, the dynamics of the system can be formulated in terms of fractional differential equations which include the nonlocal effects either in space or time. We give a brief, nonexhaustive, panoramic view of the mathematical tools associated with fractional calculus as well as a description of some fields where either it is applied or could be potentially applied.

  20. Partial differential equations of mathematical physics and integral equations

    CERN Document Server

    Guenther, Ronald B

    1996-01-01

    This book was written to help mathematics students and those in the physical sciences learn modern mathematical techniques for setting up and analyzing problems. The mathematics used is rigorous, but not overwhelming, while the authors carefully model physical situations, emphasizing feedback among a beginning model, physical experiments, mathematical predictions, and the subsequent refinement and reevaluation of the physical model itself. Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the t

  1. Handbook of differential equations stationary partial differential equations

    CERN Document Server

    Chipot, Michel

    2006-01-01

    This handbook is volume III in a series devoted to stationary partial differential quations. Similarly as volumes I and II, it is a collection of self contained state-of-the-art surveys written by well known experts in the field. The topics covered by this handbook include singular and higher order equations, problems near critically, problems with anisotropic nonlinearities, dam problem, T-convergence and Schauder-type estimates. These surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics. Ke

  2. Using fundamental equations to describe basic phenomena

    DEFF Research Database (Denmark)

    Jakobsen, Arne; Rasmussen, Bjarne D.

    1999-01-01

    When the fundamental thermodynamic balance equations (mass, energy, and momentum) are used to describe the processes in a simple refrigeration system, then one finds that the resulting equation system will have a degree of freedom equal to one. Further investigations reveal that it is the equation...... and subcooling are introduced. Since the degree of freedom was equal to one, using both the superheat and subcooling require that one of the fundamental equations must be omitted from the equation system.The main purpose of the paper is to clarify the relation between the fundamental balance equations...

  3. Partial differential equations for scientists and engineers

    CERN Document Server

    Farlow, Stanley J

    1993-01-01

    Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing th

  4. Mathematical physics with partial differential equations

    CERN Document Server

    Kirkwood, James

    2011-01-01

    Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field - the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform

  5. International Workshop on Elliptic and Parabolic Equations

    CERN Document Server

    Schrohe, Elmar; Seiler, Jörg; Walker, Christoph

    2015-01-01

    This volume covers the latest research on elliptic and parabolic equations and originates from the international Workshop on Elliptic and Parabolic Equations, held September 10-12, 2013 at the Leibniz Universität Hannover. It represents a collection of refereed research papers and survey articles written by eminent scientist on advances in different fields of elliptic and parabolic partial differential equations, including singular Riemannian manifolds, spectral analysis on manifolds, nonlinear dispersive equations, Brownian motion and kernel estimates, Euler equations, porous medium type equations, pseudodifferential calculus, free boundary problems, and bifurcation analysis.

  6. EQUATIONS FOR GRAIN SIZE DISTRIBUTION CURVE

    Institute of Scientific and Technical Information of China (English)

    Prabhata K.SWAMEE; Nimisha SWAMEE

    2004-01-01

    The grain size distribution of particulate material is of particular interest in the field of sediment transport. The size distribution is described by various equations, however no equation is flexible enough to satisfy the grain size distribution data faithfully. Presented herein are the equations for unimodal and multimodal grain size distribution curves. A graphical method has been given to evaluate the parameters involved in these equations. The size distribution equation can be used to estimate many properties of sediment sample like number of sediment particles, surface area of the particles and hydraulic conductivity. It is hoped that the equations will find many applications in studying sedimentation processes.

  7. On the Inclusion of Difference Equation Problems and Z Transform Methods in Sophomore Differential Equation Classes

    Science.gov (United States)

    Savoye, Philippe

    2009-01-01

    In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.

  8. PICARD ITERATION FOR NONSMOOTH EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    Song-bai Sheng; Hui-fu Xu

    2001-01-01

    This paper presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. It is proved that the radius of the weak Jacobian (RGJ) of Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results can be obtained provided that RGJ of the Picard iteration function at a solution point is less than one or equal to zero. As for applications, it is pointed out that the approximate Newton methods, the generalized Newton method for piecewise C1problems and splitting methods can be explained uniformly with the same viewpoint.

  9. Advanced lab on Fresnel equations

    Science.gov (United States)

    Petrova-Mayor, Anna; Gimbal, Scott

    2015-11-01

    This experimental and theoretical exercise is designed to promote students' understanding of polarization and thin-film coatings for the practical case of a scanning protected-metal coated mirror. We present results obtained with a laboratory scanner and a polarimeter and propose an affordable and student-friendly experimental arrangement for the undergraduate laboratory. This experiment will allow students to apply basic knowledge of the polarization of light and thin-film coatings, develop hands-on skills with the use of phase retarders, apply the Fresnel equations for metallic coating with complex index of refraction, and compute the polarization state of the reflected light.

  10. Anyon Equation on a Torus

    Science.gov (United States)

    Ho, Choon-Lin; Hosotani, Yutaka

    Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.

  11. BMN correlators by loop equations

    International Nuclear Information System (INIS)

    In the BMN approach to N=4 SYM a large class of correlators of interest are expressible in terms of expectation values of traces of words in a zero-dimensional gaussian complex matrix model. We develop a loop-equation based, analytic strategy for evaluating such expectation values to any order in the genus expansion. We reproduce the expectation values which were needed for the calculation of the one-loop, genus one correction to the anomalous dimension of BMN-operators and which were earlier obtained by combinatorial means. Furthermore, we present the expectation values needed for the calculation of the one-loop, genus two correction. (author)

  12. Experimental determination of circuit equations

    CERN Document Server

    Shulman, Jason; Widjaja, Matthew; Gunaratne, Gemunu H

    2013-01-01

    Kirchhoff's laws offer a general, straightforward approach to circuit analysis. Unfortunately, use of the laws becomes impractical for all but the simplest of circuits. This work presents a novel method of analyzing direct current resistor circuits. It is based on an approach developed to model complex networks, making it appropriate for use on large, complicated circuits. It is unique in that it is not an analytic method. It is based on experiment, yet the approach produces the same circuit equations obtained by more traditional means.

  13. On oscillatory solutions of certain difference equations

    Directory of Open Access Journals (Sweden)

    Grzegorz Grzegorczyk

    2006-01-01

    Full Text Available Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed.

  14. On oscillatory solutions of certain difference equations

    OpenAIRE

    Grzegorz Grzegorczyk; Jarosław Werbowski

    2006-01-01

    Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed.

  15. The Spin-2 Equation on Minkowski Background

    CERN Document Server

    Beyer, Florian; Frauendiener, Jörg; Whale, Ben

    2014-01-01

    The linearised general conformal field equations in their first and second order form are used to study the behaviour of the spin-2 zero-rest-mass equation on Minkowski background in the vicinity of space-like infinity.

  16. Hydrodynamic Equations for Microscopic Phase Densities

    OpenAIRE

    Gerasimenko, V. I.; Shtyk, V. O.; Zagorodny, A. G.

    2009-01-01

    The evolution equations for the generalized microscopic phase densities are introduced. The evolution equations of average values of microscopic phase densities are derived and a solution of the initial-value problem of the obtained hydrodynamic type hierarchy is constructed.

  17. Introduction to linear algebra and differential equations

    CERN Document Server

    Dettman, John W

    1986-01-01

    Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.

  18. Exact Solutions to Short Pulse Equation

    Institute of Scientific and Technical Information of China (English)

    FU Zun-Tao; ZHENG Ming-Hua; LIU Shi-Kuo

    2009-01-01

    In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.

  19. Exact Solutions to Degasperis-Procesi Equation

    Institute of Scientific and Technical Information of China (English)

    ZHANG Lin-Na; FU Zun-Tao; LIU Shi-Kuo

    2008-01-01

    In this paper,dependent and independent variable transformations are introduced to solve the Degasperis-Procesi equation.It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.

  20. Simultaneous Independent Linear Equations and Goldbach Conjecture

    CERN Document Server

    Linggen, Song

    2007-01-01

    It was verified that if Goldbach conjecture was a fault, the number of simultaneous independent linear equations educed from this assumption would be unreasonably at least one more than the number of unknowns involved in these equations.

  1. A new class of variational equation problems

    Institute of Scientific and Technical Information of China (English)

    2003-01-01

    Applying an analysis method to a group of multivariable equations, a new class of variational equations are proved. This method is more concise and more direct than the others. This result can be applied to some stochastic control models.

  2. Functional differential equations of third order

    Directory of Open Access Journals (Sweden)

    Tuncay Candan

    2005-04-01

    Full Text Available In this paper, we consider the third-order neutral functional differential equation with distributed deviating arguments. We give sufficient conditions for the oscillatory behavior of this functional differential equation.

  3. OSCILLATION CRITERIA FOR FORCED SUPERLINEAR DIFFERENCE EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    2008-01-01

    Using Riccati transformation techniques,some oscillation criteria for the forced second-order superlinear difference equations are established.These criteria are dis- crete analogues of the criteria for differential equations proposed by Yan.

  4. An axiomatic approach to Maxwell's equations

    CERN Document Server

    Heras, José A

    2016-01-01

    This paper suggests an axiomatic approach to Maxwell's equations. The basis of this approach is a theorem formulated for two sets of functions localized in space and time. If each set satisfies a continuity equation then the theorem provides an integral representation for each function. A corollary of this theorem yields Maxwell's equations with magnetic monopoles. It is pointed out that the causality principle and the conservation of electric and magnetic charges are the most fundamental physical axioms underlying these equations. Another application of the corollary yields Maxwell's equations in material media. The theorem is also formulated in the Minkowski space-time and applied to obtain the covariant form of Maxwell's equations with magnetic monopoles and the covariant form of Maxwell's equations in material media. The approach makes use of the infinite-space Green function of the wave equation and is therefore suitable for an advanced course in electrodynamics.

