Distribution kinetics of polymer crystallization and the Avrami equation
Yang, Jiao; McCoy, Benjamin J; Madras, Giridhar
2005-01-01
Cluster distribution kinetics is adopted to explore the kinetics of polymer crystallization. Population balance equations based on crystal size distribution and concentration of amorphous polymer segments are solved numerically and the related dynamic moment equations are also solved. The model accounts for heterogeneous or homogeneous nucleation and crystal growth. Homogeneous nucleation rates follow the classical surface-energy nucleation theory. Different mass dependences of growth and dis...
P Agarwal; Goel, S; Kumar, A.(State University of New York at Buffalo, Buffalo, USA)
1991-01-01
Conductivity measurements are done to study the kinetics of crystallization in glassy Se100-x Tex, system. Amorphous to Crystalline (a-c) transformations are studied during isothermal annealing at various temperatures between glass transition and melting temperature. Avrami's equation of isothermal transformation is used to calculate the activation energy of crystallization and the order parameter.
A test of the Johnson-Mehl-Avrami equation. [for phase transformations
Weinberg, Michael C.
1987-01-01
The accuracy of the Johnson-Mehl-Avrami (JMA) equation is evaluated for the special case of two-dimensional crystallization. The volume fraction transformed is determined directly by computer modeling, and the evaluation of the secondary phase obtained is compared with predictions of the JMA equation. The JMA equation if found to be highly acccurate over virtually the entire range of the transformation process.
Farjas, Jordi; Roura, Pere
2008-01-01
Avrami's model describes the kinetics of phase transformation under the assumption of spatially random nucleation. In this paper we provide a quasi-exact analytical solution of Avrami's model when the transformation takes place under continuous heating. This solution has been obtained with different activation energies for both nucleation and growth rates. The relation obtained is also a solution of the so-called Kolmogorov-Johnson-Mehl-Avrami transformation rate equation. The corresponding n...
A.A. Burbelko
2008-01-01
To forccast rhc kinc~icso f phasc transformations which consist in nucleation and growth of rhc grains of a ncw phnsc, rhc wcll-knownKolmogorov-lohson-Mchl-Avrami (KJM A) equation has becn used. It is gencr;llly known that in lhc case of pnrsllcl gr;cin prowlh indiffcrcnt phascs proceeding at difrcrcnl vclocirics. tbc rcsulis OF thc ralcularions arc burdcncd with an crror. 111 this study. applying rhcassumpfions of n stnist ical thcory or thc scrccncd grain growlh. an attcrnpt has bccn madc t...
Directory of Open Access Journals (Sweden)
A. A. Burbelko
2008-03-01
Full Text Available To forccast rhc kinc~icso f phasc transformations which consist in nucleation and growth of rhc grains of a ncw phnsc, rhc wcll-knownKolmogorov-lohson-Mchl-Avrami (KJM A equation has becn used. It is gencr;llly known that in lhc case of pnrsllcl gr;cin prowlh indiffcrcnt phascs proceeding at difrcrcnl vclocirics. tbc rcsulis OF thc ralcularions arc burdcncd with an crror. 111 this study. applying rhcassumpfions of n stnist ical thcory or thc scrccncd grain growlh. an attcrnpt has bccn madc to cstimatc this crror ror thc casc or prowl11 ofthc sphcroidnl grains of two rypcs. Thc ohiaincd rcsulls indicate that thc vatuc of an crror in typical equations incrcsscs with iacrcasingdilrfercncc in the growth vclocity of thc particlcs of bath typcs.
Feng, Xin; Ye, Xingyou; Park, Jun-Bom; Lu, Wenli; Morott, Joe; Beissner, Brad; Lian, Zhuoyang John; Pinto, Elanor; Bi, Vivian; Porter, Stu; Durig, Tom; Majumdar, Soumyajit; Repka, Michael A
2015-01-01
The recrystallization of an amorphous drug in a solid dispersion system could lead to a loss in the drug solubility and bioavailability. The primary objective of the current research was to use an improved kinetic model to evaluate the recrystallization kinetics of amorphous structures and to further understand the factors influencing the physical stability of amorphous solid dispersions. Amorphous solid dispersions of fenofibrate with different molecular weights of hydroxypropylcellulose, HPC (Klucel™ LF, EF, ELF) were prepared utilizing hot-melt extrusion technology. Differential scanning calorimetry was utilized to quantitatively analyze the extent of recrystallization in the samples stored at different temperatures and relative humidity (RH) conditions. The experimental data were fitted into the improved kinetics model of a modified Avrami equation to calculate the recrystallization rate constants. Klucel LF, the largest molecular weight among the HPCs used, demonstrated the greatest inhibition of fenofibrate recrystallization. Additionally, the recrystallization rate (k) decreased with increasing polymer content, however exponentially increased with higher temperature. Also k increased linearly rather than exponentially over the range of RH studied. PMID:25224341
Modeling Crystallization Dynamics when the Avrami Model Fails
Terry Gough; Reinhard Illner
1999-01-01
Recent experiments on the formation of crystalline CO2 from a newly discovered binary phase consisting of CO2 and C2H2 at 90° K fail to be adequately simulated by Avrami equations. The purpose of this note is to develop an alternative to the Avrami model which can make accurate predictions for these experiments. The new model uses empirical approximations to the distribution densities of the volumes of three-dimensional Voronoi cells defined by Poisson-generated crystallization...
Avrami exponent under transient and heterogeneous nucleation transformation conditions
Sinha, I.; Mandal, R. K.
2010-01-01
The Kolmogorov-Johnson-Mehl-Avrami model for isothermal transformation kinetics is universal under specific assumptions. However, the experimental Avrami exponent deviates from the universal value. In this context, we study the effect of transient heterogeneous nucleation on the Avrami exponent for bulk materials and also for transformations leading to nanostructured materials. All transformations are assumed to be polymorphic. A discrete version of the KJMA model is modified for this purpose...
Avrami behavior of magnetite nanoparticles formation in co-precipitation process
Ahmadi R; Hosseini Madaah H.R.; Masoudi A.
2011-01-01
In this work, magnetite nanoparticles (mean particle size about 20 nm) were synthesized via coprecipitation method. In order to investigate the kinetics of nanoparticle formation, variation in the amount of reactants within the process was measured using pH-meter and atomic absorption spectroscopy (AAS) instruments. Results show that nanoparticle formation behavior can be described by Avrami equations. Transmission electron microscopy (TEM) and X-ray diffraction (XRD) were performed to ...
J. F. Joson; L. T. Davila; Z. B. Domingo
2003-01-01
Kinetics of non-isothermal crystallization of coconut-based cholesteryl ester was performed by differentialscanning calorimetry under various heating rates. Different analysis methods were used to describe theprocess of non-isothermal crystallization. The results showed that the Avrami equation could describe thesystem very well. However, the Ozawa analysis failed. A probable reason is the difference in the crystallizationkinetics at high and low relative crystallization. The phase transition...
Choi, H W; Kim, Y H; Rim, Y H; Yang, Y S
2013-06-28
The formation of crystalline LiNbO3 (LN) from LN glass has been studied by means of differential scanning calorimetry and in situ synchrotron X-ray diffraction. The LN glass with no glass former was prepared by the polymerized complex method. The isothermal kinetics of the crystallization process is described using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation and the Avrami exponent n is found to be ~2.0, indicating that the crystallization mechanism is diffusion-controlled growth with a decreasing nucleation rate. The effective activation energy of crystallization calculated from isothermal measurements is 6.51 eV. It is found that the LN glass directly transforms into a rhombohedral LN crystal without any intermediate crystalline phase and most crystal grains are confined within the size of ~40 nm irrespective of different isothermal temperatures. Application of JMAK theory to the non-isothermal thermoanalytical study of crystallization of LN glass is discussed. PMID:23677338
Avrami behavior of magnetite nanoparticles formation in co-precipitation process
Directory of Open Access Journals (Sweden)
Ahmadi R.
2011-01-01
Full Text Available In this work, magnetite nanoparticles (mean particle size about 20 nm were synthesized via coprecipitation method. In order to investigate the kinetics of nanoparticle formation, variation in the amount of reactants within the process was measured using pH-meter and atomic absorption spectroscopy (AAS instruments. Results show that nanoparticle formation behavior can be described by Avrami equations. Transmission electron microscopy (TEM and X-ray diffraction (XRD were performed to study the chemical and morphological characterization of nanoparticles. Some simplifying assumptions were employed for estimating the nucleation and growth rate of magnetite nanoparticles.
Directory of Open Access Journals (Sweden)
J. F. Joson
2003-06-01
Full Text Available Kinetics of non-isothermal crystallization of coconut-based cholesteryl ester was performed by differentialscanning calorimetry under various heating rates. Different analysis methods were used to describe theprocess of non-isothermal crystallization. The results showed that the Avrami equation could describe thesystem very well. However, the Ozawa analysis failed. A probable reason is the difference in the crystallizationkinetics at high and low relative crystallization. The phase transitions of the coconut-based cholesterylester were also observed through optical polarizing microscopy
On the validity of Avrami formalism in primary crystallization
Bruna Escuer, Pere; Crespo Artiaga, Daniel; González Cinca, Ricardo; Pineda Soler, Eloi
2006-01-01
Calorimetric data of primary crystallization is usually interpreted in the framework of the Kolmogorov Dokl. Akad. Nauk SSSR 1, 355 1937 , Johnson and Mehl Trans. AIME 135, 416 1939 , and Avrami J. Chem. Phys. 7, 1103 1939 ; 8, 212 1940 ; 9, 177 1941 KJMA theory. However, while the KJMA theory assumes random nucleation and exhaustion of space by direct impingement, primary crystallization is usually driven by diffusion-controlled growth with soft impingement betwe...
Finite Volume Kolmogorov-Johnson-Mehl-Avrami Theory
Berg, Bernd A.; Dubey, Santosh
2008-01-01
We study Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory of phase conversion in finite volumes. For the conversion time we find the relationship $\\tau_{\\rm con} = \\tau_{\\rm nu} [1+f_d(q)]$. Here $d$ is the space dimension, $\\tau_{\\rm nu}$ the nucleation time in the volume $V$, and $f_d(q)$ a scaling function. Its dimensionless argument is $q=\\tau_{\\rm ex}/ \\tau_{\\rm nu}$, where $\\tau_{\\rm ex}$ is an expansion time, defined to be proportional to the diameter of the volume divided by expansion spe...
The Indicators’ Inadequacy and the Predictions’ Accuracy
Directory of Open Access Journals (Sweden)
Constantin Mitruț
2013-08-01
Full Text Available In this article, we proposed the introduction in literature of a new source of uncertainty in modeling and forecasting: the indicators’ inadequacy. Even if it was observed, a specific nominalization in the context of forecasting procedure has not been done yet. The inadequacy of indicators as a supplementary source of uncertainty generates a lower degree of accuracy in forecasting. This assumption was proved using empirical data related to the prediction of unemployment rate in Romania on the horizon 2011-2013. Four strategies of modeling and predicting the unemployment rate were proposed, observing two types of indicators’ inadequacy: the use of transformed variables in order to get stationary data set (the difference between the unemployment rates registered in two successive periods was used instead of the unemployment rate and the utilization of macro-regional unemployment rates whose predictions are aggregated in order to forecast the overall unemployment rate in Romania. The results put in evidence that the predictions of the total unemployment rate using moving average models of order 2 are the most accurate, being followed by the forecasts based on the predictions of active civil population and number of unemployed people. The strategies based on the aggregation of the predictions for the four macro-regional unemployment rates imply a higher inadequacy and consequently a lower degree of forecasts’ accuracy.
Nucleation and growth in one dimension, part I: The generalized Kolmogorov-Johnson-Mehl-Avrami model
Jun, Suckjoon; Zhang, Haiyang; Bechhoefer, John
2004-01-01
Motivated by a recent application of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) model to the study of DNA replication, we consider the one-dimensional version of this model. We generalize previous work to the case where the nucleation rate is an arbitrary function $I(t)$ and obtain analytical results for the time-dependent distributions of various quantities (such as the island distribution). We also present improved computer simulation algorithms to study the 1D KJMA model. The analytical res...
Model for crystallization kinetics: Deviations from Kolmogorov-Johnson-Mehl-Avrami kinetics
Castro, Mario; Domínguez-Adame Acosta, Francisco; Sánchez, A; Rodriguez, T.
1999-01-01
We propose a simple and versatile model to understand the deviations from the well-known Kolmogorov-Johnson-Mehl-Avrami kinetics theory found in metal recrystallization and amorphous semiconductor crystallization. We analyze the kinetics of the transformation and the grain-size distribution of the product material, finding a good overall agreement between our model and available experimental data. The information so obtained could help to relate the mentioned experimental deviations due to pr...
Beyond the Kolmogorov Johnson Mehl Avrami kinetics: inclusion of the spatial correlation
Fanfoni, M.; Tomellini, M.
2003-01-01
The Kolmogorov-Johnson-Mehl-Avrami model, which is a nucleation and growth poissonian process in space, has been implemented by taking into account spatial correlation among nuclei. This is achieved through a detailed study of a system of distinguishable and correlated dots (nuclei). The probability that no dots be in a region of the space has been evaluated in terms of correlation functions. The theory has been applied to describe nucleation and growth in two dimensions under constant nuclea...
Beyond the constraints underlying Kolmogorov-Johnson-Mehl-Avrami theory related to the growth laws
Tomellini, M.; Fanfoni, M.
2012-01-01
The theory of Kolmogorov-Johnson-Mehl-Avrami (KJMA) for phase transition kinetics is subjected to severe limitations concerning the functional form of the growth law. This paper is devoted to side step this drawback through the use of correlation function approach. Moreover, we put forward an easy-to-handle formula, written in terms of the experimentally accessible actual extended volume fraction, which is found to match several types of growths. Computer simulations have been done for corrob...
Daytime Sleepiness and Sleep Inadequacy as Risk Factors for Dementia
Directory of Open Access Journals (Sweden)
Angeliki Tsapanou
2015-07-01
Full Text Available Background/Aims: To examine the association between self-reported sleep problems and incidence of dementia in community-dwelling elderly people. Methods: 1,041 nondemented participants over 65 years old were examined longitudinally. Sleep problems were estimated using the RAND Medical Outcomes Study Sleep Scale examining sleep disturbance, snoring, sleep short of breath or with a headache, sleep adequacy, and sleep somnolence. Cox regression analysis was used to examine the association between sleep problems and risk for incident dementia. Age, gender, education, ethnicity, APOE-ε4, stroke, heart disease, hypertension, diabetes, and depression were included as covariates. Results: Over 3 years of follow-up, 966 (92.8% participants remained nondemented, while 78 (7.2% developed dementia. In unadjusted models, sleep inadequacy (‘Get the amount of sleep you need' at the initial visit was associated with increased risk of incident dementia (HR = 1.20; 95% CI 1.02-1.42; p = 0.027. Adjusting for all the covariates, increased risk of incident dementia was still associated with sleep inadequacy (HR = 1.20; 95% CI 1.01-1.42; p = 0.040, as well as with increased daytime sleepiness (‘Have trouble staying awake during the day' (HR = 1.24; 95% CI 1.00-1.54; p = 0.047. Conclusion: Our results suggest that sleep inadequacy and increased daytime sleepiness are risk factors for dementia in older adults, independent of demographic and clinical factors.
