Heating-Cooling Asymmetry in the δ-γ Transformation in Plutonium: Clausius-Clapeyron Considerations
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.
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.
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.
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.
Ivancic, Timothy J.; Shaw, Stephen B.
2016-04-01
Numerous papers have shown links between >99th percentile hourly precipitation and daily temperature (Pextreme versus T), often explained using the Clausius-Clapeyron (CC) relationship. The CC relationship predicts an approximately 7% increase in precipitation intensity per degree celsius. However, recent analyses indicate that the Pextreme versus T rate can be larger than the CC prediction. In this work, we analyze the Pextreme versus T rate with an automated method across the contiguous U.S. using station data aggregated on a 161 km grid. To evaluate controls on Pextreme versus T, we isolate convective storms to evaluate whether greater than CC rates are due to the transition between storm types or are a feature of convective storms at high T. We repeat the analysis using dew point to assess whether T control on extreme P is indeed a matter of moisture availability. When evaluated using both T and dew point, the northeastern U.S. is most likely to exhibit a greater than predicted Pextreme versus T rate (57% of the region when using T). At 56% of these points, the > CC rates appeared to occur entirely because of a transition between frontal and convective storms. At 30% of these sites, a greater than CC relationship appeared to occur entirely because of greater than CC scaling in convective intensity. At 11% of sites neither was found to be significant, and at 3% both were found to contribute significantly. This analysis suggests that > CC scaling is not prevalent everywhere in the contiguous U.S., and in regions where it does occur, it can be due to multiple causes.
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
Hydrostatic pressure dependence of transformation temperatures of Ti-Ni-Cu alloys
The transformation start temperatures of shape memory Ti50Ni50-xCux (x = 2, 3, 5, 12.5) alloys under hydrostatic pressures were obtained by resistivity measurements, and their thermodynamical analysis was made by using Clausius-Clapeyron equation. The obtained results are the followings. (1) As hydrostatic pressure increases, transformation temperatures (Ms and Af) of the B2 B19' transformation increase linearly for the 2Cu and 3Cu alloys, but decrease linearly for the 5Cu alloy. (2) For the 12.5Cu alloy, Ms and Af of B2 B19 transformation increase but those of B19 B19' transformation decrease as pressure increases. (3) The copper content dependence of volume change associated with martensitic transformation, which is calculated by Clausius-Clapeyron equation, suggests that the 5Cu alloy transforms in two steps B2 → B19 → B19' although it is believed to transform in one step. (orig.)
HIGH ENERGY RATE EXTRUSION OF URANIUM
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)
Isosteric Sorption Heat Determination for some Starchy Grains
ÖZTEKİN, Serdar; SOYSAL, Yurtsever
2000-01-01
InIn this study, isosteric sorption heat for 8 different starchy grains was determined by the application of the Clausius- Clapeyron equation to sorption isotherms. The latent heat of vaporisation of free water was not significantly different from the isosteric sorption heats of barley, durum wheat, oats, sorghum (31) and sorghum above 14% d.b. moisture content. There was no significant difference among the latent heat of vaporisation of free water and isosteric sorption heats of soft whea...
Understanding the Greenhouse Effect Using Clear vs Cloudy Sky Diurnal Temperature Observations
Tayor, S. V.
2006-12-01
Standard meteorological observations from local airports can provide a tangible example of how the greenhouse effect is a part of everyday life. In the exercise outlined here, students plot diurnal temperature observations to compare the relative magnitude of the greenhouse effect under clear and cloudy-sky conditions, gaining insight into the strength of the greenhouse effect. Contemplation of the relation of surface temperature and humidity with cloud cover leads to a further understading of important atmospheric processes involving the Clausius-Clapeyron equation and terrestrial and solar radiation effects.
First order phase transition at the irreversibility line of Bi2Sr2CaCu2O8
Pastoriza, H.; Goffman, M. F.; Arribére, A.; de La Cruz, F.
1994-05-01
Magnetization and susceptibility measurements show a first order phase transition in the magnetic flux structure at the irreversibility line of Bi2Sr2CaCu2O8+δ single crystals, in the field range 0
Thermochemical study of the monobromonitrobenzene isomers
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.
Thermodynamic Characteristics of Adsorption-Desorption of Methane in 3^{#} Coal Seam of Sihe
Dongmin Ma; Jianchang Zhang; Jianping Bai; Hui Zhang
2014-01-01
A series of methane adsorption-desorption isotherm experiments on anthracite of No. 3 Sihe coal mine were conducted at 20°C, 25°C, 30°C, 35°C and 40°C respectively. Based on Clausius-Clapeyron equation, isosteric heat of adsorption and maximum heat of adsorption has been calculated. These calculations indicate that the maximum heat of adsorption in process of elevated pressure (adsorption) and lowered stress (desorption) is 23.31 KJ/mol and 24.02 KJ/mol, so it belongs to physical adsorption. ...
Pressure Model of Soft Body Simulation
Maciej, Matyka; Mark, Ollila
2004-01-01
Motivated by existing models used for soft body simulation which are rather complex to implement, we present a novel technique which is based on simple laws of physics and gives high quality results in real-time. We base the implementation on simple thermodynamics laws and use the Clausius-Clapeyron state equation for pressure calculation. In addition, this provides us with a pressure force that is accumulated into a force accumulator of a 3D mesh object by using an existing spring-mass engin...
Pressure Model of Soft Body Simulation
Maciej, M; Maciej, Matyka; Mark, Ollila
2004-01-01
Motivated by existing models used for soft body simulation which are rather complex to implement, we present a novel technique which is based on simple laws of physics and gives high quality results in real-time. We base the implementation on simple thermodynamics laws and use the Clausius-Clapeyron state equation for pressure calculation. In addition, this provides us with a pressure force that is accumulated into a force accumulator of a 3D mesh object by using an existing spring-mass engine. Finally after integration of Newtons second law we obtain the behavior of a soft body with fixed or non-fixed air pressure inside of it.
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...
Thermodynamics analysis of aluminum plasma transition induced by hypervelocity impact
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.
Hydrogen vapor pressures from 4 to 30 K: a review
The following properties for the hydrogens from 4 to 300K are reviewed: liquid-gas and solid-gas vapor pressures, triple-point values, virial coefficients, liquid and solid densities, and heats of transformation. We have correlated these properties with as much consistency as the data allow. Empirical equations are given for virial coefficients and densities. Clausius-Clapeyron equations relate vapor pressure and temperature with the other properties. Data are frequently available with H2 and D2 and are sometimes available for HD and T2. We have, therefore, estimated HT and DT values to complete the set. We have also reviewed work on binary H-D mixtures and have estimated the expected behavior for H-T and D-T systems
Methane Hydrate Formation and Dissociation in the Presence of Silica Sand and Bentonite Clay
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.
Ceriani, Roberta; Gani, Rafiqul; Meirelles, A.J.A.