  5. Dirac and Maxwell equations in Split Octonions

    CERN Document Server

    Beradze, Revaz

    2016-01-01

    The split octonionic form of Dirac and Maxwell equations are found. In contrast with the previous attempts these equations are derived from the octonionic analyticity condition and also we use different basis of the 8-dimensional space of split octonions.

  6. Linear superposition solutions to nonlinear wave equations

    Institute of Scientific and Technical Information of China (English)

    Liu Yu

    2012-01-01

    The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.

  7. Fractional Complex Transform for Fractional Differential Equations

    OpenAIRE

    Li, Zheng-Biao; He, Ji-Huan

    2010-01-01

    Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Two examples are given.

  8. Metal-insulator transition in Nd{sub 1−x}Eu{sub x}NiO{sub 3}: Entropy change and electronic delocalization

    Energy Technology Data Exchange (ETDEWEB)

    Jardim, R. F., E-mail: rjardim@if.usp.br; Andrade, S. [Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo 05315-970 (Brazil); Barbeta, V. B. [Departamento de Física, Centro Universitário da FEI, São Bernardo do Campo 09850-901 (Brazil); Escote, M. T. [Centro de Engenharia, Modelagem e Ciências Sociais Aplicadas, Universidade Federal do ABC, Santo André 09210-170 (Brazil); Cordero, F. [CNR-ISC, Istituto dei Sistemi Complessi, Area della Ricerca di Roma - Tor Vergata, Via del Fosso del Cavaliere 100, I-00133 Rome (Italy); Torikachvili, M. S. [Department of Physics, San Diego State University, San Diego, California 92182-1233 (United States)

    2015-05-07

    The metal-insulator (MI) phase transition in Nd{sub 1–x}Eu{sub x}NiO{sub 3}, 0 ≤ x ≤ 0.35, has been investigated through the pressure dependence of the electrical resistivity ρ(P, T) and measurements of specific heat C{sub P}(T). The MI transition temperature (T{sub MI}) increases with increasing Eu substitution and decreases with increasing pressure. Two distinct regions for the Eu dependence of dT{sub MI}/dP were found: (i) for x ≤ 0.15, dT{sub MI}/dP is nearly constant and ∼−4.3 K/kbar; (ii) for x ≥ 0.15, dT{sub MI}/dP increases with x and it seems to reach a saturation value ∼−6.2 K/kbar for the x = 0.35 sample. This change is accompanied with a strong decrease in the thermal hysteresis in ρ(P, T) between the cooling and warming cycles, observed in the vicinity of T{sub MI}. The entropy change (ΔS) at T{sub MI} for the sample x = 0, estimated by using the dT{sub MI}/dP data and the Clausius-Clapeyron equation, resulted in ΔS ∼ 1.2 J/mol K, a value in line with specific heat measurements. When the Eu concentration is increased, the antiferromagnetic (AF) and the MI transitions are separated in temperature, permitting that an estimate of the entropy change due to the AF/Paramagnetic transition be carried out, yielding ΔS{sub M} ∼ 200 mJ/mol K. This value is much smaller than that expected for a s = 1/2 spin system. The analysis of ρ(P, T) and C{sub P}(T) data indicates that the entropy change at T{sub MI} is mainly due to the electronic delocalization and not related to the AF transition.

  9. Enthalpy-entropy compensation based on isotherms of mango Compensação entalpia-entropia baseada nas isotermas de mango

    Directory of Open Access Journals (Sweden)

    Javier Telis-Romero

    2005-06-01

    Full Text Available Moisture equilibrium data of mango pulp were determined using the static gravimetric method. Adsorption and desorption isotherms were obtained in the range of 30-70 ºC, to water activities (a w from 0.02 to 0.97. The application of the GAB model to the experimental results, using direct nonlinear regression analysis, provided agreement between experimental and calculated values. The net isosteric heat of sorption was estimated from equilibrium sorption data, using the Clausius-Clapeyron equation. Isosteric heats of sorption were found to increase with increasing temperature and could be well adjusted by an exponential relationship. The enthalpy-entropy compensation theory was applied to sorption isotherms and plots of deltaH versus deltaS provided the isokinetic temperatures, indicating an enthalpy controlled sorption process.Dados de equilíbrio da umidade da polpa de manga foram determinados utilizando-se o método estático gravimétrico. As isotermas de adsorção e dessorção foram obtidas na faixa de 30-70 ºC e as atividades de água (a w de 0,02 a 0,97. A utilização do modelo de GAB nos resultados experimentais, através da análise de regressão não linear, proporcionou um bom ajuste entre os dados experimentais e os valores calculados. O calor isostérico de sorção foi estimado a partir dos dados de equilíbrio de sorção, utilizando-se a equação de Clausius-Clayperon. Notou-se que os calores isostéricos de sorção crescem com o aumento da temperatura e pode ser bem ajustado através de uma relação exponencial. A teoria da compensação entalpia-entropia foi aplicada às isotermas de sorção e gráficos deltaH versus deltaS forneceram as temperaturas isocinéticas, indicando um processo de sorção entalpicamente controlado.

  10. Ⅱ型二元氢气水合物相边界条件测定及分解焓计算%Measurement of Phase Boundary Conditions and Calculation of Dissociation Enthalpy of sⅡ Binary Hydrogen Hydrate

    Institute of Scientific and Technical Information of China (English)

    杜建伟; 李栋梁; 陈玉凤; 梁德青; 李新军

    2011-01-01

    Experiments on Sii binary hydrogen hydrate phase boundary conditions of the cyclo-pentane/tetrahydrofuran/tert-butylamine/acetone-hydrogen-water system were conducted in a high pressure visual sapphire reactor to obtain the phase equilibrium data for hydrogen hydrate. On the basis of the experimental data, the dissociation enthalpy of Sii binary hydrogen hydrate was calculated with the Clausius-Clapeyron equation. The results show that the phase equilibrium pressure of Sii binary hydrogen hydrate is one-tenth or even more lower that of hydrogen hydrate without additives. The dissociation enthalpy of Sii binary hydrogen hydrate is related to the additives and the phase equilibrium temperature, and with the same kind of additives the dissociation enthalpy of the binary hydrogen hydrate increases with the phase equilibrium temperature. The results of this study may lead to the development of hydrogen storage by hydrate technique.%针对水合物法储氢存在相平衡数据较少及其分解焓直接实验测定非常困难的问题,在已有的高压可视蓝宝石反应釜中测定了环戊烷/丙酮/叔丁胺/四氢呋喃-水-氢气三组分体系中的水合物相边界条件,并结合Clausius-Clapeyron方程计算确定了4种Ⅱ型二元氢气水合物的分解焓.结果表明:含环戊烷/丙酮/叔丁胺/四氢呋喃的Ⅱ型二元氢气水合物的相平衡压力可降低至纯氢气水合物同温度下相平衡压力的1/10甚至更多.二元氢气水合物的分解焓与添加剂和相平衡温度有关,对于同一种添加剂,随着相平衡温度的升高,所得到的二元氢气水合物分解焓增加.二元氢气水合物相平衡边界和分解焓的测定对进一步研究和开发水合物法储氢技术具有重要价值.

  11. Etude de fibres nickel-titane superelastiques et mise en forme d'un composite avec ces fibres

    Science.gov (United States)

    Piedboeuf, Marie Christine

    1997-11-01

    Generally, in a matrix composite, the introduction of the fibre reinforcement, while increasing the strength, reduces the energy absorption of the matrix. The use of a certain amount of SMA reinforcement, with a damping coefficient to some of the polymers, should give better damping capacities while maintaining a high strength. The main goal of this research is thus to study this possibility while gaining a better understanding of the SMA fibers and of their behaviour while embedded in a polymeric matrix. The first part of this project is the throughout study of the behaviour of the SMA fibers while the second part is the preparation and the study of the NiTi/PU composite. In order to be used as fiber reinforcement, very small diameter wires are needed and 100 mum NiTi wires were chosen. These wires are austenitic at room temperature, and thus have a superelastic behaviour. The first part of the research is the study of the behaviour of these wires. They are tested in tensile solicitation, this being the simplest solicitation to study. All the samples are precycled to stabilize the superelastic effect. The results show that the effect of an increase in temperature is to increase the transformation stresses as predicted by the Clausius-Clapeyron equation. Strain rate effects observed in shape memory alloy are usually attributed to a temperature effect resulting from the exo-, endothermic phase transformation. To determine the contribution of temperature effect to the observed strain rate effect, a thermal analysis is performed in chapter 3. To study their damping capacity, these wires are tested in harmonic tensile solicitation and the dissipated energy as well as the loss factor are measured. An extensive study of the effect of frequency and strain amplitude as well as the temperature on these parameters is done in chapter 4. For the last part of the research, NiTi/PU composites are prepared and tested in tensile solicitation to the same strain rate as well as

  12. Thermal dissociation behavior and dissociation enthalpies of methane-carbon dioxide mixed hydrates

    Energy Technology Data Exchange (ETDEWEB)

    Kwon, T.H.; Kneafsey, T.J.; Rees, E.V.L.