Cell Dynamics Simulation of Kolmogorov-Johnson-Mehl-Avrami Kinetics of Phase Transformation
Iwamatsu, Masao; Nakamura, Masato
2005-01-01
In this study, we use the cell dynamics method to test the validity of the Kormogorov-Johnson-Mehl-Avrami (KJMA) theory of phase transformation. This cell dynamics method is similar to the well-known phase-field model, but it is a more simple and efficient numerical method for studying various scenarios of phase transformation in a unified manner. We find that the cell dynamics method reproduces the time evolution of the volume fraction of the transformed phase predicted by the KJMA theory. S...
Folate inadequacy in the diet of pregnant women
Directory of Open Access Journals (Sweden)
Lívia de Castro Crivellenti
2014-06-01
Full Text Available OBJECTIVE: To estimate food and dietary folate inadequacies in the diets of adult pregnant women. METHODS: A prospective study was conducted with 103 healthy pregnant adult users of the Public Health Care System of Ribeirão Preto, São Paulo, Brazil. The present study included the 82 women with complete food intake data during pregnancy, which were collected by three 24-hour dietary recalls. Food folate (folate naturally present in foods and dietary folate (food folate plus folate from fortified wheat flour and cornmeal inadequacies were determined, using the Estimated Average Requirement as cutoff. RESULTS: The diets of 100% and 94% of the pregnant women were inadequate in food folate and dietary folate, respectively. However, fortified foods increased the medium availability of the nutrient by 87%. CONCLUSION: The large number of pregnant women consuming low-folate diets was alarming. Nationwide population studies are needed to confirm the hypothesized high prevalence of low-folate diets among pregnant women.
Vitamin D inadequacy in Belgian postmenopausal osteoporotic women
Directory of Open Access Journals (Sweden)
Collette Julien
2007-04-01
Full Text Available Abstract Background Inadequate serum vitamin D [25(OHD] concentrations are associated with secondary hyperparathyroidism, increased bone turnover and bone loss, which increase fracture risk. The objective of this study is to assess the prevalence of inadequate serum 25(OHD concentrations in postmenopausal Belgian women. Opinions with regard to the definition of vitamin D deficiency and adequate vitamin D status vary widely and there are no clear international agreements on what constitute adequate concentrations of vitamin D. Methods Assessment of 25-hydroxyvitamin D [25(OHD] and parathyroid hormone was performed in 1195 Belgian postmenopausal women aged over 50 years. Main analysis has been performed in the whole study population and according to the previous use of vitamin D and calcium supplements. Four cut-offs of 25(OHD inadequacy were fixed : Results Mean (SD age of the patients was 76.9 (7.5 years, body mass index was 25.7 (4.5 kg/m2. Concentrations of 25(OHD were 52.5 (21.4 nmol/L. In the whole study population, the prevalence of 25(OHD inadequacy was 91.3 %, 87.5 %, 43.1 % and 15.9% when considering cut-offs of 80, 75, 50 and 30 nmol/L, respectively. Women who used vitamin D supplements, alone or combined with calcium supplements, had higher concentrations of 25(OHD than non-users. Significant inverse correlations were found between age/serum PTH and serum 25(OHD (r = -0.23/r = -0.31 and also between age/serum PTH and femoral neck BMD (r = -0.29/r = -0.15. There is a significant positive relation between age and PTH (r = 0.16, serum 25(OHD and femoral neck BMD (r = 0.07. (P Vitamin D concentrations varied with the season of sampling but did not reach statistical significance (P = 0.09. Conclusion This study points out a high prevalence of vitamin D inadequacy in Belgian postmenopausal osteoporotic women, even among subjects receiving vitamin D supplements.
40 CFR 52.32 - Sanctions following findings of SIP inadequacy.
2010-07-01
... 40 Protection of Environment 3 2010-07-01 2010-07-01 false Sanctions following findings of SIP inadequacy. 52.32 Section 52.32 Protection of Environment ENVIRONMENTAL PROTECTION AGENCY (CONTINUED) AIR... following findings of SIP inadequacy. For purposes of the SIP revisions required by § 51.120, EPA may make...
Avilés, Mauricio González; Campuzano, Hermelinda Servín
2013-01-01
It applies a mathematical model of solid formation, the model of Avrami-Kolgomorov [Ausloos & Petroni, 2007] to model the time evolution of percentage of adherents of the major religions practiced in Mexico, adjusting the corresponding parameters with available records in the period from 1950 to 2000 [Molina-Hernandez, 2003; INEGI, 2005]. A comparison is made with the application of the model to global trends and concludes that Catholicism is in a marked disaggregation and trends of Christian...
Tran, Thu Hien; Govin, Alexandre; Guyonnet, René; Grosseau, Philippe; Lors, Christine; Damidot, Denis; Devès, Olivier; Ruot, Bertrand
2013-01-01
International audience The aim of this research was to modelize the colonization of mortar surface by green algae using Avrami's law. The resistance of mortars, with different intrinsic characteristics (porosity, roughness, carbonation state), to the biofouling was studied by means of an accelerated lab-scale test. A suspension of green alga Klebsormidium flaccidum, was performed to periodically sprinkle the mortar surfaces. The covered surface rate followed a sigmoidal type curve versus t...
Kooi, BJ
2006-01-01
An analytical theory has been developed, based on Monte Carlo (MC) simulations, describing the kinetics of isothermal phase transformations proceeding by nucleation and subsequent growth for d-1 dimensional growth in d dimensional space (with d 2 or 3). This type of growth is of interest since it is generally anisotropic, leads to hard impingement, and obtains strong deviations from the traditional Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory. Within the MC simulations 1D growth can occur wit...
Vitamin E Inadequacy in Humans: Causes and Consequences12
Traber, Maret G.
2014-01-01
It is estimated that >90% of Americans do not consume sufficient dietary vitamin E, as α-tocopherol, to meet estimated average requirements. What are the adverse consequences of inadequate dietary α-tocopherol intakes? This review discusses health aspects where inadequate vitamin E status is detrimental and additional vitamin E has reversed the symptoms. In general, plasma α-tocopherol concentrations <12 μmol/L are associated with increased infection, anemia, stunting of growth, and poor outcomes during pregnancy for both the infant and the mother. When low dietary amounts of α-tocopherol are consumed, tissue α-tocopherol needs exceed amounts available, leading to increased damage to target tissues. Seemingly, adequacy of human vitamin E status cannot be assessed from circulating α-tocopherol concentrations, but inadequacy can be determined from “low” values. Circulating α-tocopherol concentrations are very difficult to interpret because, as a person ages, plasma lipid concentrations also increase and these elevations in lipids increase the plasma carriers for α-tocopherol, leading to higher circulating α-tocopherol concentrations. However, abnormal lipoprotein metabolism does not necessarily increase α-tocopherol delivery to tissues. Additional biomarkers of inadequate vitamin E status are needed. Urinary excretion of the vitamin E metabolite α-carboxy-ethyl-hydroxychromanol may fulfill this biomarker role, but it has not been widely studied with regard to vitamin E status in humans or with regard to health benefits. This review evaluated the information available on the adverse consequences of inadequate α-tocopherol status and provides suggestions for avenues for research. PMID:25469382
Avilés, Mauricio González
2013-01-01
It applies a mathematical model of solid formation, the model of Avrami-Kolgomorov [Ausloos & Petroni, 2007] to model the time evolution of percentage of adherents of the major religions practiced in Mexico, adjusting the corresponding parameters with available records in the period from 1950 to 2000 [Molina-Hernandez, 2003; INEGI, 2005]. A comparison is made with the application of the model to global trends and concludes that Catholicism is in a marked disaggregation and trends of Christianity in Mexico are similar to global.
Energy Technology Data Exchange (ETDEWEB)
Dill, Eric D.; Folmer, Jacob C.W.; Martin, James D. [NCSU
2013-12-05
A series of simulations was performed to enable interpretation of the material and physical significance of the parameters defined in the Kolmogorov, Johnson and Mehl, and Avrami (KJMA) rate expression commonly used to describe phase boundary controlled reactions of condensed matter. The parameters k, n, and t_{0} are shown to be highly correlated, which if unaccounted for seriously challenge mechanistic interpretation. It is demonstrated that rate measurements exhibit an intrinsic uncertainty without precise knowledge of the location and orientation of nucleation with respect to the free volume into which it grows. More significantly, it is demonstrated that the KJMA rate constant k is highly dependent on sample size. However, under the simulated conditions of slow nucleation relative to crystal growth, sample volume and sample anisotropy correction affords a means to eliminate the experimental condition dependence of the KJMA rate constant, k, producing the material-specific parameter, the velocity of the phase boundary, v_{pb}.
Anemia in postmenopausal women: dietary inadequacy or non-dietary factors
Postmenopausal women are disproportionately affected by anemia, and the prevalence in females > 65 years of age in the United States is approximately 10%. The manifestation of anemia in older populations is associated with dietary inadequacy, blood loss, genetics, alterations in bioavailability, ren...
Borg, ter S.; Verlaan, S.; Hemsworth, J.; Mijnarends, D.; Schols, J.M.G.A.; Luiking, Y.C.; Groot, de C.P.G.M.
2015-01-01
Micronutrient deficiencies and low dietary intakes among community-dwelling older adults are associated with functional decline, frailty and difficulties with independent living. As such, studies that seek to understand the types and magnitude of potential dietary inadequacies might be beneficial fo
Interplay between Kolmogorov-Johnson-Mehl-Avrami kinetics and Poisson-Voronoi tessellation
Tomellini, M.; Fanfoni, M.
2016-05-01
In this paper we investigate the connection between Voronoi tessellation and the KJMA approach of space filling. In particular, we study how nuclei, in their growth, cover a given Voronoi cell. This approach leads to an integral equation for the cell-size distribution function. Starting from the 1D case, that is solved exactly, we extend the results to the dD case. The analysis allows to find a rationale to the phenomenological parameter entering the Gamma distribution function and to improve the description of the transformation through the knowledge of the kinetics of grain formation. Moreover, the nucleus size distribution function has been calculated as a function of the transformed fraction.
Directory of Open Access Journals (Sweden)
Nicole A Huijgen
Full Text Available The composition of the diet is of increasing importance for the development and maturation of the ovarian follicles. In Polycystic Ovary Syndrome (PCOS healthy dietary interventions improve the clinical spectrum. We hypothesized that dieting and diet inadequacy in the reproductive life course is associated with impaired programming of ovarian follicles and contributes to the severity of the PCOS phenotype.To determine associations between the use of a self-initiated diet and diet inadequacy and the severity of the PCOS phenotype, we performed an explorative nested case control study embedded in a periconception cohort of 1,251 patients visiting the preconception outpatient clinic. 218 patients with PCOS and 799 subfertile controls were selected from the cohort and self-administered questionnaires, anthropometric measurements and blood samples were obtained. The Preconception Dietary Risk Score (PDR score, based on the Dutch dietary guidelines, was used to determine diet inadequacy in all women. The PDR score was negatively associated to cobalamin, serum and red blood cell folate and positively to tHcy. PCOS patients (19.9%, in particular the hyperandrogenic (HA phenotype (22.5% reported more often the use of a self-initiated diet than controls (13.1%; p = 0.023. The use of an inadequate diet was also significantly higher in PCOS than in controls (PDR score 3.7 vs 3.5; p = 0.017 and every point increase was associated with a more than 1.3 fold higher risk of the HA phenotype (adjusted OR 1.351, 95% CI 1.09-1.68. Diet inadequacy was independently associated with the anti-Müllerian Hormone (AMH concentration (β 0.084; p = 0.044; 95% CI 0.002 to 0.165 and free androgen index (β 0.128; p = 0.013; 95% CI 0.028 to 0.229 in PCOS patients.The use of a self-initiated diet and diet inadequacy is associated with PCOS, in particular with the severe HA phenotype. This novel finding substantiated by the association between diet inadequacy and AMH needs
Wing Woo
2004-01-01
The Washington Consensus suffers from fundamental inadequacies, and that a more comprehensive framework of the economic process is needed to guide the formulation of country-specific development strategies. The following five propositions summarise the set of interrelated arguments made in this paper: 1. The Washington Consensus was based on a wrong reading of the East Asian growth experience. This explains why some observers have called the trade regimes of Korea and Taiwan in the 1965- 1980...
Farjas Silva, Jordi; Roura Grabulosa, Pere
2008-01-01
The space subdivision in cells resulting from a process of random nucleation and growth is a subject of interest in many scientific fields. In this paper, we deduce the expected value and variance of these distributions while assuming that the space subdivision process is in accordance with the premises of the Kolmogorov-Johnson-Mehl-Avrami model. We have not imposed restrictions on the time dependency of nucleation and growth rates. We have also developed an approximate analytical cell size ...
GENERAL EQUATIONS OF CARBONIZATION OF EUCALYPTUS SPP KINETIC MECHANISMS
Túlio Jardim Raad; Paulo César da C. Pinheiro; Maria Irene Yoshida
2006-01-01
In the present work, a set of general equations related to kinetic mechanism of wood compound carbonization: hemicelluloses, cellulose and lignin was obtained by Avrami-Eroffev and Arrhenius equations and Thermogravimetry of Eucalyptus cloeziana, Eucalyptus camaldulensis, Corymbia citriodora, Eucalyptus urophylla and Eucalyptus grandis samples, TG-Isothermal and TG-Dynamic. The different thermal stabilities and decomposition temperature bands of those species compounds were applied as strateg...
The (in)adequacy of applicative use of quantum cryptography in wireless sensor networks
Turkanović, Muhamed; Hölbl, Marko
2014-10-01
Recently quantum computation and cryptography principles are exploited in the design of security systems for wireless sensor networks (WSNs), which are consequently named as quantum WSN. Quantum cryptography is presumably secure against any eavesdropper and thus labeled as providing unconditional security. This paper tries to analyze the aspect of the applicative use of quantum principles in WSN. The outcome of the analysis elaborates a summary about the inadequacy of applicative use of quantum cryptography in WSN and presents an overview of all possible applicative challenges and problems while designing quantum-based security systems for WSN. Since WSNs are highly complex frameworks, with many restrictions and constraints, every security system has to be fully compatible and worthwhile. The aim of the paper was to contribute a verdict about this topic, backed up by equitable facts.
Prevalence of vitamin D inadequacy in European women aged over 80 years.