2009-01-01
In the present work a group contribution method is proposed for the estimation of the heat capacity of organic liquids as a function of temperature for fatty compounds found in edible oil and biofuels industries. The data bank used for regression of the group contribution parameters (1395 values......-2085] and the Rowlinson-Bondi equation. Also, the predictive performance of general correlations of heats of vaporization based on the corresponding-states method, such as Carruth and Kobayashi [G.F. Carruth, R. Kobayashi, Ind. Eng. Chem. Fundam. 11 (1972) 509-516], Sivaraman et al. [A. Sivaraman, J.W. Magee, R...... in the prediction of heats of vaporization of fatty compounds based on the vapor pressure model of Ceriani and Meirelles [R. Ceriani. A.J.A. Meirelles, Fluid Phase Equilib. 215 (2004) 227-236] and its combination with the Clausius-Clapeyron equation has been Studied. (C) 2009 Elsevier B.V. All rights reserved....
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.
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.
The microstructure and thermomechanical behavior of Ti50Ni47Fe2.5Nd0.5 shape memory alloys
This work presents a study of a novel Ti50Ni47Fe2.5Nd0.5 shape memory alloy by EPMA, E-R analysis and tensile tests. Structural characterization shows the alloy consists of TiNiFe matrix, Nd3Ni and Ti2Ni intermetallic compounds with Fe solute. The latter two phases are well distributed in the matrix. Compared with Ti50Ni47.5Fe2.5 alloys without rare earth Nd, the novel alloy does not change the two-stage martensitic transformation behavior, however, characteristic temperatures increase rapidly. The critical stress needed for inducing martensitic transformation is near linear relationship with the increasing of the testing temperature above Ms, which is in good agreement with Clausius-Clapeyron equation. The novel alloy exhibits an improved shape memory effect as a consequence of precipitation strengthening and the maximum recoverable strain attains to 7.8%.
Pastoriza, H.; Goffman, M. F.; Arribére, A.; de la Cruz, F.
1994-12-01
Magnetization and susceptibility measurements show a first order phase transition in the magnetic flux structure at the irreversibility line of Bi 2Sr 2CaCu 2O 8 single crystals, in the field range 0 < H < 360 Oe. The resistivity in the c direction drops six orders of magnitude in less that 0.1 K, and there is a discontinuous change in the magnetization at the same temperature. The change in magnetization together with the Clausius-Clapeyron equation gives a variation of entropy of 0.06k B per vortex layer, at the transition. Above 360 Oe the H-T phase diagram is determined by the laminar nature of superconductivity and the irreversibility line is frequency dependent.
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)
Effect of hydrostatic pressure on closed-loop phase behavior of block copolymers
The effect of hydrostatic pressure (P) on closed-loop phase behavior of deuterated polystyrene-block-poly(n-pentyl methacrylate) copolymers [dPS-PnPMA] was investigated by using small-angle neutron scattering and birefringence. For PLDOT) at 175 deg. C, and then an upper order-to-disorder transition temperature (TUODT) at 255 deg. C. With increasing pressure both TLDOT and TUODT were markedly changed, where dTLDOT/dP was 725 deg. C/kbar and dTUODT/dP was -725 deg. C/kbar. These are consistent with predictions by the Clausius-Clapeyron equation using measured values of the volume and enthalpy changes of both transitions. The large pressure coefficients imply that the closed-loop phase behavior observed for PS-PnPMA is an entropic-driven phase transition
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...
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.
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...
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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.
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.
String equation from field equation
Gurovich, V T
1996-01-01
It is shown that the string equation can be obtain from field equations. Such work is performed to scalar field. The equation obtained in nonrelativistic limit describes the nonlinear string. Such string has the effective elasticity connencted with the local string curvature. Some examples of the movement such nonlinear elastic string are considered.
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
Tricomi, FG
2012-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and differential
Various microporous materials such as activated carbons, nano-tubes, synthetic microporous carbons as well as metal organic framework materials are being considered for hydrogen storage applications by means of physical adsorption. To develop materials of practical significance for hydrogen storage it is important to understand the relationships between pore sizes, adsorption energies and adsorption capacities. The pore size distribution (PSD) characterization is traditionally obtained from the analysis of nitrogen adsorption isotherms measured at 77 K. However, a portion of the pores accessible to H2 may not be accessible to N2 at this temperature. Therefore, it was recently proposed to use the DFT analysis of H2 adsorption isotherms to characterize pore structure of materials considered for hydrogen storage applications. In present work, adsorption isotherms of H2 and N2 at cryogenic temperatures are used for the characterization of carbon materials. Adsorption measurements were performed with Autosorb 1 MP (Quantachrome Instruments, Boynton Beach, Florida, USA). As an example, Fig 1 compares PSDs calculated for the activated carbon sample (F400, Calgon Carbon) using combined H2 and N2 data, and using N2 isotherm only. The nitrogen derived PSD does not include certain amount of micropores which are accessible to H2 but not to N2 molecules. Obviously, the difference in the calculated PSDs by the two methods will depend on the actual content of small micropores in a given sample. Carbon adsorption properties can also be characterized by the isosteric heat of adsorption, Qst, related to the adsorption energy and dependent on the carbon pore/surface structure. Fig 2 shows Qst data calculated using the Clausius-Clapeyron equation from H2 isotherms measured at 77 K and 87 K for the carbon molecular sieve CMS 5A (Takeda), oxidized single wall nano-tubes (SWNT), and graphitized carbon black (Supelco). The Qst values decrease with increasing pore sizes. The highest Qst is
Thermodynamic properties of HFO-1234yf (2,3,3,3-tetrafluoropropene)
Tanaka, Katsuyuki; Higashi, Yukihiro [Department of Mechanical Systems and Design Engineering, Iwaki Meisei University, 5-5-1 Iino, Chuodai, Iwaki, Fukushima 970-8551 (Japan)
2010-05-15
Measurements of the thermodynamic properties of HFO-1234yf were conducted. The critical temperature T{sub C}, critical density {rho}{sub C}, and critical pressure P{sub C}, were measured by the visual observation of the meniscus disappearance, and were determined to be 367.85 {+-} 0.01 K, 478 {+-} 3 kg/m{sup 3}, and 3382 {+-} 3 kPa, respectively. Vapor pressures were measured by a batch-type calorimeter with a metal-bellows in the temperature range from 310 K to 360 K and correlated by the Wagner-type equation. Surface tensions were measured by the differential capillary-rise method in the temperature range from 273 K to 340 K and correlated by the van der-Waals type equation. The acentric factor was determined to be 0.280 with the vapor pressure correlation. Based on the critical parameters and acentric factor, saturated vapor and liquid densities were estimated by the Peng-Robinson equation and the Hankinson-Thomson equation, respectively. The heat of vaporization was also calculated from the Clausius-Clapeyron equation. (author)
Hochstadt, Harry
2012-01-01
Modern approach to differential equations presents subject in terms of ideas and concepts rather than special cases and tricks which traditional courses emphasized. No prerequisites needed other than a good calculus course. Certain concepts from linear algebra used throughout. Problem section at end of each chapter.
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).