    2011-02-15

    Replacement of methane with carbon dioxide in hydrate has been proposed as a strategy for geologic sequestration of carbon dioxide (CO{sub 2}) and/or production of methane (CH{sub 4}) from natural hydrate deposits. This replacement strategy requires a better understanding of the thermodynamic characteristics of binary mixtures of CH{sub 4} and CO{sub 2} hydrate (CH{sub 4}-CO{sub 2} mixed hydrates), as well as thermophysical property changes during gas exchange. This study explores the thermal dissociation behavior and dissociation enthalpies of CH{sub 4}-CO{sub 2} mixed hydrates. We prepared CH{sub 4}-CO{sub 2} mixed hydrate samples from two different, well-defined gas mixtures. During thermal dissociation of a CH{sub 4}-CO{sub 2} mixed hydrate sample, gas samples from the head space were periodically collected and analyzed using gas chromatography. The changes in CH{sub 4}-CO{sub 2} compositions in both the vapor phase and hydrate phase during dissociation were estimated based on the gas chromatography measurements. It was found that the CO{sub 2} concentration in the vapor phase became richer during dissociation because the initial hydrate composition contained relatively more CO{sub 2} than the vapor phase. The composition change in the vapor phase during hydrate dissociation affected the dissociation pressure and temperature; the richer CO{sub 2} in the vapor phase led to a lower dissociation pressure. Furthermore, the increase in CO{sub 2} concentration in the vapor phase enriched the hydrate in CO{sub 2}. The dissociation enthalpy of the CH{sub 4}-CO{sub 2} mixed hydrate was computed by fitting the Clausius-Clapeyron equation to the pressure-temperature (PT) trace of a dissociation test. It was observed that the dissociation enthalpy of the CH{sub 4}-CO{sub 2} mixed hydrate lays between the limiting values of pure CH{sub 4} hydrate and CO{sub 2} hydrate, increasing with the CO{sub 2} fraction in the hydrate phase.

  13. Solving Equations of Multibody Dynamics

    Science.gov (United States)

    Jain, Abhinandan; Lim, Christopher

    2007-01-01

    Darts++ is a computer program for solving the equations of motion of a multibody system or of a multibody model of a dynamic system. It is intended especially for use in dynamical simulations performed in designing and analyzing, and developing software for the control of, complex mechanical systems. Darts++ is based on the Spatial-Operator- Algebra formulation for multibody dynamics. This software reads a description of a multibody system from a model data file, then constructs and implements an efficient algorithm that solves the dynamical equations of the system. The efficiency and, hence, the computational speed is sufficient to make Darts++ suitable for use in realtime closed-loop simulations. Darts++ features an object-oriented software architecture that enables reconfiguration of system topology at run time; in contrast, in related prior software, system topology is fixed during initialization. Darts++ provides an interface to scripting languages, including Tcl and Python, that enable the user to configure and interact with simulation objects at run time.

  14. Classical equations for quantum systems

    Energy Technology Data Exchange (ETDEWEB)

    Gell-Mann, M. (Theoretical Astrophysics Group (T-6), Los Alamos National Laboratory, Los Alamos, New Mexico 87545) (United States) (Santa Fe Institute, 1660 Old Pecos Trail, Santa Fe, New Mexico 87501); Hartle, J.B. (Department of Physics, University of California enSanta Barbara, Santa Barbara, (California) 93106)

    1993-04-15

    The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that [ital decohere], which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models.

  15. Quantization of Equations of Motion

    Directory of Open Access Journals (Sweden)

    D. Kochan

    2007-01-01

    Full Text Available The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential. A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail. 

  16. Partial differential equations of mathematical physics

    CERN Document Server

    Sobolev, S L

    1964-01-01

    Partial Differential Equations of Mathematical Physics emphasizes the study of second-order partial differential equations of mathematical physics, which is deemed as the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. The book discusses in detail a wide spectrum of topics related to partial differential equations, such as the theories of sets and of Lebesgue integration, integral equations, Green's function, and the proof of the Fourier method. Theoretical physicists, experimental physicists, mathematicians engaged in pure and applied math

  17. BOUNDARY CONTROL OF MKDV-BURGERS EQUATION

    Institute of Scientific and Technical Information of China (English)

    TIAN Li-xin; ZHAO Zhi-feng; WANG Jing-feng

    2006-01-01

    The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1]. The existence of the solution of MKdV-Burgers equation with the feedback control law was proved. On the base, priori estimates for the solution was given. At last, the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given.

  18. Nonlinear SCHRÖDINGER-PAULI Equations

    Science.gov (United States)

    Ng, Wei Khim; Parwani, Rajesh R.

    2011-11-01

    We obtain novel nonlinear Schrüdinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.

  19. Weierstrass solutions for dissipative BBM equation

    OpenAIRE

    Mancas, Stefan C.; Spradlin, Greg; Khanal, Harihar

    2013-01-01

    In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified BBM equation. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant make the equation integrable in terms of Weiers...

  20. A Note on Indefinite Stochastic Riccati Equations

    CERN Document Server

    Qian, Zhongmin

    2012-01-01

    An indefinite stochastic Riccati Equation is a matrix-valued, highly nonlinear backward stochastic differential equation together with an algebraic, matrix positive definiteness constraint. We introduce a new approach to solve a class of such equations (including the existence of solutions) driven by one-dimensional Brownian motion. The idea is to replace the original equation by a system of BSDEs (without involving any algebraic constraint) whose existence of solutions automatically enforces the original algebraic constraint to be satisfied.

  1. On a complex differential Riccati equation

    Energy Technology Data Exchange (ETDEWEB)

    Khmelnytskaya, Kira V; Kravchenko, Vladislav V [Department of Mathematics, CINVESTAV del IPN, Unidad Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 Mexico (Mexico)], E-mail: vkravchenko@qro.cinvestav.mx

    2008-02-29

    We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schroedinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation such as the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical 'one-dimensional' results, we discuss new features of the considered equation including an analogue of the Cauchy integral theorem.

  2. Solutions manual to accompany Ordinary differential equations

    CERN Document Server

    Greenberg, Michael D

    2014-01-01

    Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps

  3. Homogenization of ordinary and linear transport equations

    OpenAIRE

    Peirone, Roberto

    1996-01-01

    The homogenization of first order ordinary differential equations in $\\mathbb{R}^N$ and associated linear transport equations are studied. We prove the equivalence between $G$-convergence and strong $G$-convergence for the ordinary equations. We give a sufficient condition, which is also necessary in the autonomous case, for the weak homogenization of the linear transport equations. This condition is satisfied when div$_x f=0$.

  4. Some constant solutions to Zamolodchikov's tetrahedron equations

    CERN Document Server

    Hietarinta, Jarmo

    1992-01-01

    In this letter we present constant solutions to the tetrahedron equations proposed by Zamolodchikov. In general, from a given solution of the Yang-Baxter equation there are two ways to construct solutions to the tetrahedron equation. There are also other kinds of solutions. We present some two-dimensional solutions that were obtained by directly solving the equations using either an upper triangular or Zamolodchikov's ansatz.

  5. Introduction to differential equations with dynamical systems

    CERN Document Server

    Campbell, Stephen L

    2011-01-01

    Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Cam

  6. Onsager equations and time dependent neutron transport

    International Nuclear Information System (INIS)

    The diffusion of neutrons following an abrupt, localized temperature fluctuation can be conducted in the framework of Onsager-type transport equations. Considering Onsager equations as a generalized Fick's law, time-dependent particle and energy 'generalized diffusion equations' can be obtained. Aim of the present paper is to obtain the time-dependent diffusion Onsager-type equations for the diffusion of neutrons and to apply them to simple trial cases to gain a feeling for their behaviour. (author)

  7. Some Aspects of Extended Kinetic Equation

    Directory of Open Access Journals (Sweden)

    Dilip Kumar

    2015-09-01

    Full Text Available Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.

  8. Notes on the infinity Laplace equation

    CERN Document Server

    Lindqvist, Peter

    2016-01-01

    This BCAM SpringerBriefs is a treaty of the Infinity-Laplace Equation, which has inherited many features from the ordinary Laplace Equation, and is based on lectures by the author. The Infinity.Laplace Equation has delightful counterparts to the Dirichlet integral, the mean value property, the Brownian motion, Harnack's inequality, and so on. This "fully non-linear" equation has applications to image processing and to mass transfer problems, and it provides optimal Lipschitz extensions of boundary values.

  9. Symmetry Breaking for Black-Scholes Equations

    Institute of Scientific and Technical Information of China (English)

    YANG Xuan-Liu; ZHANG Shun-Li; QU Chang-Zheng

    2007-01-01

    Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.

  10. Symmetry Breaking for Black-Scholes Equations

    Science.gov (United States)

    Yang, Xuan-Liu; Zhang, Shun-Li; Qu, Chang-Zheng

    2007-06-01

    Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.

  11. Symmetry Breaking for Black-Scholes Equations

    International Nuclear Information System (INIS)

    Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.

  12. The Modified Enskog Equation for Mixtures

    NARCIS (Netherlands)

    Beijeren, H. van; Ernst, M.H.

    1973-01-01

    In a previous paper it was shown that a modified form of the Enskog equation, applied to mixtures of hard spheres, should be considered as the correct extension of the usual Enskog equation to the case of mixtures. The main argument was that the modified Enskog equation leads to linear transport coe

  13. Some Functional Equations Originating from Number Theory

    Indian Academy of Sciences (India)

    Soon-Mo Jung; Jae-Hyeong Bae

    2003-05-01

    We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.

  14. Errors in equations for galaxy rotation speeds

    OpenAIRE

    Nicholson, Kenneth F.

    2003-01-01

    Shown are the errors and difficulties of the equations used for galaxy rotation speeds in the book "Galactic Dynamics" (Binney and Tremaine). A usable and accurate set of equations is then presented. The new equations allow easy determination of galaxy mass distribution from the rotation profile with no need for dark matter or any knowledge of galaxy surface light.

  15. Differential Galois Theory of Linear Difference Equations

    OpenAIRE

    Hardouin, Charlotte; Singer, Michael F.

    2008-01-01

    We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions.

  16. Difference Galois theory of linear differential equations

    OpenAIRE

    Di Vizio, Lucia; Hardouin, Charlotte; Wibmer, Michael

    2013-01-01

    We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.

  17. The Schrodinger equation and negative energies

    OpenAIRE

    Bruce, S

    2008-01-01

    We present a nonrelativistic wave equation for the electron in (3+1)-dimensions which includes negative-energy eigenstates. We solve this equation for three well-known instances, reobtaining the corresponding Pauli equation (but including negative-energy eigenstates) in each case.