Bruyère, Olivier; Slomian, Justine; Beaudart, Charlotte; Buckinx, Fanny; Cavalier, Etienne; Gillain, Sophie; Petermans, Jean; Reginster, Jean-Yves
2014-01-01
Inadequate vitamin D status is associated with secondary hyperparathyroidism and increased bone turnover and bone loss, which in turn increases fracture risk. The objective of this study is to assess the prevalence of inadequate vitamin D status in European women aged over 80 years. Assessments of serum 25-hydroxyvitamin D levels (25(OH)D) were performed on 8532 European women with osteoporosis or osteopenia of which 1984 were aged over 80 years. European countries included in the study were: France, Belgium, Denmark, Italy, Poland, Hungary, United Kingdom, Spain and Germany. Two cut-offs of 25(OH)D inadequacy were fixed: nmol/L (30 ng/ml) and nmol/L (20 ng/ml). Mean (SD) age of the patients was 83.4 (2.9) years, body mass index was 25.0 (4.0) kg/m(2) and level of 25(OH)D was 53.3 (26.7) nmol/L (21.4 [10.7] ng/ml). There was a highly significant difference of 25(OH)D level across European countries (pnmol/L, respectively. In the 397 (20.0%) patients taking supplemental vitamin D with or without supplemental calcium, the mean serum 25(OH)D level was significantly higher than in the other patients (65.2 (29.2) nmol/L vs. 50.3 (25.2) nmol/L; Pvitamin D (25(OH)D) inadequacy in old European women. The prevalence could be even higher in some particular countries. PMID:24784761
International Nuclear Information System (INIS)
Highlights: • The Avrami exponent increases with increasing annealing temperature. • Recrystallization occurrence is most likely at low temperatures. • Heat of fusion and critical nucleus radius are evaluated from the MD calculations. - Abstract: In this work, molecular dynamics simulation is carried out to investigate the crystallization kinetics at low cooling rate during solidification and at different annealing temperature from amorphous phase during annealing of Pt–Pd (Pt50–Pd50) model alloy system. The interfacial free energies, critical nucleus radius, total free energy from high temperatures to low temperatures during solidification of alloy system are also determined by molecular dynamics. At the same time, in order to define the nucleation rate, it is suggested a model based on nucleation theory. The local atomic bonded pairs and short range order properties in the model alloy have been analyzed using Honeycutt–Andersen (HA) method. The kinetic of the crystallization is described by Johnson, Mehl and Avrami (JMA) model, which has been analyzed with MD method by using the crystalline-type bonded pairs during annealing process. The results demonstrated that the crystal kinetics is very important to understand the process of homogenous nucleation formation and also, the results are consistent with the classical nucleation theory
Energy Technology Data Exchange (ETDEWEB)
Celik, Fatih Ahmet, E-mail: facelik@beu.edu.tr
2015-05-25
Highlights: • The Avrami exponent increases with increasing annealing temperature. • Recrystallization occurrence is most likely at low temperatures. • Heat of fusion and critical nucleus radius are evaluated from the MD calculations. - Abstract: In this work, molecular dynamics simulation is carried out to investigate the crystallization kinetics at low cooling rate during solidification and at different annealing temperature from amorphous phase during annealing of Pt–Pd (Pt{sub 50}–Pd{sub 50}) model alloy system. The interfacial free energies, critical nucleus radius, total free energy from high temperatures to low temperatures during solidification of alloy system are also determined by molecular dynamics. At the same time, in order to define the nucleation rate, it is suggested a model based on nucleation theory. The local atomic bonded pairs and short range order properties in the model alloy have been analyzed using Honeycutt–Andersen (HA) method. The kinetic of the crystallization is described by Johnson, Mehl and Avrami (JMA) model, which has been analyzed with MD method by using the crystalline-type bonded pairs during annealing process. The results demonstrated that the crystal kinetics is very important to understand the process of homogenous nucleation formation and also, the results are consistent with the classical nucleation theory.
Maternal child-feeding practices and dietary inadequacy of 4-year-old children.
Durão, Catarina; Andreozzi, Valeska; Oliveira, Andreia; Moreira, Pedro; Guerra, António; Barros, Henrique; Lopes, Carla
2015-09-01
This study aimed to evaluate the association between maternal perceived responsibility and child-feeding practices and dietary inadequacy of 4-year-old children. We studied 4122 mothers and children enrolled in the population-based birth cohort - Generation XXI (Porto, Portugal). Mothers self-completed the Child Feeding Questionnaire and a scale on covert and overt control, and answered to a food frequency questionnaire in face-to-face interviews. Using dietary guidelines for preschool children, adequacy intervals were defined: fruit and vegetables (F&V) 4-7 times/day; dairy 3-5 times/day; meat and eggs 5-10 times/week; fish 2-4 times/week. Inadequacy was considered as below or above these cut-points. For energy-dense micronutrient-poor foods and beverages (EDF), a tolerable limit was defined (<6 times/week). Associations between maternal perceived responsibility and child-feeding practices (restriction, monitoring, pressure to eat, overt and covert control) and children's diet were examined by logistic regression models. After adjustment for maternal BMI, education, and diet, and children's characteristics (sex, BMI z-scores), restriction, monitoring, overt and covert control were associated with 11-18% lower odds of F&V consumption below the interval defined as adequate. Overt control was also associated with 24% higher odds of their consumption above it. Higher perceived responsibility was associated with higher odds of children consuming F&V and dairy above recommendations. Pressure to eat was positively associated with consumption of dairy above the adequate interval. Except for pressure to eat, maternal practices were associated with 14-27% lower odds of inadequate consumption of EDF. In conclusion, children whose mothers had higher levels of covert control, monitoring, and restriction were less likely to consume F&V below recommendations and EDF above tolerable limits. Higher overt control and pressure to eat were associated, respectively, with higher
Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
Directory of Open Access Journals (Sweden)
Constantino Tsallis
2015-05-01
Full Text Available It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\\(S_{BG}\\ is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form \\(S_{BG}=-k\\sum_i p_i \\ln p_i\\; the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the \\(S_q=k \\frac{1-\\sum_i p_i^q}{q-1}\\ one (\\(q \\in {\\cal R}; \\,S_1=S_{BG}\\, basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate \\(S_q\\. We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization.
Directory of Open Access Journals (Sweden)
N. Lumen
2008-01-01
Full Text Available Objectives. Severe penile inadequacy in adolescents is rare. Phallic reconstruction to treat this devastating condition is a major challenge to the reconstructive surgeon. Phallic reconstruction using the free radial forearm flap (RFF or the pedicled anterolateral thigh flap (ALTF has been routinely used in female-to-male transsexuals. Recently we started to use these techniques in the treatment of severe penile inadequacy. Methods. Eleven males (age 15 to 42 years were treated with a phallic reconstruction. The RFF is our method of choice; the ALTF is an alternative when a free flap is contraindicated or less desired by the patient. The RFF was used in 7 patients, the ALTF in 4 patients. Mean followup was 25 months (range: 4–49 months. Aesthetic and functional results were evaluated. Results. There were no complications related to the flap. Aesthetic results were judged as “good” in 9 patients and “moderate” in 2 patients. Sensitivity in the RFF was superior compared to the ALTF. Four patients developed urinary complications (stricture and/or fistula. Six patients underwent erectile implant surgery. In 2 patients the erectile implant had to be removed due to infection or erosion. Conclusion. In case of severe penile inadequacy due to whatever condition, a phalloplasty is the preferred treatment nowadays. The free radial forearm flap is still the method of choice. The anterolateral thigh flap can be a good alternative, especially when free flaps are contraindicated, but sensitivity is markedly inferior in these flaps.
Korobov, A.
2011-08-01
Discrete uniform Poisson-Voronoi tessellations of two-dimensional triangular tilings resulting from the Kolmogorov-Johnson-Mehl-Avrami (KJMA) growth of triangular islands have been studied. This shape of tiles and islands, rarely considered in the field of random tessellations, is prompted by the birth-growth process of Ir(210) faceting. The growth mode determines a triangular metric different from the Euclidean metric. Kinetic characteristics of tessellations appear to be metric sensitive, in contrast to area distributions. The latter have been studied for the variant of nuclei growth to the first impingement in addition to the conventional case of complete growth. Kiang conjecture works in both cases. The averaged number of neighbors is six for all studied densities of random tessellations, but neighbors appear to be mainly different in triangular and Euclidean metrics. Also, the applicability of the obtained results for simulating birth-growth processes when the 2D nucleation and impingements are combined with the 3D growth in the particular case of similar shape and the same orientation of growing nuclei is briefly discussed.
International Nuclear Information System (INIS)
Hysteresis is studied for a two-dimensional, spin- (1) /(2) , nearest-neighbor, kinetic Ising ferromagnet in a sinusoidally oscillating field, using Monte Carlo simulations and analytical theory. Attention is focused on large systems and moderately strong field amplitudes at a temperature below Tc. In this parameter regime, the magnetization switches through random nucleation and subsequent growth of many droplets of spins aligned with the applied field. Using a time-dependent extension of the Kolmogorov-Johnson-Mehl-Avrami theory of metastable decay, we analyze the statistical properties of the hysteresis-loop area and the correlation between the magnetization and the field. This analysis enables us to accurately predict the results of extensive Monte Carlo simulations. The average loop area exhibits an extremely slow approach to an asymptotic, logarithmic dependence on the product of the amplitude and the field frequency. This may explain the inconsistent exponent estimates reported in previous attempts to fit experimental and numerical data for the low-frequency behavior of this quantity to a power law. At higher frequencies we observe a dynamic phase transition. Applying standard finite-size scaling techniques from the theory of second-order equilibrium phase transitions to this nonequilibrium transition, we obtain estimates for the transition frequency and the critical exponents (β/ν∼0.11,thinspγ/ν∼1.84, and ν∼1.1). In addition to their significance for the interpretation of recent experiments on switching in ferromagnetic and ferroelectric nanoparticles and thin films, our results provide evidence for the relevance of universality and finite-size scaling to dynamic phase transitions in spatially extended nonstationary systems. copyright 1999 The American Physical Society
Folate is required for biological methylation and nucleotide synthesis, and it is aberrations in these processes that are thought to be the mechanisms that enhance colorectal carcinogenesis produced by folate inadequacy. These functions of folate also depend on availability of other B-vitamins that ...
Bezrati, Ikram; Ben Fradj, Mohamed Kacem; Ouerghi, Nejmeddine; Feki, Moncef; Chaouachi, Anis; Kaabachi, Naziha
2016-01-01
Background: Vitamin D inadequacy is widespread in children and adolescents worldwide. The present study was undertaken to assess the vitamin D status in active children living in a sunny climate and to identify the main determinants of the serum concentration of 25-hydroxyvitamin D (25-OHD).Methods: This cross-sectional study included 225 children aged 7–15 years practicing sports in a football academy. Anthropometric measures were performed to calculate body mass index (BMI), fat mass, and m...
Directory of Open Access Journals (Sweden)
Ikram Bezrati
2016-04-01
Full Text Available Background: Vitamin D inadequacy is widespread in children and adolescents worldwide. The present study was undertaken to assess the vitamin D status in active children living in a sunny climate and to identify the main determinants of the serum concentration of 25-hydroxyvitamin D (25-OHD. Methods: This cross-sectional study included 225 children aged 7–15 years practicing sports in a football academy. Anthropometric measures were performed to calculate body mass index (BMI, fat mass, and maturity status. A nutritional enquiry was performed including 3-day food records and food frequency questionnaire. Plasma 25-OHD and insulin were assessed by immunoenzymatic methods ensuring categorization of vitamin D status and calculation of insulin sensitivity/resistance indexes. A logistic regression model was applied to identify predictors for vitamin D inadequacy. Results: Vitamin D deficiency (25-OHD<12 µg/L was observed in 40.9% of children and insufficiency (12<25-OHD<20 µg/L was observed in 44% of children. In a multivariate analysis, vitamin D deficiency and insufficiency were associated with a lower dietary intake of vitamin D, proteins, milk, red meat, fish, and eggs. However, no significant relationship was observed with maturation status, adiposity, or insulin resistance. Conclusions: Tunisian children and adolescents are exposed to a high risk of vitamin D inadequacy despite living in a sunny climate. Circulating 25-OHD concentrations are related to the intake of vitamin D food sources but not to maturation status or body composition. Ensuring sufficient and safe sun exposure and adequate vitamin D intake may prevent vitamin D inadequacy in children from sunny environments.
Institute of Scientific and Technical Information of China (English)
无
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.
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
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).
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.
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
Some fundamental considerations of the equation of radiative transfer
International Nuclear Information System (INIS)
The radiation transfer of the vector electromagnetic field was first formulated by Chandrasekhar while deriving the polarization characteristics of a sunlit sky. There are two subtle problems underlying this treatment. The first concerns the crucial identification of a Stokes parameter with the specific intensity of radiation. While both depend on position in 3-D space, the latter has, intrinsic to it, an additional angular dependence defining the flow of the radiation field. How can this inadequacy be remedied without damaging the results obtained heretofore from Chandrasekhar's formalism. The second problem arises from the fact that the radiative transfer equation describes the transport of an incoherent radiation field through space. This, however, seems to contradict the results of the Van Cittert-Zernike-Wolf theorem which implies that an incoherent field develops coherence as it passes through free space implying, of course, that the radiative transfer equation must involve not incoherent but partially coherent fields. The vector transfer equation of the direct beam (Beer's law) is derived from first principles. The analysis of this equation provides a satisfactory resolution of these two problems. The result also shows that the Beer's law will have to be modified to a matrix law to accommodate systems that are not spherically symmetric. 13 references
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.
Microcencapsulation and release kinetic equation of compound oil%复合精油微胶囊化及释放动力学研究
Institute of Scientific and Technical Information of China (English)
俞露; 谭书明; 王贝贝; 谢国芳
2013-01-01
The compound oil, including garlic oil, pepper oil,ginger oil and clove oil, was microencapsulated using β-cyclodextrin as composite wall material,the microcapsule preparation technology condition was optimized through orthogonal experiment, and the optimum technology conditions for compound oil microencapsulation was Mα:Mβ-cyclodexmn= 1:1 0,Mα,:Methand = 1:10,the temperature of microcapsule was 50℃,and the time was 1h. The release kinetics of microcapsules was investigated under different temperature conditions by using Avrami's formula, and release kinetic equation of microcapsules was built at 20℃.%以大蒜、花椒、生姜、丁香四种复配精油为芯材,β-环糊精为壁材,通过正交实验优化微胶囊制备工艺为:M精油∶Mβ-环糊精=1∶10,M精油∶M乙醇=1∶10,包合温度50℃,包合时间1h.用Avrami's公式和一级动力学对其在不同温度条件下的释放性能进行对比研究,建立微胶囊产品在20℃条件下的释放动力学方程.
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.
Explicit solutions of the generalized Ginzburg-Landau equation from linear systems
International Nuclear Information System (INIS)
The singularity structure of the complex one-dimensional Ginzburg-Landau equation in the complex (x,t) plane shows the inadequacy of the representations (Re A,Im A), (A,A-bar) for A, even in the nonlinear Schroedinger (NLS) limit. The elementary representation consists of two fields (Z,Θ), respectively complex and real, uniquely defined by an explicit expression for Θ and A. The four famous solutions of Nozaki and Bekki are then represented by two linear partial differential equations with constant coefficients and a finite set of constants. It is also shown how the invariance by parity on A increases the class of expected solutions. (authors) 24 refs
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...
The Pharmacokinetics of Fentanyl Family Drugs in Patients with Renal Inadequacy%肾功能不全患者芬太尼族药物的药动学
Institute of Scientific and Technical Information of China (English)
邵伟栋
2013-01-01
芬太尼族药物包括芬太尼、瑞芬太尼、舒芬太尼和阿芬太尼等,已广泛用于临床麻醉及术后镇痛.肾功能不全患者常伴有贫血、低血容量和低蛋白血症,许多药物不良反应发生率明显高于肾功能正常者.肾功能不全时芬太尼族药物的吸收、分布、生物转化和排泄等都可能发生改变.总结分析对肾功能不全患者芬太尼族药物药动学研究的进展,有助于指导临床用药,减少药物不良反应.%Fentanyl,remifentanil,sufentanil,and alfentanil are the fentanyl family of drugs. They have been widely used in clinical anesthesia and postoperative analgesia. Patients with renal inadequacy often have anemia,hypovolemia and hypoproteinemia, the incidence of adverse drug reactions in patients with renal inadequacy is significantly higher than in patients with the normal renal function. The absorption, distribution, biotransformation and excretion of fentanyl family of drugs may change in patients with renal failure. Here is to make a review of pharmacokinetics of the fentanyl family of drugs in patients with renal failure and provide guidance for clinical medication,reducing adverse drug reactions.
The Modified Magnetohydrodynamical Equations
Institute of Scientific and Technical Information of China (English)
EvangelosChaliasos
2003-01-01
After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similar fashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is done by replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vector potential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vector analysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHD equations.