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
USINT, High Temperature Heat and Mass Transfer on Concrete Surfaces in LMFBR
1 - Description of program or function: USINT was developed to model the thermal response of concrete to very high heating rates such as might occur from sodium spills on concrete surfaces in a breeder reactor. The major phenomena treated are conductive energy transport; chemical decomposition of concrete; and two-phase, three-component heat and mass transfer of the decomposition products: steam, liquid water, and carbon dioxide. The USINT model provides for porosity to increase as water and carbon-dioxide are formed from the concrete. The concrete is treated generally as divided into two basic regions, wet and dry. In the wet region, steam, carbon-dioxide, and liquid water may co-exist, but in the dry region, there is no liquid water. There is also the possibility of a third region in which there is only liquid water and no gases. 2 - Method of solution: The finite difference representations of the partial differential equations include several models for Darcy's two-phase flow. The equations for energy and mass transfer of water and carbon-dioxide are solved by a general implicit procedure that contains the Crank-Nicolson approximation. A modified Clausius-Clapeyron equation is used as the equation of state in the wet region. 3 - Restrictions on the complexity of the problem: Maxima of 150 times for experimental temperature histories, 150 times for experimental pressure histories, 150 times for experimental water release, 20 thermal conductivities, 5 locations for experimental temperature histories, 4 locations for experimental pressure histories
Analysis on two-phase flow behavior for natural circulation system in 5 MW nuclear heating reactor
The experiment was carried out on a test loop HRTL-5 simulating the geometry and system design of the 5 MW Nuclear heating reactor, while 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 differential equation, describing 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. 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. The propagation behavior of flow oscillation was also studied. Calculated results agree with experimental data well at the 5 MW nuclear reactor conditions
1998-09-21
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
Soil temperature effect in calculating attenuation and retardation factors.
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
Melt Flow and Heat Transfer in Laser Drilling
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...
Tricomi, Francesco Giacomo
1957-01-01
This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient. The book is divided into four chapters, with two useful
Martensitic phase transformation in shape-memory alloys
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
Structural and thermodynamic signatures of marine microlayer surfactant films
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.
El-Sharkawy, I.I. [Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga-koen 6-1, Kasuga-shi, Fukuoka 816-8580 (Japan); Mechanical Power Engineering Department, Faculty of Engineering, Mansoura University, El-Mansoura (Egypt); Saha, B.B.; Koyama, S. [Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga-koen 6-1, Kasuga-shi, Fukuoka 816-8580 (Japan); He, J.; Ng, K.C.; Yap, C. [Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent (Singapore)
2008-12-15
Adsorption equilibrium uptake of ethanol onto a highly porous activated carbon based adsorbent, namely Maxsorb III, has been experimentally investigated using a thermo-gravimetric analyzer (TGA) unit over adsorption temperatures ranging from 20 to 60 C. The Dubinin-Astakhov (D-A) equation has been used to correlate the experimental data. Isosteric heat of adsorption is also estimated by using the Clausius-Clapeyron equation. Employing a thermodynamically equilibrium model, the performance of the ideal adsorption cooling cycle has also been studied and compared to that of activated carbon fiber (ACF)-ethanol pair. Experimental results show that Maxsorb III can adsorb up to 1.2 kg of ethanol per kilogram of adsorbent. Theoretical calculations show that, the Maxsorb III-ethanol adsorption cycle can achieve a specific cooling effect of about 420 kJ kg{sup -1} at an evaporator temperature of 7 C along with a heat source of temperature 80 C and thus the pair is recommended for solar cooling applications. (author)
Laser-solid interaction and dynamics of the laser-ablated materials
Rapid transformations through the liquid and vapor phases induced by laser-solid interactions are described by the authors' thermal model with the Clausius-Clapeyron equation to determine the vaporization temperature under different surface pressure condition. Hydrodynamic behavior of the vapor during and after ablation is described by gas dynamic equations. These two models are coupled. Modeling results show that lower background pressure results lower laser energy density threshold for vaporization. The ablation rate and the amount of materials removed are proportional to the laser energy density above its threshold. The authors also demonstrate a dynamic source effect that accelerates the unsteady expansion of laser-ablated material in the direction perpendicular to the solid. A dynamic partial ionization effect is studied as well. A self-similar theory shows that the maximum expansion velocity is proportional to csα, where 1 - α is the slope of the velocity profile. Numerical hydrodynamic modeling is in good agreement with the theory. With these effects, α is reduced. Therefore, the expansion front velocity is significantly higher than that from conventional models. The results are consistent with experiments. They further study how the plume propagates in high background gas condition. Under appropriate conditions, the plume is slowed down, separates with the background, is backward moving, and hits the solid surface. Then, it splits into two parts when it rebounds from the surface. The results from the modeling will be compared with experimental observations where possible
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.
Carbon dioxide captured by multi-walled carbon nanotube and activated charcoal: A comparative study
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.
Adsorbed Methane Film Properties in Nanoporous Carbon Monoliths
Soo, Yuchoong; Chada, Nagaraju; Beckner, Matthew; Romanos, Jimmy; Burress, Jacob; Pfeifer, Peter
2013-03-01
Carbon briquetting can increase methane storage capacity by reducing the useless void volume resulting in a better packing density. It is a robust and efficient space-filling form for an adsorbed natural gas vehicle storage tank. To optimize methane storage capacity, we studied three fabrication process parameters: carbon-to-binder ratio, compaction temperature, and pyrolysis temperature. We found that carbon-to-binder ratio and pyrolysis temperature both have large influences on monolith uptakes. We have been able to optimize these parameters for high methane storage. All monolith uptakes (up to 260 bar) were measured by a custom-built, volumetric, reservoir-type instrument. The saturated film density and the film thickness was determined using linear extrapolation on the high pressure excess adsorption isotherms. The saturated film density was also determined using the monolayer Ono-Kondo model. Film densities ranged from ca. 0.32 g/cm3 - 0.37 g/cm3.The Ono-Kondo model also determines the binding energy of methane. Binding energies were also determined from isosteric heats calculated from the Clausius-Clapeyron equation and compared with the Ono-Kondo model method. Binding energies from Ono-Kondo were ca. 7.8 kJ/mol - 10 kJ/mol. Work funded by California Energy Commission Contract #500-08-022.
Adsorption properties of a natural zeolite-water pair for use in adsorption cooling cycles
Solmus, ismail; Yamali, Cemil; Baker, Derek; Caglar, Ahmet [Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara (Turkey); Kaftanoglu, Bilgin [Department of Manufacturing Engineering, Atilim University, 06836 Ankara (Turkey)
2010-06-15
The equilibrium adsorption capacity of water on a natural zeolite has been experimentally determined at different zeolite temperatures and water vapor pressures for use in an adsorption cooling system. The Dubinin-Astakhov adsorption equilibrium model is fitted to experimental data with an acceptable error limit. Separate correlations are obtained for adsorption and desorption processes as well as a single correlation to model both processes. The isosteric heat of adsorption of water on zeolite has been calculated using the Clausius-Clapeyron equation as a function of adsorption capacity. The cyclic adsorption capacity swing for different condenser, evaporator and adsorbent temperatures is compared with that for the following adsorbent-refrigerant pairs: activated carbon-methanol; silica gel-water; and, zeolite 13X-water. Experimental results show that the maximum adsorption capacity of natural zeolite is nearly 0.12 kg{sub w}/kg{sub ad} for zeolite temperatures and water vapor pressures in the range 40-150 C and 0.87-7.38 kPa. (author)
Experimental thermochemical study of 3-acetyl-2-methyl-5-phenylthiophene
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.
Experimental thermochemical study of two chlorodinitroaniline isomers
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.
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.
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.
Experimental thermochemical study of 2,5- and 2,6-dichloro-4-nitroanilines
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.
Experimental thermochemical study of 4,5-dichloro-2-nitroaniline
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.