  18. Symmetry properties of fractional diffusion equations

    Energy Technology Data Exchange (ETDEWEB)

    Gazizov, R K; Kasatkin, A A; Lukashchuk, S Yu [Ufa State Aviation Technical University, Karl Marx strausse 12, Ufa (Russian Federation)], E-mail: gazizov@mail.rb.ru, E-mail: alexei_kasatkin@mail.ru, E-mail: lsu@mail.rb.ru

    2009-10-15

    In this paper, nonlinear anomalous diffusion equations with time fractional derivatives (Riemann-Liouville and Caputo) of the order of 0-2 are considered. Lie point symmetries of these equations are investigated and compared. Examples of using the obtained symmetries for constructing exact solutions of the equations under consideration are presented.

  19. The Effect of Repeaters on Equating

    Science.gov (United States)

    Kim, HeeKyoung; Kolen, Michael J.

    2010-01-01

    Test equating might be affected by including in the equating analyses examinees who have taken the test previously. This study evaluated the effect of including such repeaters on Medical College Admission Test (MCAT) equating using a population invariance approach. Three-parameter logistic (3-PL) item response theory (IRT) true score and…

  20. Contact Structures of Partial Differential Equations

    NARCIS (Netherlands)

    Eendebak, P.T.

    2007-01-01

    We study the geometry of contact structures of partial differential equations. The main classes we study are first order systems of two equations in two independent and two dependent variables and the second order scalar equations in two independent variables. The contact distribution in these two c

  1. Solving Absolute Value Equations Algebraically and Geometrically

    Science.gov (United States)

    Shiyuan, Wei

    2005-01-01

    The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.

  2. TWO PROBLEMS OF HERMITE ELLIPTIC EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    Huaug Feirain

    2009-01-01

    In this article, the author investigates some Hermite elliptic equations in a modified Sobolev space introduced by X. Ding [2]. First, the author shows the existence of a ground state solution of semilinear Hermite elliptic equation. Second, the author studies the eigenvalue problem of linear Hermite elliptic equation in a bounded or unbounded domain.

  3. Some new modular equations and their applications

    Science.gov (United States)

    Yi, Jinhee; Sim, Hyo Seob

    2006-07-01

    Ramanujan derived 23 beautiful eta-function identities, which are certain types of modular equations. We found more than 70 of certain types of modular equations by using Garvan's Maple q-series package. In this paper, we prove some new modular equations which we found by employing the theory of modular form and we give some applications for them.

  4. Non RG logarithms via RG equations

    OpenAIRE

    Malyshev, Dmitry

    2004-01-01

    We compute complete leading logarithms in $\\Phi^4$ theory with the help of Connes and Kreimer RG equations. These equations are defined in the Lie algebra dual to the Hopf algebra of graphs. The results are compared with calculations in parquet approximation. An interpretation of the new RG equations is discussed.

  5. Invalidity of the spectral Fokker-Planck equation forCauchy noise driven Langevin equation

    DEFF Research Database (Denmark)

    Ditlevsen, Ove Dalager

    2004-01-01

    The standard Langevin equation is a first order stochastic differential equation where the driving noise term is a Brownian motion. The marginal probability density is a solution to a linear partial differential equation called the Fokker-Planck equation. If the Brownian motion is replaced by so......-called alpha-stable noise (or Levy noise) the Fokker-Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. In stead it has been attempted to formulate an equation for the characteristic function (the Fourier transform......) corresponding to the density function. This equation is frequently called the spectral Fokker-Planck equation. This paper raises doubt about the validity of the spectral Fokker/Planck equation in its standard formulation. The equation can be solved with respect to stationary solutions in the particular case...

  6. Dual Isomonodromic Problems and Whitham Equations

    CERN Document Server

    Takasaki, K

    1997-01-01

    The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter $\\epsilon$. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the Seiberg- Witten solutions of low energy supersymmetric gauge theories.

  7. The Dirac equation and its solutions

    CERN Document Server

    Bagrov, Vladislav G

    2014-01-01

    Dirac equations are of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly.In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.

  8. Constraint-Preserving Scheme for Maxwell's Equations

    CERN Document Server

    Tsuchiya, Takuya

    2016-01-01

    We derive the discretized Maxwell's equations using the discrete variational derivative method (DVDM), calculate the evolution equation of the constraint, and confirm that the equation is satisfied at the discrete level. Numerical simulations showed that the results obtained by the DVDM are superior to those obtained by the Crank-Nicolson scheme. In addition, we study the two types of the discretized Maxwell's equations by the DVDM and conclude that if the evolution equation of the constraint is not conserved at the discrete level, then the numerical results are also unstable.

  9. Quadratic field equations on the brane

    International Nuclear Information System (INIS)

    It is shown that four-dimensional vacuum Einstein solutions simply embedded in five dimensions obey the Gauss-Bonnet-Einstein field equations: Gab+ αGBab + δ55abαexp[-2χ/√α]GB4 = 0 and the Pauli-Einstein equations Gab - 3αPab/5 = 0, and the Bach-Einstein equations Bab = 0. General equations are calculated for which these and similar results follow. It is briefly argued that such field equations could be significant on large distance scales.

  10. Some Recent Advances in Partial Difference Equations

    CERN Document Server

    Petropoulou, Eugenia N

    2010-01-01

    Lately there is an increasing interest in partial difference equations demonstrated by the enormous amount of research papers devoted to them. The initial reason for this increasing interest was the development of computers and the area of numerical analysis, where partial difference equations arise naturally when discretizing a partial differential equation. The aim of this e-book is to provide some recent advances in the field of partial difference equations. Applications of partial difference equations in numerical analysis and systems theory are also presented. This e-book will be of use t

  11. Generalised connections and higher-spin equations

    CERN Document Server

    Francia, Dario

    2012-01-01

    We consider high-derivative equations obtained setting to zero the divergence of the higher-spin curvatures in metric-like form, showing their equivalence to the second-order equations emerging from the tensionless limit of open string field theory, which propagate reducible spectra of particles with different spins. This result can be viewed as complementary to the possibility of setting to zero a single trace of the higher-spin field strengths, which yields an equation known to imply Fronsdal's equation in the compensator form. Higher traces and divergences of the curvatures produce a whole pattern of high-derivative equations whose systematics is also presented.

  12. Classical Equations for Quantum Systems

    CERN Document Server

    Gell-Mann, Murray; Gell-Mann, Murray; Hartle, James B.

    1993-01-01

    The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e. such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of t...

  13. Stochastic integration and differential equations

    CERN Document Server

    Protter, Philip E

    2003-01-01

    It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...

  14. The equations of medieval cosmology

    Science.gov (United States)

    Buonanno, Roberto; Quercellini, Claudia

    2009-04-01

    In Dantean cosmography the Universe is described as a series of concentric spheres with all the known planets embedded in their rotation motion, the Earth located at the centre and Lucifer at the centre of the Earth. Beyond these "celestial spheres", Dante represents the "angelic choirs" as other nine spheres surrounding God. The rotation velocity increases with decreasing distance from God, that is with increasing Power (Virtù). We show that, adding Power as an additional fourth dimension to space, the modern equations governing the expansion of a closed Universe (i.e. with the density parameter Ω0 > 1) in the space-time, can be applied to the medieval Universe as imaged by Dante in his Divine Comedy. In this representation, the Cosmos acquires a unique description and Lucifer is not located at the centre of the hyperspheres.

  15. The equations of medieval cosmology

    CERN Document Server

    Buonanno, Roberto

    2008-01-01

    In Dantean cosmography the Universe is described as a series of concentric spheres with all the known planets embedded in their rotation motion, the Earth located at the centre and Lucifer at the centre of the Earth. Beyond these "celestial spheres", Dante represents the "angelic choirs" as other nine spheres surrounding God. The rotation velocity increases with decreasing distance from God, that is with increasing Power (Virtu'). We show that, adding Power as an additional fourth dimension to space, the modern equations governing the expansion of a closed Universe (i. e. with the density parameter \\Omega_0>1) in the space-time, can be applied to the medieval Universe as imaged by Dante in his Divine Comedy. In this representation the Cosmos acquires a unique description and Lucifer is not located at the centre of the hyperspheres.

  16. An introduction to differential equations

    CERN Document Server

    Ladde, Anil G

    2012-01-01

    This is a twenty-first century book designed to meet the challenges of understanding and solving interdisciplinary problems. The book creatively incorporates "cutting-edge" research ideas and techniques at the undergraduate level. The book also is a unique research resource for undergraduate/graduate students and interdisciplinary researchers. It emphasizes and exhibits the importance of conceptual understandings and its symbiotic relationship in the problem solving process. The book is proactive in preparing for the modeling of dynamic processes in various disciplines. It introduces a "break-down-the problem" type of approach in a way that creates "fun" and "excitement". The book presents many learning tools like "step-by-step procedures (critical thinking)", the concept of "math" being a language, applied examples from diverse fields, frequent recaps, flowcharts and exercises. Uniquely, this book introduces an innovative and unified method of solving nonlinear scalar differential equations. This is called ...

  17. Model Equations: "Black Box" Reconstruction

    Science.gov (United States)

    Bezruchko, Boris P.; Smirnov, Dmitry A.

    Black box reconstruction is both the most difficult and the most tempting modelling problem when any prior information about an appropriate model structure is lacking. An intriguing thing is that a model capable of reproducing an observed behaviour or predicting further evolution should be obtained only from an observed time series, i.e. "from nothing" at first sight. Chances for a success are not large. Even more so, a "good" model would become a valuable tool to characterise an object and understand its dynamics. Lack of prior information causes one to utilise universal model structures, e.g. artificial neural networks, radial basis functions and algebraic polynomials are included in the right-hand sides of dynamical model equations. Such models are often multi-dimensional and involve quite many free parameters.