Indian Academy of Sciences (India)
George F R Ellis
2007-07-01
The Raychaudhuri equation is central to the understanding of gravitational attraction in astrophysics and cosmology, and in particular underlies the famous singularity theorems of general relativity theory. This paper reviews the derivation of the equation, and its significance in cosmology.
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
Directory of Open Access Journals (Sweden)
Malvina Baica
1985-01-01
Full Text Available The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are i x2−my2=±1 ii x3+my3+m2z3−3mxyz=1iii Some fifth degree diopantine equations
The Modified Magnetohydrodynamical Equations
Institute of Scientific and Technical Information of China (English)
Evangelos Chaliasos
2003-01-01
After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similarfashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is doneby replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vectorpotential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vectoranalysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHDequations.
International Nuclear Information System (INIS)
We classify (1+3)-dimensional Pauli equations for a spin-(1/2) particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the 11 classes of vector-potentials of the electro-magnetic field A(t,x(vector sign))=(A0(t,x(vector sign)), A(vector sign)(t,x(vector sign))) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x(vector sign)) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field
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.
Directory of Open Access Journals (Sweden)
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
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...
International Nuclear Information System (INIS)
A new evolution equation is proposed for the gluon density relevant (GLR) for the region of small xB. It generalizes the GLR equation and allows deeper penetration in dense parton systems than the GLR equation does. This generalization consists of taking shadowing effects more comprehensively into account by including multi gluon correlations, and allowing for an arbitrary initial gluon distribution in a hadron. We solve the new equation for fixed αs. It is found that the effects of multi gluon correlations on the deep-inelastic structure function are small. (author) 15 refs, 5 figs, 2 tabs
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
Directory of Open Access Journals (Sweden)
Nikita Bokarev
2014-09-01
Full Text Available An attempt to find common solutions complete some Diophantine equations of the second degree with three variables, traced some patterns, suggest a common approach, which being elementary, however, lead to a solution of such equations. Using arithmetic functions allowed to write down the solutions in a single formula with no restrictions on the parameters used.
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
International Nuclear Information System (INIS)
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.)
International Nuclear Information System (INIS)
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
International Nuclear Information System (INIS)
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...
International Nuclear Information System (INIS)
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)
Nonlinear gyrokinetic equations
International Nuclear Information System (INIS)
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
Energy Technology Data Exchange (ETDEWEB)
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.
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
International Nuclear Information System (INIS)
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
Inadequacy of margin in qualification tests
International Nuclear Information System (INIS)
Dose-rate effects are now widely known. One nuclear industry response is to use a large margin (i.e., over-dose) in qualification tests to account for this. We have carried out investigations of polymer radiation degradation behaviors which have brought to light a number of reasons why this concept of margin can break down. First of all, we have found that dose-rate effects vary greatly in magnitude. Thus, based on high dose-rate testing, poor materials with large dose-rate effects may be selected over better materials with small effects. Also, in certain cases, material properties have been found to level out (as with PVC) or reverse trend (as with buna-n) at high doses, so that margin may be ineffective, misleading, or counterproductive. For Viton, the material properties were found to change in opposite directions at high and low dose rates, making margin inappropriate. The underlying problem with the concept of margin is that differences in aging conditions can lead to fundamental differences in degradation mechanisms
Inadequacies of TPR and Krashen's Input Hypothesis
Institute of Scientific and Technical Information of China (English)
Meng Meng; LI Laifa
2008-01-01
In this paper,the rationale of TPR and the Input Hypothesis of Krashen which justifies practices of TPR are reviewed and criticized in the light of evidence from teachers'observation of a long-term TPR project.It is argued that the effectiveness of TPR is compromised by its inadequate attention to the complexity of classroom interactions and children's cognition.The Input Hypothesis is believed that it oversimplified the cognitive dynamics of language learning.
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
International Nuclear Information System (INIS)
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics
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
Energy Technology Data Exchange (ETDEWEB)
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
Energy Technology Data Exchange (ETDEWEB)
White, R. B. [PPPL; Gobbin, M. [Euratom-ENEA Association
2014-10-01
In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.
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.
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
Institute of Scientific and Technical Information of China (English)
LIU Shi-Kuo; FU Zun-Tao; LIU Shi-Da; REN Kui
2003-01-01
An iterated function system (IFS) is constructed. It is shown that the invariant measure of IFS satisfies the same equation as scaling equation for wavelet transform (WT). Obviously, IFS and scaling equation of WT both have contraction mapping principle.
Integral equations and computation problems
International Nuclear Information System (INIS)
Volterra's Integral Equations and Fredholm's Integral Equations of the second kind are discussed. Computational problems are found in the derivations and the computations. The theorem of the solution of the Fredholm's Integral Equation is discussed in detail. (author)
Transport equation solving methods
International Nuclear Information System (INIS)
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
DEFF Research Database (Denmark)
Bredmose, Henrik; Schaffer, H.; Madsen, Per A.
2004-01-01
This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic...... Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over...... a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model...
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.
Directory of Open Access Journals (Sweden)
Franciane Rocha Faria
2013-01-01
Full Text Available The aim of this study was to analyze body fat anthropometric equations and electrical bioimpedance analysis (BIA in the prediction of cardiovascular risk factors in eutrophic and overweight adolescents. 210 adolescents were divided into eutrophic group (G1 and overweight group (G2. The percentage of body fat (% BF was estimated using 10 body fat anthropometric equations and 2 BIA. We measured lipid profiles, uric acid, insulin, fasting glucose, homeostasis model assessment-insulin resistance (HOMA-IR, and blood pressure. We found that 76.7% of the adolescents exhibited inadequacy of at least one biochemical parameter or clinical cardiovascular risk. Higher values of triglycerides (TG (P=0.001, insulin, and HOMA-IR (P<0.001 were observed in the G2 adolescents. In multivariate linear regression analysis, the % BF from equation (5 was associated with TG, diastolic blood pressure, and insulin in G1. Among the G2 adolescents, the % BF estimated by (5 and (9 was associated with LDL, TG, insulin, and the HOMA-IR. Body fat anthropometric equations were associated with cardiovascular risk factors and should be used to assess the nutritional status of adolescents. In this study, equation (5 was associated with a higher number of cardiovascular risk factors independent of the nutritional status of adolescents.
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
Directory of Open Access Journals (Sweden)
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
Institute of Scientific and Technical Information of China (English)
Ding Yi
2009-01-01
In this article, the author derives a functional equation η(s)=［(π/4)s-1/2√2/πг(1-s)sin(πs/2)]η(1-s) of the analytic function η(s) which is defined by η(s)=1-s-3-s-5-s+7-s…for complex variable s with Re s>1, and is defined by analytic continuation for other values of s. The author proves (1) by Ramanujan identity (see [1], [3]). Her method provides a new derivation of the functional equation of Riemann zeta function by using Poisson summation formula.
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
Institute of Scientific and Technical Information of China (English)
LIU Yan-hong; ZHANG Hui-ming
2007-01-01
Combining the symplectic variations theory, the homogeneous control equation and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isoparametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which are often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.
Investigation on bulk water: Integral equation approach
International Nuclear Information System (INIS)
The static structure factor of water has been investigated using the central force model (CFM). Due to inadequacy of the HNC closure in describing the complex hydrogen bond interactions, we have used a bridge function obtained by adjusting the HH bond lengths. The modification results in an improvement in the theoretical structure factors. In addition to this, we examine other water models and provide a compilation of structural and thermodynamic results obtained from them. (author)
Standardized Referente Evapotranspiration Equation
Directory of Open Access Journals (Sweden)
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.
DEFF Research Database (Denmark)
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
Czech Academy of Sciences Publication Activity Database
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.
Directory of Open Access Journals (Sweden)
Garkavenko A. S.
2011-08-01
Full Text Available The rate equations of the exciton laser in the system of interacting excitons have been obtained and the inverted population conditions and generation have been derived. The possibility of creating radically new gamma-ray laser has been shown.
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
International Nuclear Information System (INIS)
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
Directory of Open Access Journals (Sweden)
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.
Directory of Open Access Journals (Sweden)
Hatem Mejjaoli
2008-12-01
Full Text Available We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.
International Nuclear Information System (INIS)
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
Institute of Scientific and Technical Information of China (English)
SONG Xu-Xia
2013-01-01
The Ishimori equation is one of the most important (2+1)-dimensional integrable models,which is an integrable generalization of (1+1)-dimensional classical continuous Heisenberg ferromagnetic spin equations.Based on importance of Lie symmetries in analysis of differential equations,in this paper,we derive Lie symmetries for the Ishimori equation by Hirota's direct method.
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
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
Directory of Open Access Journals (Sweden)
D. Diederen
2015-06-01
Full Text Available We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave in a frictionless, prismatic, tidal channel with a horizontal bed. Assessment of a large number of numerical simulations, where an open boundary condition is posed at a certain distance landward, suggests that it can also be considered accurate in the more natural case of converging estuaries with nonlinear friction and a bed slope. The equation follows from the open boundary condition and is therefore a part of the problem formulation for an infinite tidal channel. This finding provides a practical tool for evaluating tidal wave dynamics, by reconstructing the temporal variation of the velocity based on local observations of the water level, providing a fully local open boundary condition and allowing for local friction calibration.
Directory of Open Access Journals (Sweden)
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
International Nuclear Information System (INIS)
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...
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
Czech Academy of Sciences Publication Activity Database
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
International Nuclear Information System (INIS)
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
DEFF Research Database (Denmark)
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....
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
Institute of Scientific and Technical Information of China (English)
朱鸿
1991-01-01
Based on the theory of orthogonal program expansion[8-10],the paper proposes a method to transform programs by solving program equations.By the method,transformation goals are expressed in program equations,and achieved by solving these equations.Although such equations are usually too complicated to be solved directly,the orthogonal expansion of programs makes it possible to reduce such equations into systems of equations only containing simple constructors of programs.Then,the solutions of such equations can be derived by a system of solving and simplifying rules,and algebraic laws of programs.The paper discusses the methods to simplify and solve equations and gives some examples.
On Certain Dual Integral Equations
Directory of Open Access Journals (Sweden)
R. S. Pathak
1974-01-01
Full Text Available Dual integral equations involving H-Functions have been solved by using the theory of Mellin transforms. The proof is analogous to that of Busbridge on solutions of dual integral equations involving Bessel functions.
International Nuclear Information System (INIS)
This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics
Functional equations for Feynman integrals
International Nuclear Information System (INIS)
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
Institute of Scientific and Technical Information of China (English)
左其华
2003-01-01
Based on the Navier-Stokes equation, an average wave energy equation and a generalized wave action conservation equation are presented in this paper. The turbulence effects on water particle velocity ui and wave surface elavation ξ as well as energy dissipation are included. Some simplified forms are also given.
The Schroedinger equation and spin
International Nuclear Information System (INIS)
Galilei invariance of the Schroedinger equation requires linearization of the operator by the introduction of anticommuting matrices as coefficients of the linear form. In an external field this leads directly to the Pauli equation, the non-relativistic limit of Dirac's equation. An overview of the complete argument that defines spin as a non-relativistic concept is presented. 9 refs
Resonantly coupled nonlinear evolution equations
International Nuclear Information System (INIS)
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
DEFF Research Database (Denmark)
Krenk, Steen
An important step in solving any problem by the finite element method is the solution of the global equations. Numerical solution of linear equations is a subject covered in most courses in numerical analysis. However, the equations encountered in most finite element applications have some special...
Quadratic bundle and nonlinear equations
International Nuclear Information System (INIS)
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
Indian Academy of Sciences (India)
Abdon Atangana
2014-02-01
This paper examines a possible effect of uncertainties, variability or heterogeneity of any dynamic system when being included in its evolution rule; the notion is illustrated with the advection dispersion equation, which describes the groundwater pollution model. An uncertain derivative is defined; some properties of the operator are presented. The operator is used to generalize the advection dispersion equation. The generalized equation differs from the standard equation in four properties. The generalized equation is solved via the variational iteration technique. Some illustrative figures are presented.
Reduction of infinite dimensional equations
Directory of Open Access Journals (Sweden)
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...
Institute of Scientific and Technical Information of China (English)
冯凤莲; 李明; 李若凡; 李秉章; 曹翠丽
2016-01-01
Objective To study the relationship among resilience,self feeling of inadequacy and alexithymia in depressive undergraduates,and to provide a theoretical basis for the prevention and intervention of depression among undergraduates.Methods 500 undergraduates of two universities in Hebei Province were collected by cluster sampling method.Beck depression scale,adolescent resilience scale,Toronto alexithymia scale and self feeling of inadequacy scale were applied to undergraduates.The survey results were analyzed statistically.Results The detection rate of depression was 28％(119/426) in undergraduates.Total score of Alexithymia Scale(69.99±9.43) was higher than the norm(65.70±7.98),and the difference was statistical significant (P＜0.05).The score of self feeling of inadequacy had a significant difference between male (149.88±28.00) and female(138.58±28.79) (P＜0.01).Neither gender nor grade had significant difference in depression,alexithymia and resilience(P＞0.05).Self feeling of inadequacy was positively correlated with alexithymia (P＜0.05).Resilience was negatively correlated with self feeling of inadequacy and alexithymia(P＜0.05).For the depressive undergraduates,self feeling of inadequacy served as a full mediator between the resilience and alexithymia,whereas for the non depression undergraduates it was a partial mediator.Conclusion Depression undergraduates have serious alexithymia,serious self feeling of inadequacy and poor resilience.Self feeling of inadequacy serves as a mediator between the resilience and alexithymia in depressive and non-depressive undergraduates.%目的 探讨抑郁大学生心理韧性、自我缺陷感和述情障碍的关系,为抑郁大学生的干预预防提供理论依据.方法 通过整群抽样的方法抽取河北省两所高校的500名本科生作为调查对象,采用贝克抑郁量表、青少年心理韧性量表、多伦多述情障碍量表和自我缺陷感量表对大学生进行调查,对
Comparison between characteristics of mild slope equations and Boussinesq equations
Institute of Scientific and Technical Information of China (English)
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.
Energy Technology Data Exchange (ETDEWEB)
Menikoff, Ralph [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2015-12-15
The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.
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
Directory of Open Access Journals (Sweden)
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
Directory of Open Access Journals (Sweden)
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
DEFF Research Database (Denmark)
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
International Nuclear Information System (INIS)
The aim of this paper is to find an alternative to the usual electromagnetic wave equation: that is, we want to find a different equation with the same solutions. The final goal is to solve electromagnetic problems with iterative methods. The curl curl operator that appears in the electromagnetic wave equation is difficult to invert numerically, and this cannot be done iteratively. The addition of a higher order term that emphasizes the diagonal terms in the operator may help the solution of the problem, and the new equation should be solvable by an iterative algorithm. The additional mode is suppressed by suitable boundary conditions. (author) 5 figs., 9 refs
Correct Linearization of Einstein's Equations
Directory of Open Access Journals (Sweden)
Rabounski D.
2006-06-01
Full Text Available Regularly Einstein's equations can be reduced to a wave form (linearly dependent from the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel's symbols. As shown here, the origin of the problem is that one uses the general covariant theory of measurement. Here the wave form of Einstein's equations is obtained in the terms of Zelmanov's chronometric invariants (physically observable projections on the observer's time line and spatial section. The obtained equations depend on solely the second derivatives even if gravitation, the space rotation and Christoffel's symbols. The correct linearization proves: the Einstein equations are completely compatible with weak waves of the metric.