The functionalization of porous metal-organic frameworks (Cu3(BTC)2) was achieved by incorporating Keggin-type polyoxometalates (POMs), and further optimized via alkali metal ion-exchange. In addition to thermal gravimetric analysis, IR, single-crystal X-ray diffraction, and powder X-ray diffraction, the adsorption properties were characterized by N2 and volatile organic compounds (VOCs) adsorption measurements, including short-chain alcohols (C<4), cyclohexane, benzene, and toluene. The adsorption enthalpies estimated by the modified Clausius-Clapeyron equation provided insight into the impact of POMs and alkali metal cations on the adsorption of VOCs. The introduction of POMs not only improved the stability, but also brought the increase of adsorption capacity by strengthening the interaction with gas molecules. Furthermore, the exchanged alkali metal cations acted as active sites to interact with adsorbates and enhanced the adsorption of VOCs. - Graphical Abstract: The adsorption behavior of volatile organic compounds in porous metal-organic frameworks functionalized by polyoxometalates has been systematically evaluated. Highlights: → Functionalization of MOFs was achieved by incorporating Keggin-type POMs. → Introduction of POMs improved the thermal stability and adsorption capacity. → Alkali metal ion-exchange modified the inclusion state and also enhanced the adsorption. → Adsorption enthalpies were estimated to study the impact of POMs and alkali metal cations.
Singh, Khwairakpam Shantakumar; Sharma, Ashwini Kumar
2016-05-01
We report on the effect of transverse magnetic field on laser ablation of copper and aluminum targets both experimentally and numerically. The ablation depth is found to increase with magnetic field from 0 to 0.3 T and decreases at a higher magnetic field (0.5 T). It is demonstrated that the nanosecond laser ablation is mainly due to melt ejection and it solely depends on the thermo-physical parameters of the material. The increase in ablation depth with magnetic field is attributed to the increase in heat transfer from the plasma to the target, vapor pressure, and shock pressure. The ablation due to melt ejection is also calculated using vapor pressure through simulation and compared with the experimentally measured depth. In the presence of magnetic field, we introduce the magnetic pressure in Clausius-Clapeyron vapor pressure equation to account for the combined effect of magnetic field and atmospheric pressure on the vapor pressure of plasma. The ratio of calculated ablation depth at 0.3 T with respect to the absence of magnetic field is close to the corresponding experimental depth ratios indicating that the laser ablation modeling in the present work is validated. As the magnetic field increases, we observed the scattered mass at the center and around the crater. The size of deposited mass at the center is found to decrease at higher magnetic field which is attributed to breaking of large droplets into smaller ones due to increase in instability at higher magnetic field.
TUTTLE,BRUCE A.; VOIGT,JAMES A.; SCOFIELD,TIMOTHY W.; ASELAGE,TERRENCE L.; RODRIGUEZ,MARK A.; YANG,PIN; ZEUCH,DAVID H.; OLSON,WALTER R.; SIPOLA,DIANA L.
1999-12-21
A substantial decrease in hydrostatic ferroelectric (FE) to antiferroelectric (AFE) transformation pressure was measured for Pb(Zr{sub 0.949}Ti{sub 0.051}){sub 0.989}Nb{sub 0.0182}O{sub 3} ceramics with decreasing grain size. The 150 MPa decrease in hydrostatic FE to AFE transformation pressure over the grain size range of 8.5 {micro}m to 0.7{micro}m was shown to be consistent with enhanced internal stress with decreasing grain size. Further, the Curie Point decreased and the dielectric constant measured at 25 C increased with decreasing grain size. All three properties: dielectric constant magnitude, Curie point shift and FE to AFE phase transformation pressure were shown to be semi-quantitatively consistent with internal stress differences on the order of 100 MPa. Calculations of Curie point shifts from the Clausius-Clapeyron equation, using internal stress levels derived from the hydrostatic depoling characteristics, were consistent with measured values.
Correlation vs. Causation: The Effects of Ultrasonic Melt Treatment on Cast Metal Grain Size
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.
This paper reports that equilibrium phase diagrams for the systems NdCl3---CaCl2 and NdCl3---NaCl were determined by differential thermal analysis. A simple eutectic was observed at 59 ± 1 mol% CaCl2 and 600 degrees ± 2 degrees C in the NdCl3---CaCl2 system. A compound NaCl · 3NdCl3 which melts incongruently at 545 degrees ± 5 degrees C to NdCl3 and a liquid containing approximately 47 mol% NaCl, and a eutectic at 68 mol% NaCl and 439 degrees ± 2 degrees C were found in the NdCl3---NaCl system. On the basis of agreements between the activities calculated by the Clausius-Clapeyron equation and Temkin's model using the present data for the NdCl3---CaCl2 system and the literature data for the PrCl3---CaCl2 system, the melts in the former system consist of Nd3+, Ca2+, and Cl- ions and in the latter system of Pr3+, Ca2+, and Cl- ions. The above approach indicates the presence of Na+, Cl-, and NdCl4- in the NdCl3-rich melts in the NdCl3--NaCl system
Reversible Storage of Hydrogen and Natural Gas in Nanospace-Engineered Activated Carbons
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.
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.
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
Difference equations by differential equation methods
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.
Random diophantine equations, I
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.
The Generalized Jacobi Equation
Chicone, C.; Mashhoon, B.
2002-01-01
The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analyzed in this paper. The tidal accelerat...
AX-TNT, Super Prompt Critical Excursions in Spherical Geometry, Thermohydraulics
1 - Nature of physical problem solved: AX-TNT solves: (a) the coupled hydrodynamic, thermodynamic and neutronic equations which describe a spherical, super-prompt critical reactor system during an excursion. (b) the coupled equations of motion, and ideal gas equation of state for the detonation of a spherical charge in a gas. 2 - Method of solution: (a) As in the AX1 code the Sn neutronics section of the code calculates the inverse period and the relative power distribution. The inverse period and power distribution are used for calculating the power level and for assigning the energy added to a given region. During short time intervals hydrodynamic and thermodynamic calculations determine the acceleration, velocity, position, density, pressure, internal energy, kinetic energy and temperature of individual regions or mass points. Code tests send the problem to additional thermodynamic-hydrodynamic calculations or to neutronics calculations as the problem progresses. Several different equations of state and combinations of equations of state are used in the hydrodynamic-thermodynamic section of the code, namely - the linear, Clausius-Clapeyron, and ideal gas equations of state. (b) The neutronics section of the AX-TNT code is entirely bypassed. An ideal gas equation of state is used in conjunction with the Von Neumann and Richtmyer viscous pressure in the hydrodynamics -thermodynamics sections of the code, to trace the blast wave resulting from the detonation of a spherical charge. 3 - Restrictions on the complexity of the problem: Maximum number of energy groups 7; maximum number of mass points 320; maximum number of substances + mixtures 8; the effects of delayed neutrons and heat transfer are completely neglected
The Modified Magnetohydrodynamical Equations
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.
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.
Ordinary differential equations
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
Ray C. Fair
2007-01-01
How inflation and unemployment are related in both the short run and long run is perhaps the key question in macroeconomics. This paper tests various price equations using quarterly U.S. data from 1952 to the present. Issues treated are the following. 1) Estimating price and wage equations in which wages affect prices and vice versa versus estimating "reduced form" price equations with no wage explanatory variables. 2) Estimating price equations in (log) level terms, first difference (i.e., i...