  18. Equation of state of beryllium

    International Nuclear Information System (INIS)

    A new, wide-range equation of state (EOS) has been constructed for Be. The composite theoretical model incorporates ionization equilibrium and condensed-matter and multiphase physics. It also satisfies all thermodynamic equilibrium constraints. The theoretical EOS has been compared with all available high-pressure and high-temperature Be data, and satisfactory agreement is generally achieved. The most interesting feature is the theoretical prediction of melting at just below 220 GPa (2 Mb), indicating an extremely wide pressure range for solid Be. A striking feature is the appearance of shell-structure effects in physical-process paths: 2 large loops appear on the principal Hugoniot and the behavior of release isentropes from rho = rho0 is significantly affected

  19. Structural equation modeling in epidemiology

    Directory of Open Access Journals (Sweden)

    Leila Denise Alves Ferreira Amorim

    2010-12-01

    Full Text Available Structural equation modeling (SEM is an important statistical tool for evaluating complex relations in several research areas. In epidemiology, the use and discussion of SEM have been limited thus far. This article presents basic principles and concepts in SEM, including an application using epidemiological data analysis from a study on the determinants of cognitive development in young children, considering constructs related to organization of the child's home environment, parenting style, and the child's health status. The relations between the constructs and cognitive development were measured. The results showed a positive association between psychosocial stimulus at home and cognitive development in young children. The article presents the contributions by SEM to epidemiology, highlighting the need for an a priori theoretical model for improving the study of epidemiological questions from a new perspective.

  20. Inferring Mathematical Equations Using Crowdsourcing.

    Directory of Open Access Journals (Sweden)

    Szymon Wasik

    Full Text Available Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game-so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical equations that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an equation that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players.

  1. A New Route to the Majorana Equation

    Directory of Open Access Journals (Sweden)

    Eckart Marsch

    2013-09-01

    Full Text Available In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group.

  2. The tanh-coth method combined with the Riccati equation for solving non-linear equation

    Energy Technology Data Exchange (ETDEWEB)

    Bekir, Ahmet [Dumlupinar University, Art-Science Faculty, Department of Mathematics, Kuetahya (Turkey)], E-mail: abekir@dumlupinar.edu.tr

    2009-05-15

    In this work, we established abundant travelling wave solutions for some non-linear evolution equations. This method was used to construct solitons and traveling wave solutions of non-linear evolution equations. The tanh-coth method combined with Riccati equation presents a wider applicability for handling non-linear wave equations.

  3. Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

    Indian Academy of Sciences (India)

    Yusuf Gurefe; Abdullah Sonmezoglu; Emine Misirli

    2011-12-01

    In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the (, ) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.

  4. Difference equations and cluster algebras I: Poisson bracket for integrable difference equations

    CERN Document Server

    Inoue, Rei

    2010-01-01

    We introduce the cluster algebraic formulation of the integrable difference equations, the discrete Lotka-Volterra equation and the discrete Liouville equation, from the view point of the general T-system and Y-system. We also study the Poisson structure for the cluster algebra, and give the associated Poisson bracket for the two difference equations.

  5. Similarities and Differences Between Freundlich Kinetic Equation and Two—Constant Equation

    Institute of Scientific and Technical Information of China (English)

    ZHANGZENGQIANG; ZHANGYIPING

    1999-01-01

    A mathematical expression of Freundlich kinetic equation,lnS=A'+B'lnt,is presented,and the physical meanings of its parameters are indicated.Although the Freundlich kinetic equation and the two-constant equation are the same in the form,the derivation of the Freundlich kinetic equation is precise,while the derivation of the two-constant equation has some contradictions and is unreasonable,And it is suggested that the Freundlich kinetic equation should have prority over the two-constant equation to be used.

  6. The properties of the first equation of the Vlasov chain of equations

    Science.gov (United States)

    Perepelkin, E. E.; Sadovnikov, B. I.; Inozemtseva, N. G.

    2015-05-01

    A derivation of the first Vlasov equation as a well-known Schrödinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred. A physical meaning of the phase of the wave function which is a scalar potential of the probabilistic flow velocity is demonstrated. Occurrence of the velocity potential vortex component leads to the Pauli equation for one of the spinar components. A scheme for the construction of the Schrödinger equation solution from the Vlasov equation solution and vice-versa is shown. A process of introduction of the potential to the Schrödinger equation and its interpretation are given. The analysis of the potential properties gives us the Maxwell equation, the equation of the kinematic point movement, and the equation for movement of the medium within electromagnetic fields.

  7. Dust levitation about Itokawa's equator

    Science.gov (United States)

    Hartzell, C.; Zimmerman, M.; Takahashi, Y.

    2014-07-01

    levitation about Itokawa, we must include accurate plasma and gravity models. We use a 2D PIC code (described in [8]) to model the plasma environment about Itokawa's equator. The plasma model includes photoemission and shadowing. Thus, we model the plasma environment for various solar incidence angles. The plasma model gives us the 2D electric field components and the plasma potential. We model the gravity field around the equatorial cross-section using an Interior Gravity model [9]. The gravity model is based on the shape model acquired by the Hayabusa mission team and, unlike other models, is quick and accurate close to the surface of the body. Due to the nonspherical shape of Itokawa, the electrostatic force and the gravity may not be collinear. Given our accurate plasma and gravity environments, we are able to simulate the trajectories of dust grains about the equator of Itokawa. When modeling the trajectories of the grains, the current to the grains is calculated using Nitter et al.'s formulation [10] with the plasma sheath parameters provided by our PIC model (i.e., the potential minimum, the potential at the surface, and the sheath type). Additionally, we are able to numerically locate the equilibria about which dust grains may levitate. Interestingly, we observe that equilibria exist for grains up to 20 microns in radius about Itokawa's equator when the Sun is illuminating Itokawa's 'otter tail'. This grain size is significantly larger than the stably levitating grains we observed using our 1D plasma and gravity models. Conclusions and Future Work: The possibility of dust levitation above asteroids has implications both for our understanding of their evolution and for the design of future missions to these bodies. Using detailed gravity and plasma models, we are above to propagate the trajectories of dust particles about Itokawa's equator and identify the equilibria about which these grains will levitate. Using these simulations, we see that grains up to 20 microns

  8. A new application of Riccati equation to some nonlinear evolution equations

    Energy Technology Data Exchange (ETDEWEB)

    Geng Tao [School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876 (China)], E-mail: taogeng@yahoo.com.cn; Shan Wenrui [School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876 (China)

    2008-03-03

    By means of symbolic computation, a new application of Riccati equation is presented to obtain novel exact solutions of some nonlinear evolution equations, such as nonlinear Klein-Gordon equation, generalized Pochhammer-Chree equation and nonlinear Schroedinger equation. Comparing with the existing tanh methods and the proposed modifications, we obtain the exact solutions in the form as a non-integer power polynomial of tanh (or tan) functions by using this method, and the availability of symbolic computation is demonstrated.

  9. New Type of Soliton Equation Described Some Statistical Distributions and Nonlinear Equations Unified Quantum Statistics

    OpenAIRE

    Chang, Yi-Fang

    2009-01-01

    We proposed a new type of soliton equation, whose solutions may describe some statistical distributions, for example, Cauchy distribution, normal distribution and student distribution, etc. The equation possesses two characters. Further, from an extension of this type of equation we may obtain the exponential distribution, and the Fermi-Dirac distribution in quantum statistics. Moreover, by using the method of the soliton-solution, the nonlinear Klein-Gordon equation and nonlinear Dirac equat...

  10. Renormalization group flow equations from the 4PI equations of motion

    CERN Document Server

    Carrington, M E

    2013-01-01

    The 4PI effective action provides a a hierarchy of integral equations which have the form of Bethe-Salpeter equations. The vertex functions obtained from these equations can be used to truncate the exact renormalization group flow equations. This truncation has the property that the flow is a total derivative with respect to the flow parameter and is equivalent to solving the nPI equations of motion. This result establishes a direct connection between two non-perturbative methods.

  11. Solving equations through particle dynamics

    Science.gov (United States)

    Edvardsson, S.; Neuman, M.; Edström, P.; Olin, H.

    2015-12-01

    The present work evaluates a recently developed particle method (DFPM). The basic idea behind this method is to utilize a Newtonian system of interacting particles that through dissipation solves mathematical problems. We find that this second order dynamical system results in an algorithm that is among the best methods known. The present work studies large systems of linear equations. Of special interest is the wide eigenvalue spectrum. This case is common as the discretization of the continuous problem becomes dense. The convergence rate of DFPM is shown to be in parity with that of the conjugate gradient method, both analytically and through numerical examples. However, an advantage with DFPM is that it is cheaper per iteration. Another advantage is that it is not restricted to symmetric matrices only, as is the case for the conjugate gradient method. The convergence properties of DFPM are shown to be superior to the closely related approach utilizing only a first order dynamical system, and also to several other iterative methods in numerical linear algebra. The performance properties are understood and optimized by taking advantage of critically damped oscillators in classical mechanics. Just as in the case of the conjugate gradient method, a limitation is that all eigenvalues (spring constants) are required to be of the same sign. DFPM has no other limitation such as matrix structure or a spectral radius as is common among iterative methods. Examples are provided to test the particle algorithm's merits and also various performance comparisons with existent numerical algorithms are provided.

  12. Stochastic differential equation model to Prendiville processes

    Energy Technology Data Exchange (ETDEWEB)

    Granita, E-mail: granitafc@gmail.com [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); Bahar, Arifah [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); UTM Center for Industrial & Applied Mathematics (UTM-CIAM) (Malaysia)

    2015-10-22

    The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.

  13. Numerical methods for ordinary differential equations

    CERN Document Server

    Butcher, John C

    2008-01-01

    In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author''s pioneering text is fully revised and updated to acknowledge many of these developments.  It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding.  Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler. Features of the book includeIntroductory work on differential and difference equations.A comprehensive introduction to the theory and practice of solving ordinary differential equations numeri...