Diffusion equations and turbulent transport
International Nuclear Information System (INIS)
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
International Nuclear Information System (INIS)
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
Directory of Open Access Journals (Sweden)
Nils Rutstam
2012-04-01
Full Text Available The equations of motion for the rolling and gliding Tippe Top (TT are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion. To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis 3ˆ during the inversion. We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT.
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
Directory of Open Access Journals (Sweden)
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.
Partial Completion of Equational Theories
Institute of Scientific and Technical Information of China (English)
孙永强; 林凯; 陆朝俊
2000-01-01
In this paper, the notion of partial completion of equational theories is proposed, which is a procedure to construct a confluent term rewriting system from an equational theory without requirement of termination condition. A partial completion algorithm is presented with a brief description of its application in a program development system.
Differential equations a concise course
Bear, H S
2011-01-01
Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complex-valued solutions, linear differential operators, inverse operators and variation of parameters method, the Laplace transform, Picard's existence theorem, and an exploration of various interpretations of systems of equations. Numerous clearly stated theorems and proofs, examples, and problems followed by solutions.
Differential equations and moving frames
Abib, Odinette Renée
2006-01-01
The purpose of the paper is to study the relationship between differential equations, Pfaffian systems and geometric structures, via the method of moving frames of E.Cartan. We show a local structure theorem. The Lie algebra aspects differential equations is studied too.
Enclosing Solutions of Integral Equations
DEFF Research Database (Denmark)
Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole
1996-01-01
We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because of...
Solutions to Arithmetic Convolution Equations
Czech Academy of Sciences Publication Activity Database
Glöckner, H.; Lucht, L.G.; Porubský, Štefan
2007-01-01
Roč. 135, č. 6 (2007), s. 1619-1629. ISSN 0002-9939 R&D Projects: GA ČR GA201/04/0381 Institutional research plan: CEZ:AV0Z10300504 Keywords : arithmetic functions * Dirichlet convolution * polynomial equations * analytic equations * topological algebras * holomorphic functional calculus Subject RIV: BA - General Mathematics Impact factor: 0.520, year: 2007
International Nuclear Information System (INIS)
The Boltzmann-Uhlenbeck (BUU) equation, which is the time evolution of the wigner function of the single particle Green's function, is dervied by using the closed-time Green's function approach. The quantum mechanical approximation in derving the BUU equation is discussed
Phenomenological equations for reacting fluids
International Nuclear Information System (INIS)
A nonlocal phenomenological equation is introduced for a multicomponent fluid where chemical or nuclear reactions are taking place. The reciprocity between the nonlocal linear-coefficients is examined closely. An approximation reduces the nonlocal equation to the ordinary phenomenological relation with correction terms which show clearly a coupling of the reaction with the diffusion and the thermal conduction in an isotropic system. (auth.)
Uncertainty of empirical correlation equations
Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.
2016-08-01
The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.
Saturation and linear transport equation
Energy Technology Data Exchange (ETDEWEB)
Kutak, K.
2009-03-15
We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term. (orig.)
Saturation and linear transport equation
International Nuclear Information System (INIS)
We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term. (orig.)
Kocak, M.; Gonul, B.
2007-01-01
The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given.
Wigner transforms and Liouville equations
International Nuclear Information System (INIS)
Recent works concerning the semi-classical limit (h barred tending to zero) of the Quantum Mechanics linear and non linear models or equations, are presented. The non linear case is corresponding to mean field (or self consistent) models and gives, at the limit, the Vlasov equations of the Classical Statistical Mechanics. 48 refs
Singularity: Raychaudhuri equation once again
Indian Academy of Sciences (India)
Naresh Dadhich
2007-07-01
I first recount Raychaudhuri's deep involvement with the singularity problem in general relativity. I then argue that precisely the same situation has arisen today in loop quantum cosmology as obtained when Raychaudhuri discovered his celebrated equation. We thus need a new analogue of the Raychaudhuri equation in quantum gravity.
Nonlinear evolution equations and the Painleve test
International Nuclear Information System (INIS)
In this paper a survey is given of new results of the Painleve test and nonlinear evolution equations where ordinary- and partial-differential equations are considered. The authors study the semiclassical Haynes-Cumming model, the energy-eigenvalue-level-motion equation, the Kadomtsev-Petviashvili equation, the nonlinear Klein-Gordon equation and the self-dual Yang-Mills equation
Conservation Laws of Differential Equations in Finance
Institute of Scientific and Technical Information of China (English)
QIN Mao-Chang; MEI Feng-Xiang; SHANG Mei
2005-01-01
Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.
Integrability of equations for soliton's eigenfunctions
International Nuclear Information System (INIS)
Eigenfunctions of the auxiliary linear problems for the soliton equations obey the nonlinear evolution equations. It is shown that these eigenfunction equations are integrable by the inverse spectral transform method. Eigenfunction equations are also the generating equations. Several (1+1) and (2+1) dimensional eigenfunction equations and their properties are considered. 11 refs
Conservation laws of semidiscrete canonical Hamiltonian equations
International Nuclear Information System (INIS)
There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs. The same result holds for semidiscrete Hamiltonian equations. In this paper we consider semidiscrete canonical Hamiltonian equations. Using symmetries, we find conservation laws for the semidiscretized nonlinear wave equation and Schroedinger equation. (author)
Conservation Laws of Differential Equations in Finance
International Nuclear Information System (INIS)
Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.
Transport Equations for Oscillating Neutrinos
Zhang, Yunfan
2013-01-01
We derive a suite of generalized Boltzmann equations, based on the density-matrix formalism, that incorporates the physics of neutrino oscillations for two- and three-flavor oscillations, matter refraction, and self-refraction. The resulting equations are straightforward extensions of the classical transport equations that nevertheless contain the full physics of quantum oscillation phenomena. In this way, our broadened formalism provides a bridge between the familiar neutrino transport algorithms employed by supernova modelers and the more quantum-heavy approaches frequently employed to illuminate the various neutrino oscillation effects. We also provide the corresponding angular-moment versions of this generalized equation set. Our goal is to make it easier for astrophysicists to address oscillation phenomena in a language with which they are familiar. The equations we derive are simple and practical, and are intended to facilitate progress concerning oscillation phenomena in the context of core-collapse su...
Determining dynamical equations is hard
Cubitt, Toby S; Wolf, Michael M
2010-01-01
The behaviour of any physical system is governed by its underlying dynamical equations--the differential equations describing how the system evolves with time--and much of physics is ultimately concerned with discovering these dynamical equations and understanding their consequences. At the end of the day, any such dynamical law is identified by making measurements at different times, and computing the dynamical equation consistent with the acquired data. In this work, we show that, remarkably, this process is a provably computationally intractable problem (technically, it is NP-hard). That is, even for a moderately complex system, no matter how accurately we have specified the data, discovering its dynamical equations can take an infeasibly long time (unless P=NP). As such, we find a complexity-theoretic solution to both the quantum and the classical embedding problems; the classical version is a long-standing open problem, dating from 1937, which we finally lay to rest.
Nominal Logic with Equations Only
Clouston, Ranald
2011-01-01
Many formal systems, particularly in computer science, may be captured by equations modulated by side conditions asserting the "freshness of names"; these can be reasoned about with Nominal Equational Logic (NEL). Like most logics of this sort NEL employs this notion of freshness as a first class logical connective. However, this can become inconvenient when attempting to translate results from standard equational logic to the nominal setting. This paper presents proof rules for a logic whose only connectives are equations, which we call Nominal Equation-only Logic (NEoL). We prove that NEoL is just as expressive as NEL. We then give a simple description of equality in the empty NEoL-theory, then extend that result to describe freshness in the empty NEL-theory.
Generalizing the cosmic energy equation
International Nuclear Information System (INIS)
We generalize the cosmic energy equation to the case when massive particles interact via a modified gravitational potential of the form φ(a,r), which is allowed to explicitly depend upon the cosmological time through the expansion factor a(t). Using the nonrelativistic approximation for particle dynamics, we derive the equation for the cosmological expansion which has the form of the Friedmann equation with a renormalized gravitational constant. The generalized Layzer-Irvine cosmic energy equation and the associated cosmic virial theorem are applied to some recently proposed modifications of the Newtonian gravitational interaction between dark-matter particles. We also draw attention to the possibility that the cosmic energy equation may be used to probe the expansion history of the universe thereby throwing light on the nature of dark matter and dark energy.
Some Variations on Maxwell's Equations
Ascoli, G A; Ascoli, Giorgio A.; Goldin, Gerald A.
2006-01-01
In the first sections of this article, we discuss two variations on Maxwell's equations that have been introduced in earlier work---a class of nonlinear Maxwell theories with well-defined Galilean limits (and correspondingly generalized Yang-Mills equations), and a linear modification motivated by the coupling of the electromagnetic potential with a certain nonlinear Schroedinger equation. In the final section, revisiting an old idea of Lorentz, we write Maxwell's equations for a theory in which the electrostatic force of repulsion between like charges differs fundamentally in magnitude from the electrostatic force of attraction between unlike charges. We elaborate on Lorentz' description by means of electric and magnetic field strengths, whose governing equations separate into two fully relativistic Maxwell systems---one describing ordinary electromagnetism, and the other describing a universally attractive or repulsive long-range force. If such a force cannot be ruled out {\\it a priori\\/} by known physical ...
Stochastic differential equations, backward SDEs, partial differential equations
Pardoux, Etienne
2014-01-01
This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has...
Higher derivative gravity: field equation as the equation of state
Dey, Ramit; Mohd, Arif
2016-01-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. Extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Higher derivative gravity: Field equation as the equation of state
Dey, Ramit; Liberati, Stefano; Mohd, Arif
2016-08-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. The extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher-curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism-invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Soliton equations and Hamiltonian systems
Dickey, L A
2002-01-01
The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also becau
Galois theory of difference equations
Put, Marius
1997-01-01
This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.
Equational theories of tropical sernirings
DEFF Research Database (Denmark)
Aceto, Luca; Esik, Zoltan; Ingolfsdottir, Anna
2003-01-01
of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results. Udgivelsesdato......This paper studies the equational theories of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime...
Lectures on ordinary differential equations
Hurewicz, Witold
2014-01-01
Hailed by The American Mathematical Monthly as ""a rigorous and lively introduction,"" this text explores a topic of perennial interest in mathematics. The author, a distinguished mathematician and formulator of the Hurewicz theorem, presents a clear and lucid treatment that emphasizes geometric methods. Topics include first-order scalar and vector equations, basic properties of linear vector equations, and two-dimensional nonlinear autonomous systems. Suitable for senior mathematics students, the text begins with an examination of differential equations of the first order in one unknown funct
THE ERMAKOV EQUATION: A COMMENTARY
Directory of Open Access Journals (Sweden)
P. G. L. Leach
2008-08-01
Full Text Available We present a short history of the Ermakov Equation with an emphasis on its discovery by theWest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008, 123–145 for an English translation of Ermakov’s original paper.
Loop equations from differential systems
Eynard, Bertrand; Marchal, Olivier
2016-01-01
To any differential system $d\\Psi=\\Phi\\Psi$ where $\\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\\Phi$ is a Lie algebra $\\mathfrak g$ valued 1-form on a Riemann surface $\\Sigma$, is associated an infinite sequence of "correlators" $W_n$ that are symmetric $n$-forms on $\\Sigma^n$. The goal of this article is to prove that these correlators always satisfy "loop equations", the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.
Integral equation methods for electromagnetics
Volakis, John
2012-01-01
This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo
Direct 'delay' reductions of the Toda equation
International Nuclear Information System (INIS)
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated with this equation is obtained, also by reduction. (fast track communication)
Direct "Delay" Reductions of the Toda Equation
Joshi, Nalini
2008-01-01
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painlev\\'e equations. The Lax pair associated to this equation is obtained, also by reduction.
Integral Transform Approach to Generalized Tricomi Equations
Yagdjian, Karen
2014-01-01
We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14],[41]-[46] the particular version of this transform in order to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes.
Symmetry Analysis of Telegraph Equation
Nadjafikhah, Mehdi; Hejazi, Seyed Reza
2010-01-01
Lie symmetry group method is applied to study the Telegraph equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
Overdetermined Systems of Linear Equations.
Williams, Gareth
1990-01-01
Explored is an overdetermined system of linear equations to find an appropriate least squares solution. A geometrical interpretation of this solution is given. Included is a least squares point discussion. (KR)
Correct Linearization of Einstein's Equations
Directory of Open Access Journals (Sweden)
Rabounski D.
2006-04-01
Full Text Available Routinely, Einstein’s equations are be reduced to a wave form (linearly independent of the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel’s symbols. As shown herein, the origin of the problem is the use of the general covariant theory of measurement. Herein the wave form of Einstein’s equations is obtained in terms of Zelmanov’s chronometric invariants (physically observable projections on the observer’s time line and spatial section. The equations so obtained depend solely upon the second derivatives, even for gravitation, the space rotation and Christoffel’s symbols. The correct linearization proves that the Einstein equations are completely compatible with weak waves of the metric.
Spin equation and its solutions
Bagrov, V G; Baldiotti, M C; Levin, A D
2005-01-01
The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0+1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE.
Solving Differential Equations in R
Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...
Diophantine approximations and Diophantine equations
Schmidt, Wolfgang M
1991-01-01
"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum
Geophysical interpretation using integral equations
Eskola, L
1992-01-01
Along with the general development of numerical methods in pure and applied to apply integral equations to geophysical modelling has sciences, the ability improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assu med to have a back...
Improved beam propagation method equations.
Nichelatti, E; Pozzi, G
1998-01-01
Improved beam propagation method (BPM) equations are derived for the general case of arbitrary refractive-index spatial distributions. It is shown that in the paraxial approximation the discrete equations admit an analytical solution for the propagation of a paraxial spherical wave, which converges to the analytical solution of the paraxial Helmholtz equation. The generalized Kirchhoff-Fresnel diffraction integral between the object and the image planes can be derived, with its coefficients expressed in terms of the standard ABCD matrix. This result allows the substitution, in the case of an unaberrated system, of the many numerical steps with a single analytical step. We compared the predictions of the standard and improved BPM equations by considering the cases of a Maxwell fish-eye and of a Luneburg lens. PMID:18268554
Krabuanrat, T.
2000-01-01
This exploratory study examines the communication media choice of managers. Despite a substantial body of theories on media choice, inadequacies are apparent in the literature particularly in relation to modem communication technologies. A field study approach was adopted to explore some of these inadequacies and to study the media choice of subject from a manager background. Overall, within the limitations and confines of this exploratory study, this thesis has made the following contributio...
Scattering Equations and KLT Orthogonality
Cachazo, Freddy; He, Song; Yuan, Ellis Ye
2013-01-01
Several recent developments point to the fact that rational maps from n-punctured spheres to the null cone of D dimensional momentum space provide a natural language for describing the scattering of massless particles in D dimensions. In this note we identify and study equations relating the kinematic invariants and the puncture locations, which we call the scattering equations. We provide an inductive algorithm in the number of particles for their solutions and prove a remarkable property wh...
Czech Academy of Sciences Publication Activity Database
Agarwal, R. P.; Bohner, Martin; Řehák, Pavel
Dordrecht : Kluwer Academic Publishers, 2003 - (Agarwal, R.; O´Regan, D.), s. 1-56 Grant ostatní: GA ČR(CZ) GA201/01/0079; GA ČR(CZ) GP201/01/P041 Institutional research plan: CEZ:AV0Z1019905 Keywords : dynamic equation s * time scales * half-linear equation s Subject RIV: BA - General Mathematics
On solving periodic Riccati equations
Varga, Andreas
2008-01-01
Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooti...