New unified evolution equation
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...
Goncalves, Patricia
2010-01-01
We introduce the notion of energy solutions of the KPZ equation. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, weakly asymmetric, conservative particle systems with respect to the stationary states are given by energy solutions of the KPZ equation. As a consequence, we prove that the Cole-Hofp solutions are also energy solutions of the KPZ equation.
Diophantine equations and identities
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
The Modified Magnetohydrodynamical Equations
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.
Analysis on flow characteristic of nuclear heating reactor
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.)
Thermal hydraulic modeling of a natural circulation loop
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.)
Thermal hydraulic modeling of a natural circulation loop
Jiang, S. Y.; Wu, X. X.; Zhang, Y. J.; Jia, H. J.
The experiment was carried out on the test loop HRTL-5, which simulates the geometry and system design of a 5MW 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.
Experimental analysis on thermohydraulic characteristic of nuclear heating reactor
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 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 does not reach its situation value. Thirdly, the mechanism of two-phase flow oscillation, namely, 'zero-pressure-drop', is described. In the wide range of inlet subcooling (0K<ΔT<28 K) there exists three regions for system flow condition, namely, stable two-phase flow, bulk and subcooled boiling unstable flow and subcooled boiling and single phase stable flow. The response of mass flow rate, after a small disturbance in the heat flux, are shown in the above inlet subcooling range, and based on it the instability map of the system are given through experiment and calculation
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
Functional equations with causal operators
Corduneanu, C
2003-01-01
Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with causal operators. Functional Equations with Causal Operators explains the connection between equations with causal operators and the classical types of functional equations encountered by mathematicians and engineers. It details the fundamentals of linear equations and stability theory and provides several applications and examples.
Differential equations for dummies
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.
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.
Elliptic partial differential equations
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...
Fundamental Equation of Economics
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...
Solving Ordinary Differential Equations
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.
Differential equations I essentials
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.
Ordinary differential equations
Pontryagin, Lev Semenovich
1962-01-01
Ordinary Differential Equations presents the study of the system of ordinary differential equations and its applications to engineering. The book is designed to serve as a first course in differential equations. Importance is given to the linear equation with constant coefficients; stability theory; use of matrices and linear algebra; and the introduction to the Lyapunov theory. Engineering problems such as the Watt regulator for a steam engine and the vacuum-tube circuit are also presented. Engineers, mathematicians, and engineering students will find the book invaluable.
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...
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
Linear Equations: Equivalence = Success
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…
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.
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.
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.
Some classical Diophantine equations
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.
Applied singular integral equations
Mandal, B N
2011-01-01
The book is devoted to varieties of linear singular integral equations, with special emphasis on their methods of solution. It introduces the singular integral equations and their applications to researchers as well as graduate students of this fascinating and growing branch of applied mathematics.
Alternative equations of gravitation
It is shown, trough a new formalism, that the quantum fluctuation effects of the gravitational field in Einstein's equations are analogs to the effects of a continuum medium in Maxwell's Electrodynamics. Following, a real example of the applications of these equations is studied. Qunatum fluctuations effects as perturbation sources in Minkowski and Friedmann Universes are examined. (L.C.)
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that we solve the perpendicular component of Ohm's law to conserve the physical energy while ensuring the relation ∇ · j = 0
The relativistic Pauli equation
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.
The generalized Jacobi equation
The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighbouring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analysed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed 2-1/2c ∼ 0.7c is pointed out. The astrophysical implication of this result for the terminal speed of a relativistic jet is briefly explored
Applied partial differential equations
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...
The direct use of enlarged subsets of mathematically exact equations of change in moments of the velocity distribution function, each equation corresponding to one of the macroscopic variables to be retained, produces extended MHD models. The first relevant level of closure provides 'ten moment' equations in the density ρ, velocity v, scalar pressure p, and the traceless component of the pressure tensor t. The next 'thirteen moment' level also includes the thermal flux vector q, and further extended MHD models could be developed by including even higher level basic equations of change. Explicit invariant forms for the tensor t and the heat flux vector defining q follow from their respective basic equations of change. Except in the neighbourhood of a magnetic null, in magnetised plasma these forms may be resolved into known sums of their parallel, cross (or transverse) and perpendicular components. Parallel viscosity in an electron-ion plasma is specifically discussed. (author)
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.
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.
Nonlinear gyrokinetic equations
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
A barrier of compacted bentonite clay is planned to be used in geological disposal of spent nuclear fuel. In addition to providing mechanical stability to the waste containers the barrier is supposed to prevent or delay the movement of groundwater and the consequential transport of radionuclides from the repository. Fluid flow, phase changes, mechanical behavior of the buffer, rock, and the containers, and the heat produced by the radioactive waste constitute a coupled thermo-hydro-mechanical (THM) system. The objective of the thesis is to model the coupled THM behaviour of the bentonite buffer. For this purpose a general thermomechanical and mixture theoretical model is derived and applied to the fully coupled THM description of swelling compacted bentonite. The particular form of the free energy of the system is chosen to take into account interactions of the mixture components, namely, mixing of the gaseous components (water vapor and air) and adsorption and swelling interactions between the liquid water and the solid skeleton. The mechanical part of the model is limited to reversible behavior within the limit of small strains. Numerical implementation is done with the multi-purpose finite element method software ELMER. The model is applied to various coupled experiments: two kinds of laboratory scale tests for Febex bentonite, larger scale mock-up and in-situ tests for Febex bentonite, and to three kinds of laboratory scale experiments for MX-80 bentonite. In addition, a brief consideration of the difference of the large scale Febex experiments and the real disposal situation is done by incorporating more realistic temperature evolutions of the containers. The inclusion of the mixing interaction yields Clausius-Clapeyron equations which are valid both for the total pressure (i.e. the boiling pressure) and for the partial pressure of saturated vapor. Additionally, together with an appropriate dissipation function the mixing interaction yields a common form of
Standardized Referente Evapotranspiration Equation
M.D. Mundo–Molina
2009-01-01
In this paper is presented a discussion on the necessity to standardize the Penman–Monteith equations in order to estimate ETo. The proposal is to define an accuracy and standarize equation based in Penman–Monteith. The automated weather station named CIANO (27° 22 ' 144 North latitude and 109" 55' west longitude) it was selected tomake comparisons. The compared equations we re: a) CIANO weat her station, b) Penman–Monteith ASCE (PMA), Penman–Monteith FAO 56 (PM FAO 56), Penman–Monteith estan...
Stochastic Schroedinger equations
A derivation of Belavkin's stochastic Schroedinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system's quantum state given the observations made. This estimate satisfies a stochastic Schroedinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence
Beginning partial differential equations
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
Uncertain differential equations
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.
Partial differential equations
Friedman, Avner
2008-01-01
This three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of we
Hyperbolic partial differential equations
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
Ordinary differential equations
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,
Differential equations problem solver
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
Modern introduction to differential equations
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
A Comparison of IRT Equating and Beta 4 Equating.
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…
Nonlinear differential equations
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
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.
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.
Diophantine Equations and Computation
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.
Applied partial differential equations
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...
Hedin Equations for Superconductors
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...
Resistive ballooning mode equation
Bateman, G.; Nelson, D. B.