  14. Diffusion phenomenon for linear dissipative wave equations

    KAUST Repository

    Said-Houari, Belkacem

    2012-01-01

    In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, we prove that for {equation presented} decays with the rate {equation presented} [0,1] faster than that of either u or v, where u is the solution of the linear wave equation with initial data {equation presented} [0,1], and v is the solution of the related heat equation with initial data v 0 = u 0 + u 1. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415-433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by t -γ/2. © European Mathematical Society.

  15. Exponential Attractor for a Nonlinear Boussinesq Equation

    Institute of Scientific and Technical Information of China (English)

    Ahmed Y. Abdallah

    2006-01-01

    This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space H20(0, 1) × L2(0, 1). The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space H03(0, 1) × H10(0, 1).

  16. Multigrid method for nonlinear poroelasticity equations

    OpenAIRE

    Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, Cornelis

    2016-01-01

    In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretizati...

  17. Introducing Equational Semantics for Argumentation Networks

    OpenAIRE

    Gabbay, Dov M.

    2011-01-01

    This paper provides equational semantics for Dung’s argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the “extensions” of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, etc, etc. ...

  18. Reaction diffusion equations with boundary degeneracy

    Directory of Open Access Journals (Sweden)

    Huashui Zhan

    2016-03-01

    Full Text Available In this article, we consider the reaction diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\Delta A(u,\\quad (x,t\\in \\Omega \\times (0,T, $$ with the homogeneous boundary condition. Inspired by the Fichera-Oleinik theory, if the equation is not only strongly degenerate in the interior of $\\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.

  19. Involutive reductions and solutions of differential equations

    OpenAIRE

    Engelmann, Joachim

    2003-01-01

    This work introduces the so-called involutive reduction procedure to simplify and solve differential equations. The method is based on symmetry analysis, which was developed by the Norwegian mathematician Sophus Lie. The involutive reduction procedure uses the given differential equation itself and the invariant surface condition of this differential equation. The invariant surface condition incorporates the symmetries of the problem. This coupled system of partial differential equati...

  20. Differential equations inverse and direct problems

    CERN Document Server

    Favini, Angelo

    2006-01-01

    DEGENERATE FIRST ORDER IDENTIFICATION PROBLEMS IN BANACH SPACES A NONISOTHERMAL DYNAMICAL GINZBURG-LANDAU MODEL OF SUPERCONDUCTIVITY. EXISTENCE AND UNIQUENESS THEOREMSSOME GLOBAL IN TIME RESULTS FOR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEMSFOURTH ORDER ORDINARY DIFFERENTIAL OPERATORS WITH GENERAL WENTZELL BOUNDARY CONDITIONSTUDY OF ELLIPTIC DIFFERENTIAL EQUATIONS IN UMD SPACESDEGENERATE INTEGRODIFFERENTIAL EQUATIONS OF PARABOLIC TYPE EXPONENTIAL ATTRACTORS FOR SEMICONDUCTOR EQUATIONSCONVERGENCE TO STATIONARY STATES OF SOLUTIONS TO THE SEMILINEAR EQUATION OF VISCOELASTICITY ASYMPTOTIC BEHA

  1. Horizon Thermodynamics from Einstein's Equation of State

    CERN Document Server

    Hansen, Devin; Mann, Robert

    2016-01-01

    By regarding the Einstein equations as equation(s) of state, we demonstrate that a full cohomogeneity horizon first law can be derived in horizon thermodynamics. In this approach both the entropy and the free energy are derived concepts, while the standard (degenerate) horizon first law is recovered by a Legendre projection from the more general one we derive. These results readily generalize to higher curvature gravities and establish a way of how to formulate consistent black hole thermodynamics without conserved charges.

  2. The Nonlinear Convection—Reaction—Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    ShiminTANG; MaochangCUI; 等

    1996-01-01

    A nonlinear convection-reaction-diffusion equation is used as a model equation of the El Nino events.In this model,the effects of convection,turbulent diffusion,linear feed-back and nolinear radiation on the anomaly of Sea Surface Temperature(SST) are considered.In the case of constant convection,this equation has exact kink-like travelling wave solutions,which can be used to explain the history of an El Nino event.

  3. Symmetry Coefficients of Semilinear Partial Differential Equations

    OpenAIRE

    Freire, Igor Leite; Martins, Antonio Carlos Gilli

    2008-01-01

    We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial differential equations in Analysis, Geometry and Mathematical - Physics are given in order to enlighte...

  4. Partial differential equations of parabolic type

    CERN Document Server

    Friedman, Avner

    2008-01-01

    This accessible and self-contained treatment provides even readers previously unacquainted with parabolic and elliptic equations with sufficient background to understand research literature. Author Avner Friedman - Director of the Mathematical Biosciences Institute at The Ohio State University - offers a systematic and thorough approach that begins with the main facts of the general theory of second order linear parabolic equations. Subsequent chapters explore asymptotic behavior of solutions, semi-linear equations and free boundary problems, and the extension of results concerning fundamenta

  5. Estimating Structural Change in Linear Simultaneous Equations

    OpenAIRE

    Huang Weihong; Zhang Yang

    2004-01-01

    Tests and estimation for changes in the coefficients of linear regression models, particularly the analysis of covariance and the Chow tests, are well known to econometricians and are widely used. This paper demonstrates that analogous estimation can also be constructed in simultaneous equation models when equations are estimated by common estimator like OLS, 2SLS and LIML. In the present paper, we discuss the problem of estimating structural changes in equations from a simultaneous structura...

  6. The Boltzmann equation in the difference formulation

    Energy Technology Data Exchange (ETDEWEB)

    Szoke, Abraham [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks III, Eugene D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

    2015-05-06

    First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.

  7. Equation of state of HE detonation products

    OpenAIRE

    Nadykto B.A.

    2011-01-01

    Computational analysis of steady-state HE detonation parameters is possible if one knows the equation of state of detonation products and thermal energy released at the Jouget point during detonation. There are a number of equations of state of HE detonation products that result from different assumptions concerning detonated material conditions. The paper considers one more version of the equation of state for HE detonation products.

  8. The model equation of soliton theory

    OpenAIRE

    Adler, V. E.; Shabat, A. B.

    2007-01-01

    We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most interesting result is the simple equation for the generating function of the hierarchy which defines the dynamics for the negative times and also has applications to the second order spectral problems. A rather general theory of integrable 1+1-dimensional equations c...

  9. Dynamic equations for curved submerged floating tunnel

    Institute of Scientific and Technical Information of China (English)

    2007-01-01

    In virtue of reference Cartesian coordinates, geometrical relations of spatial curved structure are presented in orthogonal curvilinear coordinates. Dynamic equations for helical girder are derived by Hamilton principle. These equations indicate that four generalized displacements are coupled with each other. When spatial structure degenerates into planar curvilinear structure, two generalized displacements in two perpendicular planes are coupled with each other. Dynamic equations for arbitrary curvilinear structure may be obtained by the method used in this paper.

  10. Geometrical and Graphical Solutions of Quadratic Equations.

    Science.gov (United States)

    Hornsby, E. John, Jr.

    1990-01-01

    Presented are several geometrical and graphical methods of solving quadratic equations. Discussed are Greek origins, Carlyle's method, von Staudt's method, fixed graph methods and imaginary solutions. (CW)

  11. Kinetic Boltzmann, Vlasov and Related Equations

    CERN Document Server

    Sinitsyn, Alexander; Vedenyapin, Victor

    2011-01-01

    Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in

  12. Fuzzy Symmetric Solutions of Fuzzy Matrix Equations

    OpenAIRE

    Xiaobin Guo; Dequan Shang

    2012-01-01

    The fuzzy symmetric solution of fuzzy matrix equation AX˜=B˜, in which A is a crisp m×m nonsingular matrix and B˜ is an m×n fuzzy numbers matrix with nonzero spreads, is investigated. The fuzzy matrix equation is converted to a fuzzy system of linear equations according to the Kronecker product of matrices. From solving the fuzzy linear system, three types of fuzzy symmetric solutions of the fuzzy matrix equation are derived. Finally, two examples are given to illustrate the proposed method....

  13. Folding transformations for quantum Painleve equations

    Energy Technology Data Exchange (ETDEWEB)

    Ramani, A [Centre de Physique Theorique, Ecole Polytechnique, CNRS, 91128 Palaiseau (France); Nagoya, H [Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914 Tokyo (Japan); Grammaticos, B [IMNC, Universite Paris VII-Paris XI, CNRS, UMR 8165, Bat. 104, 91406 Orsay (France); Tamizhmani, T [Department of Mathematics, Kanchi Mamunivar Centre for Postgraduate Studies, Puducherry (India)

    2009-03-06

    We examine a special property of Painleve equations, namely possessing folding transformations. The latter are relations of the solution of a given Painleve equation to the square of that of some other, which can be the same as the initial one. They generally exist only for special values of the parameters of a given equation. The present setting will be that of the quantum Painleve equations, which are systems where the dependent variables are noncommuting objects. Both continuous and discrete cases are analysed and the folding transformations are established in a perfect parallel between continuous and discrete systems.

  14. Folding transformations for quantum Painleve equations

    International Nuclear Information System (INIS)

    We examine a special property of Painleve equations, namely possessing folding transformations. The latter are relations of the solution of a given Painleve equation to the square of that of some other, which can be the same as the initial one. They generally exist only for special values of the parameters of a given equation. The present setting will be that of the quantum Painleve equations, which are systems where the dependent variables are noncommuting objects. Both continuous and discrete cases are analysed and the folding transformations are established in a perfect parallel between continuous and discrete systems

  15. The Raychaudhuri equations: A brief review

    Indian Academy of Sciences (India)

    Sayan Kar; Soumitra Sengupta

    2007-07-01

    We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essential features of the original article by Raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate non-Riemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion (Einstein–Cartan–Sciama–Kibble theory). Finally, we give an overview of some recent applications of these equations in general relativity, quantum field theory, string theory and the theory of relativisitic membranes. We conclude with a summary and provide our own perspectives on directions of future research.