Stochastic Geometric Partial Differential Equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Goldys, B.; Ondreját, Martin
1. Singapore : World Scientific Publishing Company, 2011 - (Zhao, H.; Truman, A.), s. 1-32 ISBN 978-981-4360-91-3. - (Interdisciplinary Mathematical Sciences. 12) R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic geometric * partial differential equations Subject RIV: BA - General Mathematics http://library.utia.cas.cz/separaty/2012/SI/ondrejat-stochastic geometric partial differential equations. pdf
Hidden Statistics of Schroedinger Equation
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
Hydrodynamic equations for granular mixtures
Garzo, V.; Dufty, J. W.
2003-01-01
Many features of granular media can be modeled by a fluid of hard spheres with inelastic collisions. Under rapid flow conditions, the macroscopic behavior of grains can be described through hydrodynamic equations accounting for dissipation among the interacting particles. A basis for the derivation of hydrodynamic equations and explicit expressions appearing in them is provided by the Boltzmann kinetic theory conveniently modified to account for inelastic binary collisions. The goal of this r...
Boltzmann equation and hydrodynamic fluctuations.
Colangeli, Matteo; Kröger, Martin; Ottinger, Hans Christian
2009-11-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics. PMID:20364972
Boltzmann equation and hydrodynamic fluctuations
Colangeli, M.; Kroger, M.; Ottinger, H. C.
2009-01-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.
A New Unified Evolution Equation
Lim, Jyh-Liong
1998-01-01
WE propose a new unified evolution equation for parton distribution functions appropriate for both large and small Bjorken x. Compared with the Ciafaloni- Catani-Fiorani-Marchesini equation, the cancellation of soft poles between virtual and real gluon emissions is made explicitly without introducing infrared cutoffs, next-to-leading contributions to the Sudakov resummation can be included systematically, and the scales of the running coupling constants are determined unambiguously.
Wave equations for pulse propagation
International Nuclear Information System (INIS)
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity. The memo discusses various ways of characterizing the polarization characteristics of plane waves, that is, of parameterizing a transverse unit vector, such as the Jones vector, the Stokes vector, and the Poincare sphere. It discusses the connection between macroscopically defined quantities, such as the intensity or, more generally, the Stokes parameters, and microscopic field amplitudes. The material presented here is a portion of a more extensive treatment of propagation to be presented separately. The equations presented here have been described in various books and articles. They are collected here as a summary and review of theory needed when treating pulse propagation
Computational partial differential equations using Matlab
Li, Jichun
2008-01-01
Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE
Equationally Compact Acts : Coproducts / Peeter Normak
Normak, Peeter
1998-01-01
In this article equational compactness of acts and its generalizations are discussed. As equational compactness does not carry over to coproducts a slight generalization of c-equational campactness is introduced. It is proved that a coproduct of acts is c-equationally compact if and only if all components are c-equationally campact
Exact solutions of the generalized Bretherton equation
Energy Technology Data Exchange (ETDEWEB)
Kudryashov, Nikolai A., E-mail: nakudr@gmail.co [Department of Applied Mathematics, National Research Nuclear University, MEPHI, 31 Kashirskoe Shosse, 115409 Moscow (Russian Federation); Sinelshchikov, Dmitry I.; Demina, Maria V. [Department of Applied Mathematics, National Research Nuclear University, MEPHI, 31 Kashirskoe Shosse, 115409 Moscow (Russian Federation)
2011-02-14
The generalized Bretherton equation is studied. The Baecklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.
Exact results for the Boltzmann equation and Smoluchowski's coagulation equation
International Nuclear Information System (INIS)
Almost no analytical solutions have been found for realistic intermolecular forces, largely due to the complicated structure of the collision term which calls for the construction of simplified models, in which as many physical properties are maintained as possible. In the first three chapters of this thesis such model Boltzmann equations are studied. Only spatially homogeneous gases with isotropic distribution functions are considered. Chapter I considers transition kernels, chapter II persistent scattering models and chapter III very hard particles. The second part of this dissertation deals with Smoluchowski's coagulation equation for the size distribution function in a coagulating system, with chapters devoted to the following topics: kinetics of gelation and universality, coagulation equations with gelation and exactly soluble models of nucleation. (Auth./C.F.)
Non-linear constitutive equations for gravitoelectromagnetism
Duplij, Steven; Di Grezia, Elisabetta; Esposito, Giampiero; Kotvytskiy, Albert
2013-01-01
This paper studies non-linear constitutive equations for gravitoelectromagnetism. Eventually, the problem is solved of finding, for a given particular solution of the gravity-Maxwell equations, the exact form of the corresponding non-linear constitutive equations.
Linear integral equations and soliton systems
International Nuclear Information System (INIS)
A study is presented of classical integrable dynamical systems in one temporal and one spatial dimension. The direct linearizations are given of several nonlinear partial differential equations, for example the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the sine-Gordon equation, the nonlinear Schroedinger equation, and the equation of motion for the isotropic Heisenberg spin chain; the author also discusses several relations between these equations. The Baecklund transformations of these partial differential equations are treated on the basis of a singular transformation of the measure (or equivalently of the plane-wave factor) occurring in the corresponding linear integral equations, and the Baecklund transformations are used to derive the direct linearization of a chain of so-called modified partial differential equations. Finally it is shown that the singular linear integral equations lead in a natural way to the direct linearizations of various nonlinear difference-difference equations. (Auth.)
Multi-Time Equations, Classical and Quantum
Petrat, Sören
2013-01-01
Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics.
ON THE EQUIVALENCE OF THE ABEL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations.
First-order partial differential equations
Rhee, Hyun-Ku; Amundson, Neal R
2001-01-01
This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of mo
Exponential function approach to parabolic equations
Lin, Chin-Yuan
2014-01-01
This volume is on initial-boundary value problems for parabolic partial differential equations of second order. It rewrites the problems as abstract Cauchy problems or evolution equations, and then solves them by the technique of elementary difference equations. Because of this, the volume assumes less background and provides an easy approach for readers to understand. Contents:Existence Theorems for Cauchy ProblemsExistence Theorems for Evolution Equations (I)Linear Autonomous Parabolic EquationsNonlinear Autonomous Parabolic EquationsLinea
How to obtain the covariant form of Maxwell's equations from the continuity equation
Energy Technology Data Exchange (ETDEWEB)
Heras, Jose A [Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Av. San Pablo No. 180, Col. Reynosa, 02200, Mexico D. F. (Mexico); Departamento de Fisica y Matematicas, Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Mexico D. F. 01210 (Mexico)
2009-07-15
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
Students' understanding of quadratic equations
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-05-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.
Scattering Equations and KLT Orthogonality
Cachazo, Freddy; Yuan, Ellis Ye
2013-01-01
Several recent developments point to the fact that rational maps from n-punctured spheres to the null cone of D dimensional momentum space provide a natural language for describing the scattering of massless particles in D dimensions. In this note we identify and study equations relating the kinematic invariants and the puncture locations, which we call the scattering equations. We provide an inductive algorithm in the number of particles for their solutions and prove a remarkable property which we call KLT Orthogonality. In a nutshell, KLT orthogonality means that "Parke-Taylor" vectors constructed from the solutions to the scattering equations are mutually orthogonal with respect to the Kawai-Lewellen-Tye (KLT) bilinear form. We end with comments on possible connections to gauge theory and gravity amplitudes in any dimension and to the high-energy limit of string theory amplitudes.
Random equations in nilpotent groups
Gilman, Robert; Romankov, Vitalii
2011-01-01
In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. We show that the set SAT(G,k) of all equations in k > 1 variables over G which are satisfiable in G has an intermediate asymptotic density in the space of all equations in k variables over G. When G is a free abelian group of finite rank, we compute this density precisely; otherwise we give some non-trivial upper and lower bounds. For k = 1 the set SAT(G,k) is negligible. Usually the asymptotic densities of interesting sets in groups are either zero or one. The results of this paper provide new examples of algebraically significant sets of intermediate asymptotic density.
Stability Analysis of Ecomorphodynamic Equations
Bärenbold, Fabian; Perona, Paolo
2014-01-01
Although riparian vegetation is present in or along many water courses of the world, its active role resulting from the interaction with flow and sediment processes has only recently become an active field of research. Especially, the role of vegetation in the process of river pattern formation has been explored and demonstrated mostly experimentally and numerically until now. In the present work, we shed light on this subject by performing a linear stability analysis on a simple model for riverbed vegetation dynamics coupled with the set of classical river morphodynamic equations. The vegetation model only accounts for logistic growth, local positive feedback through seeding and resprouting, and mortality by means of uprooting through flow shear stress. Due to the simplicity of the model, we can transform the set of equations into an eigenvalue problem and assess the stability of the linearized equations when slightly perturbated away from a spatially homogeneous solution. If we couple vegetation dynamics wi...
Confidence interval in Kirsch equations
Directory of Open Access Journals (Sweden)
Morteza Khodabin
2014-09-01
Full Text Available Rocks at depth are affected by stresses resulting from the weight of the overlying strata and tectonic stresses. When a tunnel is excavated in this rock, the stress field is locally disordered and radial, tangential and shear stresses are induced in the rock around the tunnel. Knowledge of the magnitudes and directions of these induced stresses is essential. Since the measuring of specific gravity and depth are inevitably affected by environmental noise, we consider a random version of P2 in Kirsch equations. By doing this, we define random version of the Kirsch equations. Then we introduce an algorithm to calculate confidence intervals for the Kirsch parameters. Finally we use Alborz tunnel characteristics for creating these confidence intervals as a case study. The results show that the proposed amounts of radial, tangential and shear stresses lie in desired range. Keywords: Kirsch equations; Confidence interval; confidence level; Alborz tunnel; normal distribution.
Quantum corrections for Boltzmann equation
Institute of Scientific and Technical Information of China (English)
M.; Levy; PETER
2008-01-01
We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.
The respiratory system in equations
Maury, Bertrand
2013-01-01
The book proposes an introduction to the mathematical modeling of the respiratory system. A detailed introduction on the physiological aspects makes it accessible to a large audience without any prior knowledge on the lung. Different levels of description are proposed, from the lumped models with a small number of parameters (Ordinary Differential Equations), up to infinite dimensional models based on Partial Differential Equations. Besides these two types of differential equations, two chapters are dedicated to resistive networks, and to the way they can be used to investigate the dependence of the resistance of the lung upon geometrical characteristics. The theoretical analysis of the various models is provided, together with state-of-the-art techniques to compute approximate solutions, allowing comparisons with experimental measurements. The book contains several exercises, most of which are accessible to advanced undergraduate students.
Introductory course on differential equations
Gorain, Ganesh C
2014-01-01
Introductory Course on DIFFERENTIAL EQUATIONS provides an excellent exposition of the fundamentals of ordinary and partial differential equations and is ideally suited for a first course of undergraduate students of mathematics, physics and engineering. The aim of this book is to present the elementary theories of differential equations in the forms suitable for use of those students whose main interest in the subject are based on simple mathematical ideas. KEY FEATURES: Discusses the subject in a systematic manner without sacrificing mathematical rigour. A variety of exercises drill the students in problem solving in view of the mathematical theories explained in the book. Worked out examples illustrated according to the theories developed in the book with possible alternatives. Exhaustive collection of problems and the simplicity of presentation differentiate this book from several others. Material contained will help teachers as well as aspiring students of different competitive examinations.
Dynamics of partial differential equations
Wayne, C Eugene
2015-01-01
This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation. The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equ...
On Reducing a System of Equations to a Single Equation
DEFF Research Database (Denmark)
Frandsen, G.S.; Shparlinski, I.E.
2004-01-01
For a system of polynomial equations over Q;p; we present an efficient construction of a single polynomial of quite small degree whose zero set over Q;p; coincides with the zero set over Q;p; of the original system. We also show that the polynomial has some other attractive features such as low...
Partial differential equations an introduction
Colton, David
2004-01-01
Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of
Group analysis of differential equations
Ovsiannikov, L V
1982-01-01
Group Analysis of Differential Equations provides a systematic exposition of the theory of Lie groups and Lie algebras and its application to creating algorithms for solving the problems of the group analysis of differential equations.This text is organized into eight chapters. Chapters I to III describe the one-parameter group with its tangential field of vectors. The nonstandard treatment of the Banach Lie groups is reviewed in Chapter IV, including a discussion of the complete theory of Lie group transformations. Chapters V and VI cover the construction of partial solution classes for the g
Differential equations and mathematical biology
Jones, DS; Sleeman, BD
2009-01-01
""… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions.""-SIAM Review, Sept. 2010, Vol. 52, No. 3""… Where this text stands out i
Radar equations for modern radar
Barton, David K
2012-01-01
Based on the classic Radar Range-Performance Analysis from 1980, this practical volume extends that work to ensure applicability of radar equations to the design and analysis of modern radars. This unique book helps you identify what information on the radar and its environment is needed to predict detection range. Moreover, it provides equations and data to improve the accuracy of range calculations. You find detailed information on propagation effects, methods of range calculation in environments that include clutter, jamming and thermal noise, as well as loss factors that reduce radar perfo
Basic linear partial differential equations
Treves, Francois
2006-01-01
Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their
Quantum Gross-Pitaevskii Equation
Haegeman, Jutho; Stojevic, Vid; Cirac, J Ignacio; Osborne, Tobias J; Verstraete, Frank
2015-01-01
We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum field theories. This generalization is obtained by applying the Dirac-Frenkel time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of the many body system including entanglement and correlations and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for one-dimensional systems.
Applied analysis and differential equations
Cârj, Ovidiu
2007-01-01
This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.
Geometric approach to soliton equations
International Nuclear Information System (INIS)
A class of nonlinear equations that can be solved in terms of nxn scattering problem is investigated. A systematic geometric method of exploiting conservation laws and related equations, the so-called prolongation structure, is worked out. The nxn problem is reduced to nsub(n-1)x(n-1) problems and finally to 2x2 problems, which have been comprehensively investigated recently by the author. A general method of deriving the infinite numbers of polynomial conservation laws for an nxn problem is presented. The cases of 3x3 and 2x2 problems are discussed explicitly. (Auth.)
Fundamentals of equations of state
Eliezer, Shalom; Hora, Heinrich
2002-01-01
The equation of state was originally developed for ideal gases, and proved central to the development of early molecular and atomic physics. Increasingly sophisticated equations of state have been developed to take into account molecular interactions, quantization, relativistic effects, etc. Extreme conditions of matter are encountered both in nature and in the laboratory, for example in the centres of stars, in relativistic collisions of heavy nuclei, in inertial confinement fusion (where a temperature of 10 9 K and a pressure exceeding a billion atmospheres can be achieved). A sound knowledg
Stability theory of differential equations
Bellman, Richard
2008-01-01
Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from
On a nonhomogeneous Burgers' equation
Institute of Scientific and Technical Information of China (English)
DING; Xiaqi(
2001-01-01
［1］Hopf, E., The partial differential equation ut + uux = μuxx, Comm. Pure Appl. Math., 1950, 3: 201-230.［2］Ding, X. Q. , Luo, P. Z. , Generalized expansions in Hilbert space, Acta Mathematica Scientia, 1999, 19(3): 241 250.［3］Titchmarsh, E., Introduction to the Theory of Fourier Integrals, 2nd ed., Oxford: Oxford University Press, 1948.［4］Ladyzhenskaya, O. A., Solonnikov, V. A., Ural' ceva, N. N., Linear and Quasilinear Equations of Parabolic Type,Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, 1968.