1978-10-01
A second-order ordinary differential equation on each flux surface is derived for the high mode number limit of resistive MHD ballooning modes in tokamaks with arbitrary cross section, aspect ratio, and shear. The equation is structurally similar to that used to study ideal MHD ballooning modes computationally. The model used in this paper indicates that all tokamak plasmas are unstable, with growth rate proportional to resistivity when the pressure gradient is less than the critical value needed for ideal MHD stability.
Relativistic Guiding Center Equations
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.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
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.
Thermal dissociation behavior and dissociation enthalpies of methane-carbon dioxide mixed hydrates.
Kwon, Tae-Hyuk; Kneafsey, Timothy J; Rees, Emily V L
2011-06-30
Replacement of methane with carbon dioxide in hydrate has been proposed as a strategy for geologic sequestration of carbon dioxide (CO(2)) and/or production of methane (CH(4)) from natural hydrate deposits. This replacement strategy requires a better understanding of the thermodynamic characteristics of binary mixtures of CH(4) and CO(2) hydrate (CH(4)-CO(2) mixed hydrates), as well as thermophysical property changes during gas exchange. This study explores the thermal dissociation behavior and dissociation enthalpies of CH(4)-CO(2) mixed hydrates. We prepared CH(4)-CO(2) mixed hydrate samples from two different, well-defined gas mixtures. During thermal dissociation of a CH(4)-CO(2) mixed hydrate sample, gas samples from the head space were periodically collected and analyzed using gas chromatography. The changes in CH(4)-CO(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(2) concentration in the vapor phase became richer during dissociation because the initial hydrate composition contained relatively more CO(2) than the vapor phase. The composition change in the vapor phase during hydrate dissociation affected the dissociation pressure and temperature; the richer CO(2) in the vapor phase led to a lower dissociation pressure. Furthermore, the increase in CO(2) concentration in the vapor phase enriched the hydrate in CO(2). The dissociation enthalpy of the CH(4)-CO(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(4)-CO(2) mixed hydrate lays between the limiting values of pure CH(4) hydrate and CO(2) hydrate, increasing with the CO(2) fraction in the hydrate phase. PMID:21604671
Standard molar enthalpies of formation of 3'- and 4'-nitroacetophenones
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.
Thermochemical study of some dichloroacetophenone isomers
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)
Thermodynamic study of 1,2,3-triphenylbenzene and 1,3,5-triphenylbenzene
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.
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.
Aernecke, Matthew J; Mendum, Ted; Geurtsen, Geoff; Ostrinskaya, Alla; Kunz, Roderick R
2015-11-25
A rapid method for vapor pressure measurement was developed and used to derive the vapor pressure curve of the thermally labile peroxide-based explosive hexamethylene triperoxide diamine (HMTD) over the temperature range from 28 to 80 °C. This method uses a controlled flow of vapor from a solid-phase HMTD source that is presented to an ambient-pressure-ionization mass spectrometer equipped with a secondary-electrospray-ionization (SESI) source. The subpart-per-trillion sensitivity of this system enables direct detection of HMTD vapor through an intact [M + H](+) ion in real time at temperatures near 20 °C. By calibrating this method using vapor sources of cocaine and heroin, which have known pressure-temperature (P-T) curves, the temperature dependence of HMTD vapor was determined, and a Clausius-Clapeyron plot of ln[P (Pa)] vs 1/[T (K)] yielded a straight line with the expression ln[P (Pa)] = {(-11091 ± 356) × 1/[T (K)]} + 25 ± 1 (error limits are the standard error of the regression analysis). From this equation, the sublimation enthalpy of HMTD was estimated to be 92 ± 3 kJ/mol, which compares well with the theoretical estimate of 95 kJ/mol, and the vapor pressure at 20 °C was estimated to be ∼60 parts per trillion by volume, which is within a factor of 2 of previous theoretical estimates. Thus, this method provides not only the first direct experimental determination of HMTD vapor pressure but also a rapid, near-real-time capability to quantitatively measure low-vapor-pressure compounds, which will be useful for aiding in the development of training aids for bomb-sniffing canines. PMID:26505487
Functional Equations and Fourier Analysis
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.
Scaling Equation for Invariant Measure
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.
Integral equations and computation problems
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)
Transport equation solving methods
This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method
Introduction to partial differential equations
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.
Unified derivation of evolution equations
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...
The Equations of Magnetoquasigeostrophy
Umurhan, O M
2013-01-01
The dynamics contained in magnetized layers of exoplanet atmospheres are important to understand in order to characterize what observational signatures they may provide for future observations. It is important to develop a framework to begin studying and learning the physical processes possible under those conditions and what, if any, features contained in them may be observed in future observation missions. The aims of this study is to formally derive, from scaling arguments, a manageable reduced set of equations for analysis, i.e. a magnetic formulation of the equations of quasigeostrophy appropriate for a multi-layer atmosphere. The main goal is to provide a simpler theoretical platform to explore the dynamics possible within confined magnetized layers of exoplanet atmospheres. We primarily use scaling arguments to derive the reduced equations of "magnetoquasigeostrophy" which assumes dynamics to take place in an atmospheric layer which is vertically thin compared to its horizontal scales. The derived equa...
Boussinesq evolution equations
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...
Quadratic Diophantine equations
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.
Equations of mathematical physics
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
Mirce Functionability Equation
Dr Jezdimir Knezevic
2014-08-01
Full Text Available Scientific principles and concepts expressed through the laws, equations and formulas are the bedrock for the prediction of the deign-in functionality performance of any engineering creation. However, there is no equivalent when the in-service functionability performance predictions have to be made. Hence, Mirce Mechanics has been created at the MIRCE Akademy to fulfil the roll. The main purpose of this paper is to present the development and application of Mirce Functionability Equation which is the bedrock for the prediction of the functionability performance of maintainable systems.
Obtaining Maxwell's equations heuristically
Diener, Gerhard; Weissbarth, Jürgen; Grossmann, Frank; Schmidt, Rüdiger
2013-02-01
Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light, together with Galilean invariance of the Lorentz force, allows us to finalize Maxwell's equations and to introduce arbitrary electrodynamics units naturally.
Generalized estimating equations
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
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.
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.
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.
The relativistic Pauli equation
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...
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...
The Statistical Drake Equation
Maccone, Claudio
2010-12-01
We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density
Comparison of Kernel Equating and Item Response Theory Equating Methods
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
On difference Riccati equations and second order linear difference equations
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.
Test equating methods and practices
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...
Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation
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.
Standardized Referente Evapotranspiration Equation
M.D. Mundo–Molina
2009-04-01
Full Text Available In this paper is presented a discussion on the necessity to standardize the Penman–Monteith equations in order to estimate ETo. The proposal is to define an accuracy and standarize equation based in Penman–Monteith. The automated weather station named CIANO (27° 22 ' 144 North latitude and 109" 55' west longitude it was selected tomake comparisons. The compared equations we re: a CIANO weat her station, b Penman–Monteith ASCE (PMA, Penman–Monteith FAO 56 (PM FAO 56, Penman–Monteith estandarizado ASCE (PM Std. ASCE. The results were: a There are important differences between PMA and CIANO weather station. The differences are attributed to the nonstandardization of the equation CIANO weather station, b The coefficient of correlation between both methods was of 0,92, with a standard deviation of 1,63 mm, an average quadratic error of 0,60 mm and one efficiency in the estimation of ETo with respect to the method pattern of 87%.