  16. Fractional Schrödinger equation.

    Science.gov (United States)

    Laskin, Nick

    2002-11-01

    Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrödinger equations. PMID:12513557

  17. New integrability case for the Riccati equation

    CERN Document Server

    Mak, M K

    2012-01-01

    A new integrability condition of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ is presented. By introducing an auxiliary equation depending on a generating function $f(x)$, the general solution of the Riccati equation can be obtained if the coefficients $a(x)$, $b(x)$, $c(x)$, and the function $f(x)$ satisfy a particular constraint. The validity and reliability of the method are tested by obtaining the general solutions of some Riccati type differential equations. Some applications of the integrability conditions for the case of the damped harmonic oscillator with time dependent frequency, and for solitonic wave, are briefly discussed.

  18. Mass continuity equation in the electromagnetic field

    CERN Document Server

    Weng, Ying

    2009-01-01

    A theoretical method with the quaternion algebra was presented to derive the mass continuity equation from the linear momentum. It predicts that the strength of electromagnetic field and the velocity have the impact on the mass continuity equation. In the gravitational field and electromagnetic field, the mass continuity equation will change with the electromagnetic field strength, gravitational field strength, linear momentum, electric current, and the speed of light. The deduction can explain why the field strength has an influence on the anomalous transport about the mass continuity equation in the plasma and electrolytes etc.

  19. Algebras with Parastrophically Uncancellable Quasigroup Equations

    Directory of Open Access Journals (Sweden)

    Amir Ehsani

    2016-07-01

    Full Text Available We consider 48 parastrophically uncancellable quadratic functional equations with four object variables and two quasigroup operations in two classes: balanced non-Belousov (consists of 16 equations and non-balanced non-gemini (consists of 32 equations. A linear representation of a group (Abelian group for a pair of quasigroup operations satisfying one of these parastrophically uncancellable quadratic equations is obtained. As a consequence of these results, a linear representation for every operation of a binary algebra satisfying one of these hyperidentities is obtained.

  20. Numerical Solutions of Fractional Boussinesq Equation

    Institute of Scientific and Technical Information of China (English)

    WANG Qi

    2007-01-01

    Based upon the Adomian decomposition method,a scheme is developed to obtain numerical solutions of a fractional Boussinesq equation with initial condition,which is introduced by replacing some order time and space derivatives by fractional derivatives.The fractional derivatives are described in the Caputo sense.So the traditional Adomian decomposition method for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional differential equations.The solutions of our model equation are calculated in the form of convergent series with easily computable components.

  1. Blending Brownian motion and heat equation

    CERN Document Server

    Cristiani, Emiliano

    2015-01-01

    In this short communication we present an original way to couple the Brownian motion and the heat equation. More in general, we suggest a way for coupling the Langevin equation for a particle, which describes a single realization of its trajectory, with the associated Fokker-Planck equation, which instead describes the evolution of the particle's probability density function. Numerical results show that it is indeed possible to obtain a regularized Brownian motion and a Brownianized heat equation still preserving the global statistical properties of the solutions. The results also suggest that the more macroscale leads the dynamics the more one can reduce the microscopic degrees of freedom.

  2. General particle transport equation. Final report

    International Nuclear Information System (INIS)

    The general objectives of this research are as follows: (1) To develop fundamental models for fluid particle coalescence and breakage rates for incorporation into statistically based (Population Balance Approach or Monte Carlo Approach) two-phase thermal hydraulics codes. (2) To develop fundamental models for flow structure transitions based on stability theory and fluid particle interaction rates. This report details the derivation of the mass, momentum and energy conservation equations for a distribution of spherical, chemically non-reacting fluid particles of variable size and velocity. To study the effects of fluid particle interactions on interfacial transfer and flow structure requires detailed particulate flow conservation equations. The equations are derived using a particle continuity equation analogous to Boltzmann's transport equation. When coupled with the appropriate closure equations, the conservation equations can be used to model nonequilibrium, two-phase, dispersed, fluid flow behavior. Unlike the Eulerian volume and time averaged conservation equations, the statistically averaged conservation equations contain additional terms that take into account the change due to fluid particle interfacial acceleration and fluid particle dynamics. Two types of particle dynamics are considered; coalescence and breakage. Therefore, the rate of change due to particle dynamics will consider the gain and loss involved in these processes and implement phenomenological models for fluid particle breakage and coalescence

  3. On q-Difference Riccati Equations and Second-Order Linear q-Difference Equations

    Directory of Open Access Journals (Sweden)

    Zhi-Bo Huang

    2013-01-01

    Full Text Available We consider q-difference Riccati equations and second-order linear q-difference equations in the complex plane. We present some basic properties, such as the transformations between these two equations, the representations and the value distribution of meromorphic solutions of q-difference Riccati equations, and the q-Casorati determinant of meromorphic solutions of second-order linear q-difference equations. In particular, we find that the meromorphic solutions of these two equations are concerned with the q-Gamma function when q∈ℂ such that 0<|q|<1. Some examples are also listed to illustrate our results.

  4. Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation

    Science.gov (United States)

    Naher, Hasibun; Abdullah, Farah Aini; Mohyud-Din, Syed Tauseef

    2013-05-01

    In this article, the generalized Riccati equation mapping together with the basic (G'/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G'(ϕ) = h + f G(ϕ) + g G2(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.

  5. On the Equivalence of the Massless DKP equation and the Maxwell Equations in the Shuwer

    CERN Document Server

    Salti, M; Salti, Mustafa; Havare, Ali

    2005-01-01

    In this paper, a general relativistic wave equation is written to deal with electromagnetic waves in the background of the Shuwer. We obtain the exact form of this equation in a second order form. On the other hand, by using spinor form of the Maxwell equations the propagation problem is reduced to the solution of the second order differential equation of complex combination of the electric and magnetic fields. For these two different approach, we obtain the spinors in terms of field strength tensor. We show that the Maxwell equations are the equivalence with the mDKP equation in the Shuwer.

  6. On the Existence and the Applications of Modified Equations for Stochastic Differential Equations

    KAUST Repository

    Zygalakis, K. C.

    2011-01-01

    In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive an SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed. © 2011 Society for Industrial and Applied Mathematics.

  7. An Exact Mapping from Navier-Stokes Equation to Schrodinger Equation via Riccati EquationAn Exact Mapping from Navier-Stokes Equation to Schrodinger Equation via Riccati Equation

    Directory of Open Access Journals (Sweden)

    Christianto V.

    2008-01-01

    Full Text Available In the present article we argue that it is possible to write down Schrodinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R.M.Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this proposition.

  8. SOLVABILITY OF SINGULAR CANTILEVER BEAM EQUATION

    Institute of Scientific and Technical Information of China (English)

    2008-01-01

    Two local existence theorems are established for a class of fourth-order two-point boundary value problems with all order derivatives and singularity.Main ingredients are Green function and integral equation.In mechanics,such class of problems is called cantilever beam equation which describes the deflection of an elastic beam fixed at left and freed at right.

  9. The Homoclinic Orbits in Nonlinear Schroedinger Equation

    Institute of Scientific and Technical Information of China (English)

    PengchengXU; BolingGUO; 等

    1998-01-01

    The persistence of Homoclinic orbits for perturbed nonlinear Schroedinger equation with five degree term under een periodic boundary conditions is considered.The exstences of the homoclinic orbits for the truncation equation is established by Melnikov's analysis and geometric singular perturbation theory.

  10. The Homoclinic Orbit Solution for Functional Equation

    Institute of Scientific and Technical Information of China (English)

    LIUShi-Da; FUZun-Tao; 等

    2002-01-01

    In this paper,some examples,such as iterated functional systems,scaling equation of wavelet transform,and invariant measure system,are used to show that the homoclinic orbit solutions exist in the functional equations too.And the solitary wave exists in generalized dynamical systems and functional systems.

  11. The Homoclinic Orbit Solution for Functional Equation

    Institute of Scientific and Technical Information of China (English)

    LIU Shi-Da; FU Zun-Tao; LIU Shi-Kuo; REN Kui

    2002-01-01

    In this paper, some examples, such as iterated functional systems, scaling equation of wavelet transform,and invariant measure system, are used to show that the homoclinic orbit solutions exist in the functional equations too.And the solitary wave exists in generalized dynamical systems and functional systems.

  12. The open boundary equation (discussion paper)

    NARCIS (Netherlands)

    Diederen, D.; Savenije, H.H.G.; Toffolon, M.

    2015-01-01

    We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection) under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave

  13. Multigrid method for nonlinear poroelasticity equations

    NARCIS (Netherlands)

    Luo, P.; Rodrigo, C.; Gaspar, F.J.; Oosterlee, C.W.

    2016-01-01

    In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on

  14. Semigroup methods for evolution equations on networks

    CERN Document Server

    Mugnolo, Delio

    2014-01-01

    This concise text is based on a series of lectures held only a few years ago and originally intended as an introduction to known results on linear hyperbolic and parabolic equations.  Yet the topic of differential equations on graphs, ramified spaces, and more general network-like objects has recently gained significant momentum and, well beyond the confines of mathematics, there is a lively interdisciplinary discourse on all aspects of so-called complex networks. Such network-like structures can be found in virtually all branches of science, engineering and the humanities, and future research thus calls for solid theoretical foundations.      This book is specifically devoted to the study of evolution equations – i.e., of time-dependent differential equations such as the heat equation, the wave equation, or the Schrödinger equation (quantum graphs) – bearing in mind that the majority of the literature in the last ten years on the subject of differential equations of graphs has been devoted to ellip...