Modular equations and lattice sums
Rogers, Mathew; Yuttanan, Boonrod
2010-01-01
We highlight modular equations discovered by Somos and Ramanujan, and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd's Mahler measure conjectures, and we conjecture a new formula for $L(E,2)$ of conductor 17 elliptic curves.
International Nuclear Information System (INIS)
In 1977, Dave Young published an equation-of-state (EOS) for lithium. This EOS was used by Lew Glenn in his AFTON calculations of the HYLIFE inertial-fusion-reactor hydrodynamics. In this paper, I summarize Young's development of the EOS and demonstrate a computer program (MATHSY) that plots isotherms, isentropes and constant energy lines on a P-V diagram
Quaternionic Monge-Ampere equations
Alesker, Semyon
2002-01-01
The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternionic strictly pseudoconvex bounded domains in H^n. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].
Non Monotone Stochastic Evolution Equations
Kenneth L. Kuttler; Li, Ji
2013-01-01
An approach to stochastic evolution equations based on a simple generalization of known embedding theorems is presented. It allows for the inclusion of problems which have nonlinear non monotone operators. This is used to discuss the existence of strong solutions to a stochastic Navier Stokes problem in dimension less than four.
Convexity, Differential Equations, and Games
Flåm, Sjur Didrik
2002-01-01
Theoretical and experimental studies of noncooperative games increasingly recognize Nash equilibrium as a limiting outcome of players repeated interaction. This note, while sharing that view, illustrates and advocates combined use of convex optimization and differential equations, the purpose being to render equilibrium both plausible and stable.
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus. PMID:27586766
Pendulum Motion and Differential Equations
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
Conservation laws and kinetic equations
International Nuclear Information System (INIS)
It is shown that the conservation of a magnitude in a kinetic process can be stated at two different levels (microscopic and macroscopic) determining non-equivalent conditions on the associated transition probabilities. These conditions and related examples are explicitly given in the case of the Boltzmann equation. (author). 5 refs
Stability of Functional Differential Equations
Lemm, Jeffrey M
1986-01-01
This book provides an introduction to the structure and stability properties of solutions of functional differential equations. Numerous examples of applications (such as feedback systrems with aftereffect, two-reflector antennae, nuclear reactors, mathematical models in immunology, viscoelastic bodies, aeroautoelastic phenomena and so on) are considered in detail. The development is illustrated by numerous figures and tables.
Stochastic dynamic equations on general time scales
Martin Bohner; Olexandr M. Stanzhytskyi; Anastasiia O. Bratochkina
2013-01-01
In this article, we construct stochastic integral and stochastic differential equations on general time scales. We call these equations stochastic dynamic equations. We provide the existence and uniqueness theorem for solutions of stochastic dynamic equations. The crucial tool of our construction is a result about a connection between the time scales Lebesgue integral and the Lebesgue integral in the common sense.
Exact Vacuum Solutions to the Einstein Equation
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation. This procedure of resolution is based on a canonical form of the metric. According to this procedure, the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations,which are much convenient for the resolution.
Functional Equations and Inequalities with Applications
Kannappan, Palaniappan
2009-01-01
Presents a comprehensive study of the classical topic of functional equations. This monograph explores different aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis.
On Backward Stochstic Partial Differential Equations.
2001-01-01
We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations. This is a type of equations which appear as adjoint equations in the maximum principle approach to optimal control of systems described by stochastic partial differential equations.
Algebraic entropy for differential-delay equations
Viallet, Claude M.
2014-01-01
We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.
The AGL equation from the dipole picture
International Nuclear Information System (INIS)
The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to a unitarized gluon distribution in the small x regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this paper that the AGL equation and, consequently, the GLR equation, can also be obtained from the dipole picture in the double logarithmic limit, using an evolution equation, recently proposed, which includes all multiple pomeron exchanges in the leading logarithmic approximation. Our conclusion is that the AGL equation is a good candidate for a unitarized evolution equation at small x in the DLA limit
Thermoviscous Model Equations in Nonlinear Acoustics
DEFF Research Database (Denmark)
Rasmussen, Anders Rønne
solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties of......Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared to...... solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....
More conservative governing equations in RELAP5: Derivation of equations
International Nuclear Information System (INIS)
Highlights: • The “non-conservative” numerical approximation is used in current versions of RELAP5. • Mass and energy errors increase for some transients due to non-conservativism. • This paper shows the derivation of a numerical approach to eliminate the mass error. • The second article (Fu et al., 2015) shows the (strategic) solution for the approach. - Abstract: The design and analysis of the thermal/hydraulic systems in nuclear power plants necessitate system codes that can be used in the analysis of steady state and transient conditions. RELAP5 is one of the most commonly used system codes in nuclear organizations. RELAP5 is based on a two-fluid, non-equilibrium, non-homogeneous, hydrodynamic model for the transient simulation of the two-phase system behavior. This model includes six governing equations to describe the mass, energy, and momentum of the two fluids. The “non-conservative” numerical approximation form (which is the current version of RELAP5 code) is obtained through the manipulation of selected derivative terms in the equations including the linearization of the product terms in the time derivatives of the equations. In the non-conservative technique, the truncation errors introduced in the linearization process can produce mass and energy errors for some classes of transients during time advancements, either resulting in (a) automatic reduction of time steps used in the advancement of the equations and increased run times or (b) the growth of unacceptably large errors in the transient results. To eliminate these difficulties, an optional numerical approach has been introduced in RELAP/SCDAPSIM/MOD4.0. This approach uses a more consistent set of “conservative” numerical approximations to solve non-linearized mass and energy governing equations. The RELAP/SCDAPSIM/MOD4.0 code, being developed as part of the international Severe Core Damage Analysis Package (SCDAP) Development and Training Program (SDTP), is the first version of
Dual Isomonodromic Problems and Whitham Equations
Takasaki, Kanehisa
1997-01-01
The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter $\\epsilon$. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the Seiberg- Wit...
Integrable (2k)-Dimensional Hitchin Equations
Ward, R S
2016-01-01
This letter describes a completely-integrable system of Yang-Mills-Higgs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k=2 case are described.
The Pauli equation in scale relativity
Celerier, Marie-Noelle; Nottale, Laurent
2006-01-01
In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and cont...
Maxwell's equations of electrodynamics an explanation
Ball, David W
2012-01-01
Maxwell's Equations of Electrodynamics: An Explanation is a concise discussion of Maxwell's four equations of electrodynamics - the fundamental theory of electricity, magnetism, and light. It guides readers step-by-step through the vector calculus and development of each equation. Pictures and diagrams illustrate what the equations mean in basic terms. The book not only provides a fundamental description of our universe but also explains how these equations predict the fact that light is better described as "electromagnetic radiation."
Techniques for solving Boolean equation systems
Keinänen, Misa
2006-01-01
Boolean equation systems are ordered sequences of Boolean equations decorated with least and greatest fixpoint operators. Boolean equation systems provide a useful framework for formal verification because various specification and verification problems, for instance, μ-calculus model checking can be represented as the problem of solving Boolean equation systems. The general problem of solving a Boolean equation system is a computationally hard task, and no polynomial time solution technique ...
Numerical Solution of Parabolic Equations
DEFF Research Database (Denmark)
Østerby, Ole
These lecture notes are designed for a one-semester course on finite-difference methods for parabolic equations. These equations which traditionally are used for describing diffusion and heat-conduction problems in Geology, Physics, and Chemistry have recently found applications in Finance Theory....... Among the special features of this book can be mentioned the presentation of a practical approach to reliable estimates of the global error, including warning signals if the reliability is questionable. The technique is generally applicable for estimating the discretization error in numerical...... expense. Problems in two space dimensions are effectively handled using the Alternating Direction Implicit (ADI) technique. We present a systematic way of incorporating inhomogeneous terms and derivative boundary conditions in ADI methods as well as mixed derivative terms....
Algebrization of Nonautonomous Differential Equations
Directory of Open Access Journals (Sweden)
María Aracelia Alcorta-García
2015-01-01
Full Text Available Given a planar system of nonautonomous ordinary differential equations, dw/dt=F(t,w, conditions are given for the existence of an associative commutative unital algebra A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t,w=H(te,w and the maps H1(τ=H(τ,ξ and H2(ξ=H(τ,ξ are Lorch differentiable with respect to A for all (τ,ξ∈Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ of the differential equation dξ/dτ=H(τ,ξ over A define solutions (x(t,y(t=ξ(te of the planar system.
Nonlocal higher order evolution equations
Rossi, Julio D.
2010-06-01
In this article, we study the asymptotic behaviour of solutions to the nonlocal operator ut(x, t)1/4(-1)n-1 (J*Id -1)n (u(x, t)), x ∈ ℝN, which is the nonlocal analogous to the higher order local evolution equation vt(-1)n-1(Δ)nv. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. © 2010 Taylor & Francis.
Differential Equations of Ideal Memristors
Directory of Open Access Journals (Sweden)
Z. Biolek
2015-06-01
Full Text Available Ideal memristor is a resistor with a memory, which adds dynamics to its behavior. The most usual characteristics describing this dynamics are the constitutive relation (i.e. the relation between flux and charge, or Parameter-vs-state- map (PSM, mostly represented by the memristance-to-charge dependence. One of the so far unheeded tools for memristor description is its differential equation (DEM, composed exclusively of instantaneous values of voltage, current, and their derivatives. The article derives a general form of DEM that holds for any ideal memristor and shows that it is always a nonlinear equation of the first order; the PSM forms are found for memristors which are governed by DEMs of the Bernoulli and the Riccati types; a classification of memristors according to the type of their dynamics with respect to voltage and current is carried out.
The equations icons of knowledge
Bais, Sander
2005-01-01
For thousands of years mankind has tried to understand nature. Exploring the world on all scales with instruments of ever more ingenuity, we have been able to unravel some of the great mysteries that surround us. While collecting an overwhelming multitude of observational facts, we discovered fundamental laws that govern the structure and evolution of physical reality. We know that nature speaks to us in the language of mathematics. In this language most of our basic understanding of the physical world can be expressed in an unambiguous and concise way. The most artificial language turns out to be the most natural of all. The laws of nature correspond to equations. These equations are the icons of knowledge that mark crucial turning points in our thinking about the world we happen to live in. They form the symbolic representation of most of what we know, and as such constitute an important and robust part of our culture.
Sensitivity for the Smoluchowski equation
Energy Technology Data Exchange (ETDEWEB)
Bailleul, I F, E-mail: i.bailleul@statslab.cam.ac.uk [Statistical Laboratory, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB (United Kingdom)
2011-06-17
This paper investigates the question of sensitivity of the solutions {mu}{sup {lambda}}{sub t} of the Smoluchowski equation on R{sub +}* with respect to the parameters {lambda} in the interaction kernel K{sup {lambda}}. It is proved that {mu}{sup {lambda}}{sub t} is a C{sup 1} function of (t, {lambda}) with values in a good space of measures under the hypotheses K{sup {lambda}}(x, y) {<=} {psi}(x) {psi}(y), for some sub-linear function {psi}, and {integral}{psi}{sup 4+{epsilon}}(x) {mu}{sub 0}(dx) < {infinity}, and that the derivative is the unique solution of a related equation.
Handbook of structural equation modeling
Hoyle, Rick H
2012-01-01
The first comprehensive structural equation modeling (SEM) handbook, this accessible volume presents both the mechanics of SEM and specific SEM strategies and applications. The editor, contributors, and editorial advisory board are leading methodologists who have organized the book to move from simpler material to more statistically complex modeling approaches. Sections cover the foundations of SEM; statistical underpinnings, from assumptions to model modifications; steps in implementation, from data preparation through writing the SEM report; and basic and advanced applications, inclu
Lagrange-Poincare field equations
Ellis, David C P; Holm, Darryl D; Ratiu, Tudor S
2009-01-01
The Lagrange-Poincare equations of classical mechanics are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The Kelvin-Noether theorem is formulated in this context. Applications to the isoperimetric problem, the Skyrme model for meson interaction, metamorphosis image dynamics, and molecular strands illustrate various aspects of the theory.
Some Applications of Fractional Equations
Weitzner, H.; Zaslavsky, G. M.
2002-01-01
We present two observations related to theapplication of linear (LFE) and nonlinear fractional equations (NFE). First, we give the comparison and estimates of the role of the fractional derivative term to the normal diffusion term in a LFE. The transition of the solution from normal to anomalous transport is demonstrated and the dominant role of the power tails in the long time asymptotics is shown. Second, wave propagation or kinetics in a nonlinear media with fractal properties is considere...
Eigenwavelets of the Wave equation
Kaiser, Gerald
2004-01-01
We study a class of localized solutions of the wave equation, called eigenwavelets, obtained by extending its fundamental solutions to complex spacetime in the sense of hyperfunctions. The imaginary spacetime variables y, which form a timelike vector, act as scale parameters generalizing the scale variable of wavelets in one dimension. They determine the shape of the wavelets in spacetime, making them pulsed beams that can be focused as tightly as desired around a single ray by letting y appr...
Riccati Equations and their Solution
Czech Academy of Sciences Publication Activity Database
Kučera, Vladimír
Control System Advanced Methods. Boca Raton: CRC Press, 2011 - (Lewine, W.), s. 14.1-14.21. (Electrical Engineering Handbook). ISBN 978-1-4200-7366-9 R&D Projects: GA MŠk(CZ) 1M0567 Institutional research plan: CEZ:AV0Z10750506 Institutional support: RVO:67985556 Keywords : Riccati equation * optimal control * solution Subject RIV: BC - Control Systems Theory http://library.utia.cas.cz/separaty/2011/TR/kucera-0436431.pdf
Generalized bootstrap for estimating equations
Chatterjee, Snigdhansu; Bose, Arup
2005-01-01
We introduce a generalized bootstrap technique for estimators obtained by solving estimating equations. Some special cases of this generalized bootstrap are the classical bootstrap of Efron, the delete-d jackknife and variations of the Bayesian bootstrap. The use of the proposed technique is discussed in some examples. Distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained.
Instantaneous Bethe-Salpeter equation
International Nuclear Information System (INIS)
We present a systematic algebraic and numerical investigation of the instantaneous Beth-Salpeter equation. Emphasis is placed on confining interaction kernels of the Lorentz scalar, time component vector, and full vector-types. We explore the stability of the solutions and Regge behavior for each of these interactions, and conclude that only time component vector confinement leads to normal Regge structure and stable solutions for all quark masses
Equation of State Project Overview
Energy Technology Data Exchange (ETDEWEB)
Crockett, Scott [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2015-09-11
A general overview of the Equation of State (EOS) Project will be presented. The goal is to provide the audience with an introduction of what our more advanced methods entail (DFT, QMD, etc.. ) and how these models are being utilized to better constrain the thermodynamic models. These models substantially reduce our regions of interpolation between the various thermodynamic limits. I will also present a variety example of recent EOS work.
Wave equations in higher dimensions
Dong, Shi-Hai
2011-01-01
Higher dimensional theories have attracted much attention because they make it possible to reduce much of physics in a concise, elegant fashion that unifies the two great theories of the 20th century: Quantum Theory and Relativity. This book provides an elementary description of quantum wave equations in higher dimensions at an advanced level so as to put all current mathematical and physical concepts and techniques at the reader’s disposal. A comprehensive description of quantum wave equations in higher dimensions and their broad range of applications in quantum mechanics is provided, which complements the traditional coverage found in the existing quantum mechanics textbooks and gives scientists a fresh outlook on quantum systems in all branches of physics. In Parts I and II the basic properties of the SO(n) group are reviewed and basic theories and techniques related to wave equations in higher dimensions are introduced. Parts III and IV cover important quantum systems in the framework of non-relativisti...