Calculus & ordinary differential equations
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.
Dyre, Jeppe
1995-01-01
energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk modelthe energy master equation...
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.
Stochastic nonlinear beam equations
Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan
2005-01-01
Roč. 132, č. 1 (2005), s. 119-149. ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005
On rough differential equations
Lejay, Antoine
2009-01-01
We prove that the Ito map, that is the map that gives the solution of a differential equation controlled by a rough path of finite p-variation with p in [2,3) is locally Lipschitz continuous in all its arguments and could be extended to vector fields that have only a linear growth.
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.
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.
Generalized reduced magnetohydrodynamic equations
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics
Modelling by Differential Equations
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…
Do Differential Equations Swing?
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…
Kinetic equation of sociodynamics
Володимир Олександрович Касьянов
2014-01-01
This article aims to build a theory of social dynamics, similar to the kinetic theory of gases. In general, given model is hybrid because off static mechanics ideas. In particular, Boltsman equation, Jaynes’s principle of entropy optimality have been applied to preference distribution of first and second type.
Equational binary decision diagrams
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
Kinetic equation of sociodynamics
Володимир Олександрович Касьянов
2014-08-01
Full Text Available This article aims to build a theory of social dynamics, similar to the kinetic theory of gases. In general, given model is hybrid because off static mechanics ideas. In particular, Boltsman equation, Jaynes’s principle of entropy optimality have been applied to preference distribution of first and second type.
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.
We present part of our (direct or indirect) knwoledge of the equation of state of nuclear matter in a density-temperature domain for which nucleonic effects are dominant (densities smaller than 2-4 times the saturation density and temperatures smaller than 10-20 MeV). The lectures are divided into three parts corresponding, respectiveley, to direct studies close to the saturation, to the astrophysical case and to the studies involving heavy-ion collisions. In chapter one, after a brief introduction to the concept of equation of state, we discuss the saturation property of nuclear matter. The notion of incompressibility modulus is also introduced and its value is discussed in detail. Nuclear matter calculations trying to reproduce saturation from a nucleon-nucleon interaction are also briefly presented. In chapter two we study the equation of state in the astrophysical context. The role of the nuclear component is discussed in detail for the final phase of the collapse of supernovae cores. A brief presentation of calculations of dense matter constituting neutron stars is also given. Chapter three is devoted to heavy-ion collisions below 500-600 MeV per nucleon. After a brief presentation of both theoretical and experimental frameworks, we focus on three particular aspects which could have a link with the nuclear matter equation of state: the formation of intermediate mass fragments, flow effects and subthreshold particle production
RPA equations and the instantaneous Bethe-Salpeter equation
Resag, J
1993-01-01
We give a derivation of the particle-hole RPA equations for an interacting multi-fermion system by applying the instantaneous approximation to the amputated two-fermion propagator of the system. In relativistic field theory the same approximation leads from the fermion-antifermion Bethe-Salpeter equation to the Salpeter equation. We show that RPA equations and Salpeter equation are indeed equivalent.
Lie Symmetries of Ishimori Equation
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.
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Anticipated backward stochastic differential equations
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.
Elements of partial differential equations
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
DIANA P NAVIA
2011-06-01
C para a variedade MBRA 383, enquanto o modelo de GAB foi a 35°C por MBRA 383. Calor isostérico de sorção diminuiu com o aumento do teor de umidade de equilíbrio foi encontrado um valor máximo de 87 kJ/mol e 78 kJ/mol, e mínima de 44,6 kJ/mol e 44,5 kJ/mol em amostras preparadas com o CM 7951-5 e MBRA 383, respectivamente.Water adsorption and isosteric heat were evaluated in biopolymers made from flour of two varieties of cassava (CM 7951-5 and MBRA 383, fique dust and glycerol by compression molding technique. The adsorption isotherms of polymeric samples were performed at 15,25, and 35°C in a water activity range of 0.12 to 0.98, using a gravimetric method. The adsorption experimental data were adjusted using the GAB, Caurie, Oswin, Smith, Henderson and Peleg models. The sorption isosteric heat (Qst was determined with Clausius-Clapeyron equation. The results showed that the Peleg model was adjusted appropriately to experimental values of adsorption at 15, 25 and 35°C in the samples prepared with the variety CM 7951-5 and 15 and 25°C for the variety MBRA 383, while the GAB model was at 35°C for MBRA 383. Isosteric heat of sorption decreased with increase in equilibrium moisture content finding the maximum in 87Kj/mol and 78 Kj/mol, and minimum in 44.6 Kj/mol and 44.5 Kj/mol in samples made with CM 7951-5y MBRA 383 respectively.
Stochastic differential equations and applications
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
Chaos in Partial Differential Equations
Li, Y. Charles
2009-01-01
This is a survey on Chaos in Partial Differential Equations. First we classify soliton equations into three categories: 1. (1+1)-dimensional soliton equations, 2. soliton lattices, 3. (1+n)-dimensional soliton equations (n greater than 1). A systematic program has been established by the author and collaborators, for proving the existence of chaos in soliton equations under perturbations. For each category, we pick a representative to present the results. Then we review some initial results o...
Classical Diophantine equations
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, ...
Multinomial diffusion equation
Balter, Ariel; Tartakovsky, Alexandre M.
2011-06-01
We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles N→∞, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.
Differential equations with Mathematica
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
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.
M. Paul Gough
2008-07-01
Full Text Available LandauerÃ¢Â€Â™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 Ã¢Â€Â˜Why now?Ã¢Â€Â™ question we wonder Ã¢Â€Â˜What next?Ã¢Â€Â™ as we expect the information equation of state to tend towards w = 0 in the future.c
Maxwell Equations as the One Photon Quantum Equation
Maxwell equations (Faraday and Ampere-Maxwell laws) can be presented as a three component equation in a way similar to the two component neutrino equation. However, in this case, the electric and magnetic Gauss's laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have 'also- shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum equations
Bitsadze, A V
1963-01-01
Equations of the Mixed Type compiles a series of lectures on certain fundamental questions in the theory of equations of mixed type. This book investigates the series of problems concerning linear partial differential equations of the second order in two variables, and possessing the property that the type of the equation changes either on the boundary of or inside the considered domain. Topics covered include general remarks on linear partial differential equations of mixed type; study of the solutions of second order hyperbolic equations with initial conditions given along the lines of parab
Telegrapher's equation for light derived from the transport equation
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...
Entropy: From Thermodynamics to Hydrology
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)...
Converting fractional differential equations into partial differential equations
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.
Dimensional Equations of Entropy
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.