  15. Difference methods for stiff delay differential equations

    International Nuclear Information System (INIS)

    Delay differential equations of the form y'(t) = f(y(t), z(t)), where z(t) = [y1(α1(y(t))),..., y/sub n/(α/sub n/(y(t)))]/sup T/ and α/sub i/(y(t)) less than or equal to t, arise in many scientific and engineering fields when transport lags and propagation times are physically significant in a dynamic process. Difference methods for approximating the solution of stiff delay systems require special stability properties that are generalizations of those employed for stiff ordinary differential equations. By use of the model equation y'(t) = py(t) + qy(t-1), with complex p and q, the definitions of A-stability, A( )-stability, and stiff stability have been generalize to delay equations. For linear multistep difference formulas, these properties extend directly from ordinary to delay equations. This straight forward extension is not true for implicit Runge-Kutta methods, as illustrated by the midpoint formula, which is A-stable for ordinary equations, but not for delay equations. A computer code for stiff delay equations was developed using the BDF. 24 figures, 5 tables

  16. Coupling and reduction of the HAWC equations

    DEFF Research Database (Denmark)

    Nim, E.

    2001-01-01

    This report contains a description of a general method for coupling and reduction of the so-called HAWC equations, which constitute the basis equations of motion of the aeroelastic model HAWC used widely by research institutes and industrial companies formore than the ten years. The principal aim...

  17. Acoustofluidics 1: Governing equations in microfluidics

    DEFF Research Database (Denmark)

    Bruus, Henrik

    2011-01-01

    Governing equations for microfluidics and basic flow solutions are presented. Equivalent circuit modeling for determining flow rates in microfluidic networks is introduced.......Governing equations for microfluidics and basic flow solutions are presented. Equivalent circuit modeling for determining flow rates in microfluidic networks is introduced....

  18. A comparison of two equations of state

    CERN Document Server

    Celebonovic, V

    1999-01-01

    The forms of two astrophysically applicable equations of state (EOS) are compared: the EOS proposed within the semiclassical theory of dense matter developed by P.Savic and R.Kasanin,and the universal equation of state introduced by Vinet et al.Some similarities between them are discussed and possibilities of astrophysical tests are pointed out.

  19. xRage Equation of State

    Energy Technology Data Exchange (ETDEWEB)

    Grove, John W. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

    2016-08-16

    The xRage code supports a variety of hydrodynamic equation of state (EOS) models. In practice these are generally accessed in the executing code via a pressure-temperature based table look up. This document will describe the various models supported by these codes and provide details on the algorithms used to evaluate the equation of state.

  20. Linearization of Systems of Nonlinear Diffusion Equations

    Institute of Scientific and Technical Information of China (English)

    KANG Jing; QU Chang-Zheng

    2007-01-01

    We investigate the linearization of systems of n-component nonlinear diffusion equations; such systems have physical applications in soil science, mathematical biology and invariant curve flows. Equivalence transformations of their auxiliary systems are used to identify the systems that can be linearized. We also provide several examples of systems with two-component equations, and show how to linearize them by nonlocal mappings.

  1. Solving Differential Equations Using Modified Picard Iteration

    Science.gov (United States)

    Robin, W. A.

    2010-01-01

    Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

  2. Lie algebras and linear differential equations.

    Science.gov (United States)

    Brockett, R. W.; Rahimi, A.

    1972-01-01

    Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

  3. Modeling helicity dissipation-rate equation

    CERN Document Server

    Yokoi, Nobumitsu

    2016-01-01

    Transport equation of the dissipation rate of turbulent helicity is derived with the aid of a statistical analytical closure theory of inhomogeneous turbulence. It is shown that an assumption on the helicity scaling with an algebraic relationship between the helicity and its dissipation rate leads to the transport equation of the turbulent helicity dissipation rate without resorting to a heuristic modeling.

  4. Quasi-exact Solvability of Dirac Equations

    CERN Document Server

    Ho, Choon-Lin

    2007-01-01

    We present a general procedure for determining quasi-exact solvability of the Dirac and the Pauli equation with an underlying $sl(2)$ symmetry. This procedure makes full use of the close connection between quasi-exactly solvable systems and supersymmetry. The Dirac-Pauli equation with spherical electric field is taken as an example to illustrate the procedure.

  5. The Pauli equation in scale relativity

    CERN Document Server

    Célérier, M N; Celerier, Marie-Noelle; Nottale, Laurent

    2006-01-01

    In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and continuous spacetime. Since such a generalized geometry implies the occurence of new discrete symmetry breakings, this has led us to write Dirac bi-spinors in the form of bi-quaternions (complex quaternions). In the present work, we show that, in scale relativity also, the correct Pauli equation can only be obtained from a non-relativistic limit of the relativistic geodesics equation (which, after integration, becomes the Dirac equation) and not from the non-relativistic formalism (that involves symmetry breakings in a fracta...

  6. Euler's Amazing Way to Solve Equations.

    Science.gov (United States)

    Flusser, Peter

    1992-01-01

    Presented is a series of examples that illustrate a method of solving equations developed by Leonhard Euler based on an unsubstantiated assumption. The method integrates aspects of recursion relations and sequences of converging ratios and can be extended to polynomial equation with infinite exponents. (MDH)

  7. Poisson theory of generalized Bikhoff equations

    Institute of Scientific and Technical Information of China (English)

    Shang Mei; Mei Feng-Xiang

    2009-01-01

    This paper presents a Poisson theory of the generalized Birkhoff equations,including the algebraic structure of the equations,the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.

  8. Discrete Riccati equation solutions: Distributed algorithms

    Directory of Open Access Journals (Sweden)

    D. G. Lainiotis

    1996-01-01

    Full Text Available In this paper new distributed algorithms for the solution of the discrete Riccati equation are introduced. The algorithms are used to provide robust and computational efficient solutions to the discrete Riccati equation. The proposed distributed algorithms are theoretically interesting and computationally attractive.

  9. RICCATI EQUATIONS ON NONCOMMUTATIVE BANACH ALGEBRAS

    NARCIS (Netherlands)

    Curtain, Ruth

    2011-01-01

    Conditions for the existence of a solution of a Riccati equation to be in some prescribed noncommutative involutive Banach algebras are given. The Banach algebras are inverse-closed subalgebras of the space of bounded linear operators on some Hilbert space, and the Riccati equation has an exponentia

  10. Long term dynamics of stochastic evolution equations

    NARCIS (Netherlands)

    Bierkens, Gregorius Nicolaas Johannes Cornelis

    2010-01-01

    Stochastic differential equations with delay are the inspiration for this thesis. Examples of such equations arise in population models, control systems with delay and noise, lasers, economical models, neural networks, environmental pollution and in many other situations. In such models we are often

  11. Diophantine equations related to quasicrystals: a note

    OpenAIRE

    Pelantová, E.; Perelomov, A. M.

    2001-01-01

    We give the general solution of three Diophantine equations in the ring of integer of the algebraic number field ${\\bf Q}[{\\sqr 5}]$. These equations are related to the problem of determination of the minimum distance in quasicrystals with fivefold symmetry.

  12. Regularity of the Gurtin-Pipkin equation

    CERN Document Server

    Ivanov, Sergei A

    2012-01-01

    We study regularity of the solution $\\theta$ to the Gurtin-Pipkin integral-differential equation of the first order in time. The solution smoothness in Sobolev spaces is proved. Also it is proved that the 'perturbation' part, namely, the difference of $\\theta$ and the solution to the corresponding wave equation is smoother than $\\theta$.

  13. ON ALGEBRICO-DIFFERENTIAL EQUATIONS-SOLVING

    Institute of Scientific and Technical Information of China (English)

    WU Wenjun(Wu Wen-tsun)

    2004-01-01

    The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations. As an illustration of the method, the Devil's Problem of Pommaret is solved in details.

  14. Qualitative permanence of Lotka-Volterra equations.

    Science.gov (United States)

    Hofbauer, Josef; Kon, Ryusuke; Saito, Yasuhisa

    2008-12-01

    In this paper, we consider permanence of Lotka-Volterra equations. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra equation whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.

  15. Sonar Equations for Planets and Moons

    NARCIS (Netherlands)

    Ainslie, M.A.; Leighton, T.G.

    2015-01-01

    A set of equations to describe the performance of sonar systems, collectively known as the “sonar equations”, was developed during and after the Second World War. These equations assumed that both the sonar equipment and the object to be detected (usually a submarine) would be submerged in one of Ea

  16. GLOBAL ATTRACTIVITY OF A DIFFERENCE EQUATION

    Institute of Scientific and Technical Information of China (English)

    2011-01-01

    In this paper,we investigate the global stability of all positive solutions to a difference equation.We show that the unique positive equilibrium of the equation is a global attractor with a basin under some certain conditions on the coefficient.

  17. Multisymplectic Geometry for the Seismic Wave Equation

    Institute of Scientific and Technical Information of China (English)

    CHEN Jing-Bo

    2004-01-01

    The multisymplectic geometry for the seismic wave equation is presented in this paper.The local energy conservation law,the local momentum evolution equations,and the multisymplectic form are derived directly from the variational principle.Based on the covariant Legendre transform,the multisymplectic Hamiltonian formulation is developed.Multisymplectic discretization and numerical experiments are also explored.

  18. Quasilinear evolution equations of the third order

    Directory of Open Access Journals (Sweden)

    Andrei V. Faminskii

    2007-11-01

    Full Text Available The present paper is a survey concerned with certain aspects of solvability and well-posedness of initial and initial-boundary value problems for various quasilinear evolution equations of the third order. This class includes, for example, Korteweg-de Vries (KdV and Zakharov-Kuznetsov (ZK equations.

  19. Topologies for neutral functional differential equations.

    Science.gov (United States)

    Melvin, W. R.

    1973-01-01

    Bounded topologies are considered for functional differential equations of the neutral type in which present dynamics of the system are influenced by its past behavior. A special bounded topology is generated on a collection of absolutely continuous functions with essentially bounded derivatives, and an application to a class of nonlinear neutral functional differential equations due to Driver (1965) is presented.

  20. Symbolic Solution of Linear Differential Equations

    Science.gov (United States)

    Feinberg, R. B.; Grooms, R. G.

    1981-01-01

    An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.