Partial differential equations
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
Stochastic Geometric Wave Equations
Brzezniak, Z.; Ondreját, Martin
Cham: Springer, 2015, s. 157-188. (Progress in Probability. 68). ISBN 978-3-0348-0908-5. ISSN 1050-6977. [Stochastic analysis and applications at the Centre Interfacultaire Bernoulli, Ecole Polytechnique Fédérale de Lausanne. Lausanne (CH), 09.01.2012-29.6.2012] R&D Projects: GA ČR GAP201/10/0752 Institutional research plan: CEZ:AV0Z10750506 Institutional support: RVO:67985556 Keywords : Stochastic wave equation * Riemannian manifold * homogeneous space Subject RIV: BA - General Mathematics http://library.utia.cas.cz/separaty/2015/SI/ondrejat-0447803.pdf
The nonlinear fragmentation equation
We study the kinetics of nonlinear irreversible fragmentation. Here, fragmentation is induced by interactions/collisions between pairs of particles and modelled by general classes of interaction kernels, for several types of breakage models. We construct initial value and scaling solutions of the fragmentation equations, and apply the 'non-vanishing mass flux' criterion for the occurrence of shattering transitions. These properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters. (fast track communication)
Elliptic differential equations
Hackbusch, Wolfgang; Ion, PDF
2010-01-01
The book offers a simultaneous presentation of the theory and of the numerical treatment of elliptic problems. The author starts with a discussion of the Laplace equation in the classical formulation and its discretisation by finite differences and deals with topics of gradually increasing complexity in the following chapters. He introduces the variational formulation of boundary value problems together with the necessary background from functional analysis and describes the finite element method including the most important error estimates. A more advanced chapter leads the reader into the th
Dimensional Equations of Entropy
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 ent...
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
Makkonen, Lasse
2016-04-01
Young's construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young's equation can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line. PMID:26940644
Differential Equations as Actions
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....
Conservational PDF Equations of Turbulence
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
Program Transformation by Solving Equations
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.
On Certain Dual Integral Equations
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.
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
Functional equations for Feynman integrals
New types of equations for Feynman integrals are found. It is shown that Feynman integrals satisfy functional equations connecting integrals with different kinematics. A regular method is proposed for obtaining such relations. The derivation of functional equations for one-loop two-, three- and four-point functions with arbitrary masses and external momenta is given. It is demonstrated that functional equations can be used for the analytic continuation of Feynman integrals to different kinematic domains
Growth Equation with Conservation Law
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.
Successfully Transitioning to Linear Equations
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…
Hyperbolic Methods for Einstein's Equations
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.
An Extented Wave Action Equation
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.
The Schroedinger equation and spin
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
Resonantly coupled nonlinear evolution equations
A differential matrix eigenvalue problem is used to generate systems of nonlinear evolution equations. They model triad, multitriad, self-modal, and quartet wave interactions. A nonlinear string equation is also recovered as a special case. A continuum limit of the eigenvalue problem and associated evolution equations are discussed. The initial value solution requires an investigation of the corresponding inverse-scattering problem. (auth)
Solving Nonlinear Coupled Differential Equations
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.
Solution of Finite Element Equations
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...
Quadratic bundle and nonlinear equations
The paper is aimed at giving an exhaustive description of the nonlinear evolution equations (NLEE), connected with the quadratic bundle (the spectral parameter lambda, which enters quadratically into the equations) and at describing Hamiltonian structure of these equations. The equations are solved through the inverse scattering method (ISM). The basic formulae for the scattering problem are given. The spectral expansion of the integrodifferential operator is used so that its eigenfunctions are the squared solutions of the equation. By using the notions of Hamiltonian structure hierarchy and gauge transformations it is shown how to single out physically interesting NLEE
Generalized Klein-Kramers equations
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.
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...
A generalized advection dispersion equation
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.
Reduction of infinite dimensional equations
Zhongding Li
2006-02-01
Full Text Available In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.
Integral equations and their applications
Rahman, M
2007-01-01
For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of in
Discovering evolution equations with applications
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
$\\Lambda$ Scattering Equations
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.
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.
Scaling of differential equations
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...
Comparison between characteristics of mild slope equations and Boussinesq equations
无
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.
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.
Differential equations methods and applications
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. .
Spinor wave equation of photon
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.
Quaternion Dirac Equation and Supersymmetry
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...
Differential Equations for Algebraic Functions
Bostan, Alin; Chyzak, Frédéric; Salvy, Bruno; Lecerf, Grégoire; Schost, Éric
2007-01-01
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove ...
Perturbed linear rough differential equations
Coutin, Laure; Lejay, Antoine
2014-01-01
We study linear rough differential equations and we solve perturbed linear rough differential equation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebra more general than tensor algebra and to consider extensions of classical results such as the Mag...
THE ERMAKOV EQUATION: A COMMENTARY
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), ...
Hyperbolic Methods for Einstein's Equations
Reula Oscar
1998-01-01
Full Text Available 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.
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 ...
The generalized Airy diffusion equation
Frank M. Cholewinski
2003-08-01
Full Text Available Solutions of a generalized Airy diffusion equation and an associated nonlinear partial differential equation are obtained. Trigonometric type functions are derived for a third order generalized radial Euler type operator. An associated complex variable theory and generalized Cauchy-Euler equations are obtained. Further, it is shown that the Airy expansions can be mapped onto the Bessel Calculus of Bochner, Cholewinski and Haimo.
Introduction to ordinary differential equations
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
Equation with the many fathers
Kragh, Helge
1984-01-01
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 of the us...... electrodynamics. Although this ambitious attempt attracted some interest in 1926, its impact on the mainstream of development in quantum mechanics was virtually nil....
Temporal Fokker-Planck Equations
Boon, Jean Pierre; Lutsko, James F.
2016-01-01
The temporal Fokker-Plank equation [{\\it J. Stat. Phys.}, {\\bf 3/4}, 527 (2003)] or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. %\\cite{boon-grosfils-lutsko}. We present two generalizations of the temporal Fokker-Plank equation for the first passage distribution function $f_j(r,t)$ of a particle moving on a substrate with time delays $\\tau_j$. Both generalizations follow from the ...
A modified electromagnetic wave equation
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
Correct Linearization of Einstein's Equations
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.
Diffusion equations and turbulent transport
One scrutinized transport equations differing essentially in form from the classical diffusion one. Description of diffusion under strong nonequilibrium and turbulence involved application of equations that took account of transport nonlocality and memory effects. One analyzed ways to derive the mentioned equations starting from quasi-linear approximation and up to equations with fractional derivatives. One points out the generality of the applied theoretical concepts in spite of the essential difference of the exact physical problems. One demonstrated the way of application of the theoretical and probabilistic ideas
Diffusion equations and turbulent transport
Diffusion equations are considered that differ substantially in structure from classical ones. A description of diffusion under strongly nonequilibrium conditions in a highly turbulent plasma requires the use of equations that take into account memory effects and the nonlocal nature of transport. Different methods are developed for constructing such equations, ranging from those in the quasilinear approximation to those with fractional derivatives. It is emphasized that the theoretical concepts underlying the equations proposed are common for a very wide variety of specific physical problems. The ways of applying theoretical probabilistic ideas are demonstrated
Electronic representation of wave equation
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.
ON A CORRELATION BETWEEN DIFFERENTIAL EQUATIONS AND THEIR CHARACTERISTIC EQUATIONS
Boro M. Piperevski
2007-01-01
Abstract: The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.
Tippe Top Equations and Equations for the Related Mechanical Systems
Rutstam, Nils
2012-01-01
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 $\\mathbf{\\hat{3}}$ 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.
Tippe Top Equations and Equations for the Related Mechanical Systems
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.
On asymptotics for difference equations
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
Solving equations by topological methods
Lech Górniewicz
2005-01-01
Full Text Available In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.
Solving equations by topological methods
Lech Górniewicz
2005-01-01
In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.