The complex chemical Langevin equation
International Nuclear Information System (INIS)
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE’s main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE’s predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE’s accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the “complex CLE” predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation
Torsion Effects and LLG Equation
Ferreira, Cristine N; Neto, J A Helayël
2016-01-01
Based on the non-relativistic regime of the Dirac equation coupled to a torsion pseudo-vector, we study the dynamics of magnetization and how it is affected by the presence of torsion. We consider that torsion interacting terms in Dirac equation appear in two ways one of these is thhrough the covariant derivative considering the spin connection and gauge magnetic field and the other is through a non-minimal spin torsion coupling. We show within this framework, that it is possible to obtain the most general Landau, Lifshitz and Gilbert (LLG) equation including the torsion effects, where we refer to torsion as a geometric field playing an important role in the spin coupling process. We show that the torsion terms can give us two important landscapes in the magnetization dynamics: one of them related with damping and the other related with the screw dislocation that give us a global effect like a helix damping sharped. These terms are responsible for changes in the magnetization precession dynamics.
Effective Schroedinger equations on submanifolds
International Nuclear Information System (INIS)
In this thesis the time dependent Schroedinger equation is considered on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When the potential is scaled in the directions normal to C by a small parameter epsilon, the solutions concentrate in an epsilon-neighborhood of the submanifold. An effective Schroedinger equation on the submanifold C is derived and it is shown that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order ε3 vertical stroke t vertical stroke at time t. Furthermore, it is proved that, under reasonable conditions, the eigenvalues of the corresponding Hamiltonians below a certain energy coincide upto errors of order ε3. These results holds in the situation where tangential and normal energies are of the same order, and where exchange between normal and tangential energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were needed, to ensure that the separation of energies is maintained during the time evolution. The most important consequence of this thesis is that now constraining potentials that change their shape along the submanifold can be treated, which is the typical situation in applications like molecular dynamics and quantum waveguides.
Effective Schroedinger equations on submanifolds
Energy Technology Data Exchange (ETDEWEB)
Wachsmuth, Jakob
2010-02-11
In this thesis the time dependent Schroedinger equation is considered on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When the potential is scaled in the directions normal to C by a small parameter epsilon, the solutions concentrate in an epsilon-neighborhood of the submanifold. An effective Schroedinger equation on the submanifold C is derived and it is shown that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order {epsilon}{sup 3} vertical stroke t vertical stroke at time t. Furthermore, it is proved that, under reasonable conditions, the eigenvalues of the corresponding Hamiltonians below a certain energy coincide upto errors of order {epsilon}{sup 3}. These results holds in the situation where tangential and normal energies are of the same order, and where exchange between normal and tangential energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were needed, to ensure that the separation of energies is maintained during the time evolution. The most important consequence of this thesis is that now constraining potentials that change their shape along the submanifold can be treated, which is the typical situation in applications like molecular dynamics and quantum waveguides.
Dynamical equations and approximation methods
International Nuclear Information System (INIS)
The integral equations approach to the three-body problem, decisively stimulated by Faddeev's formulation, provides the most powerful tool for studying the internal structure of this system. An essential step towards a detailed understanding of composite particle dynamics has been done in this way. The search for adequate extensions to the general N-body situation therefore represented, and still represents a natural challenge. For various reasons this transition is non-trivial and non-unique. Emphasizing different aspects of the three-body theory, different generalizations have been found. In particular, it was the concept of connectedness of the (iterated) integral kernel which allows for an arbitrary number of formulations, many of them being presumably only mathematically correct, but physically rather unsatisfactory. Therefore, the present status of the N-body theory is reviewed in a less technical way. Starting from the basic, physically convincing definitions of scattering states, the defining equations are replaced by more appropriate matrix relations. This is done in a reversible way, thus preserving in every step the original structure and information. In order to be as close as possible to the basic definitions, all relations are first derived for scattering states or half-on-shell transition amplitudes. The ambiguity in going over to corresponding operator identities (fully off-shell equations) is demonstrated. (Auth.)
Neutron star equation of state
International Nuclear Information System (INIS)
Experimental information concerning the equation of state in neutron stars is lacking, because of the necessary extrapolations in both density and neutron excess from the nearly symmetric nuclear matter observed in nuclei. However, the combination of new developments in the theory of neutron star structure and in astronomical observations provides important constraints. From a theoretical perspective, it is argued that the extrapolation in neutron excess is more crucial for neutron star structure than is the density extrapolation. For example, the radius of neutron stars is primarily a function of the pressure of matter in the vicinity of nuclear matter density, which is essentially determined by the isospin properties of dense matter. In the absence of extreme softening in the dense matter equation of state, a measurement of the radius of a neutron star more accurate than about 1 km will usefully constrain the equation of state. In addition, the moment of inertial and the binding energy of neutron stars are nearly universal functions of the star's compactness. The potential constraints that can be deduced from observations of thermal emission from young neutron stars, neutrinos from newly born neutron stars, Quasi-Periodic Oscillations from X-ray emitting neutron stars in binaries, and glitches from pulsars are discussed
ADVANCED WAVE-EQUATION MIGRATION
Energy Technology Data Exchange (ETDEWEB)
L. HUANG; M. C. FEHLER
2000-12-01
Wave-equation migration methods can more accurately account for complex wave phenomena than ray-tracing-based Kirchhoff methods that are based on the high-frequency asymptotic approximation of waves. With steadily increasing speed of massively parallel computers, wave-equation migration methods are becoming more and more feasible and attractive for imaging complex 3D structures. We present an overview of several efficient and accurate wave-equation-based migration methods that we have recently developed. The methods are implemented in the frequency-space and frequency-wavenumber domains and hence they are called dual-domain methods. In the methods, we make use of different approximate solutions of the scalar-wave equation in heterogeneous media to recursively downward continue wavefields. The approximations used within each extrapolation interval include the Born, quasi-Born, and Rytov approximations. In one of our dual-domain methods, we use an optimized expansion of the square-root operator in the one-way wave equation to minimize the phase error for a given model. This leads to a globally optimized Fourier finite-difference method that is a hybrid split-step Fourier and finite-difference scheme. Migration examples demonstrate that our dual-domain migration methods provide more accurate images than those obtained using the split-step Fourier scheme. The Born-based, quasi-Born-based, and Rytov-based methods are suitable for imaging complex structures whose lateral variations are moderate, such as the Marmousi model. For this model, the computational cost of the Born-based method is almost the same as the split-step Fourier scheme, while other methods takes approximately 15-50% more computational time. The globally optimized Fourier finite-difference method significantly improves the accuracy of the split-step Fourier method for imaging structures having strong lateral velocity variations, such as the SEG/EAGE salt model, at an approximately 30% greater
Estimation of saturation and coherence effects in the KGBJS equation - a non-linear CCFM equation
Deak, Michal
2012-01-01
We solve the modified non-linear extension of the CCFM equation - KGBJS equation - numerically for certain initial conditions and compare the resulting gluon Green functions with those obtained from solving the original CCFM equation and the BFKL and BK equations for the same initial conditions. We improve the low transversal momentum behaviour of the KGBJS equation by a small modification.
Kinetic equations for an unstable plasma
International Nuclear Information System (INIS)
In this work, we establish the plasma kinetic equations starting from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations. We demonstrate that relations existing between correlation functions may help to justify the truncation of the hierarchy. Then we obtain the kinetic equations of a stable or unstable plasma. They do not reduce to an equation for the one-body distribution function, but generally involve two coupled equations for the one-body distribution function and the spectral density of the fluctuating electric field. We study limiting cases where the Balescu-Lenard equation, the quasi-linear theory, the Pines-Schrieffer equations and the equations of weak turbulence in the random phase approximation are recovered. At last we generalise the H-theorem for the system of equations and we define conditions for irreversible behaviour. (authors)
Gibbs adsorption and the compressibility equation
International Nuclear Information System (INIS)
A new approach for deriving the equation of state is developed. It is shown that the integral in the compressibility equation is identical to the isotherm for Gibbs adsorption in radial coordinates. The Henry, Langmuir, and Frumkin adsorption isotherms are converted into equations of state. It is shown that using Henry's law gives an expression for the second virial coefficient that is identical to the result from statistical mechanics. Using the Langmuir isotherm leads to a new analytic expression for the hard-sphere equation of state which can be explicit in either pressure or density. The Frumkin isotherm results in a new equation of state for the square-well potential fluid. Conversely, new adsorption isotherms can be derived from equations of state using the compressibility equation. It is shown that the van der Waals equation gives an adsorption isotherm equation that describes both polymolecular adsorption and the unusual adsorption behavior observed for supercritical fluids. copyright 1995 American Institute of Physics
From Newton's Equation to Fractional Diffusion and Wave Equations
Directory of Open Access Journals (Sweden)
Vázquez Luis
2011-01-01
Full Text Available Fractional calculus represents a natural instrument to model nonlocal (or long-range dependence phenomena either in space or time. The processes that involve different space and time scales appear in a wide range of contexts, from physics and chemistry to biology and engineering. In many of these problems, the dynamics of the system can be formulated in terms of fractional differential equations which include the nonlocal effects either in space or time. We give a brief, nonexhaustive, panoramic view of the mathematical tools associated with fractional calculus as well as a description of some fields where either it is applied or could be potentially applied.
Handbook of differential equations stationary partial differential equations
Chipot, Michel
2006-01-01
This handbook is volume III in a series devoted to stationary partial differential quations. Similarly as volumes I and II, it is a collection of self contained state-of-the-art surveys written by well known experts in the field. The topics covered by this handbook include singular and higher order equations, problems near critically, problems with anisotropic nonlinearities, dam problem, T-convergence and Schauder-type estimates. These surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics. Ke
Partial differential equations of mathematical physics and integral equations
Guenther, Ronald B
1996-01-01
This book was written to help mathematics students and those in the physical sciences learn modern mathematical techniques for setting up and analyzing problems. The mathematics used is rigorous, but not overwhelming, while the authors carefully model physical situations, emphasizing feedback among a beginning model, physical experiments, mathematical predictions, and the subsequent refinement and reevaluation of the physical model itself. Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the t
International Workshop on Elliptic and Parabolic Equations
Schrohe, Elmar; Seiler, Jörg; Walker, Christoph
2015-01-01
This volume covers the latest research on elliptic and parabolic equations and originates from the international Workshop on Elliptic and Parabolic Equations, held September 10-12, 2013 at the Leibniz Universität Hannover. It represents a collection of refereed research papers and survey articles written by eminent scientist on advances in different fields of elliptic and parabolic partial differential equations, including singular Riemannian manifolds, spectral analysis on manifolds, nonlinear dispersive equations, Brownian motion and kernel estimates, Euler equations, porous medium type equations, pseudodifferential calculus, free boundary problems, and bifurcation analysis.
Mathematical physics with partial differential equations
Kirkwood, James
2011-01-01
Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field - the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform
Partial differential equations for scientists and engineers
Farlow, Stanley J
1993-01-01
Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing th
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
BMN Correlators by Loop Equations
Eynard, B.; Kristjansen, C.
2002-01-01
In the BMN approach to N=4 SYM a large class of correlators of interest are expressible in terms of expectation values of traces of words in a zero-dimensional Gaussian complex matrix model. We develop a loop-equation based, analytic strategy for evaluating such expectation values to any order in the genus expansion. We reproduce the expectation values which were needed for the calculation of the one-loop, genus one correction to the anomalous dimension of BMN-operators and which were earlier...
PICARD ITERATION FOR NONSMOOTH EQUATIONS
Institute of Scientific and Technical Information of China (English)
Song-bai Sheng; Hui-fu Xu
2001-01-01
This paper presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. It is proved that the radius of the weak Jacobian (RGJ) of Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results can be obtained provided that RGJ of the Picard iteration function at a solution point is less than one or equal to zero. As for applications, it is pointed out that the approximate Newton methods, the generalized Newton method for piecewise C1problems and splitting methods can be explained uniformly with the same viewpoint.
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.
Sensitivity of rough differential equations
Coutin, Laure; Lejay, Antoine
2015-01-01
In the context of rough paths theory, we study the regularity of the Itô map with respect to the starting point, the coefficients and perturbation of the driving rough paths. In particular, we show that the Itô mapis differentiable with a Hölder or Lipschitz continuousderivatives under general hypotheses. With respect to the current literatureon the subject, our proof relies on perturbation of linearRough Differential Equations, and the Hölder regularity of the Itô map is established.
Introduction to Piecewise Differentiable Equations
Scholtes, Stefan
2012-01-01
This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations. In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the non smooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop
Advanced lab on Fresnel equations
Petrova-Mayor, Anna; Gimbal, Scott
2015-11-01
This experimental and theoretical exercise is designed to promote students' understanding of polarization and thin-film coatings for the practical case of a scanning protected-metal coated mirror. We present results obtained with a laboratory scanner and a polarimeter and propose an affordable and student-friendly experimental arrangement for the undergraduate laboratory. This experiment will allow students to apply basic knowledge of the polarization of light and thin-film coatings, develop hands-on skills with the use of phase retarders, apply the Fresnel equations for metallic coating with complex index of refraction, and compute the polarization state of the reflected light.
Experimental determination of circuit equations
Shulman, Jason; Widjaja, Matthew; Gunaratne, Gemunu H
2013-01-01
Kirchhoff's laws offer a general, straightforward approach to circuit analysis. Unfortunately, use of the laws becomes impractical for all but the simplest of circuits. This work presents a novel method of analyzing direct current resistor circuits. It is based on an approach developed to model complex networks, making it appropriate for use on large, complicated circuits. It is unique in that it is not an analytic method. It is based on experiment, yet the approach produces the same circuit equations obtained by more traditional means.
BMN correlators by loop equations
International Nuclear Information System (INIS)
In the BMN approach to N=4 SYM a large class of correlators of interest are expressible in terms of expectation values of traces of words in a zero-dimensional gaussian complex matrix model. We develop a loop-equation based, analytic strategy for evaluating such expectation values to any order in the genus expansion. We reproduce the expectation values which were needed for the calculation of the one-loop, genus one correction to the anomalous dimension of BMN-operators and which were earlier obtained by combinatorial means. Furthermore, we present the expectation values needed for the calculation of the one-loop, genus two correction. (author)
Ising models and soliton equations
International Nuclear Information System (INIS)
Several new results for the critical point of correlation functions of the Hirota equation are derived within the two-dimensional Ising model. The recent success of the conformal-invariance approach in the determination of a critical two-spin correration function is analyzed. The two-spin correlation function is predicted to be rotationally invariant and to decay with a power law in this approach. In the approach suggested here systematic corrections due to the underlying lattice breaking the rotational invariance are obtained
On the connection of the quadratic Lienard equation with an equation for the elliptic functions
Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I.
2015-07-01
The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear second-order differential equation has been discussed. Here we show that the whole family of quadratic Lienard equations can be transformed into an equation for the elliptic functions. We demonstrate that this connection can be useful for finding explicit forms of general solutions of the quadratic Lienard equation. We provide several examples of application of our approach.
Asymptotic-group analysis of algebraic equations
Shamrovskii, A. D.; I. V. Andrianov; J. Awrejcewicz
2004-01-01
Both the method of asymptotic analysis and the theory of extension group are applied to study the Descates equation. The proposed algorithm allows to obtain various variants of simplification and can be easily generalized to their algebraic and differential equations.
Linear superposition solutions to nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
Liu Yu
2012-01-01
The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.
Simultaneous Independent Linear Equations and Goldbach Conjecture
Linggen, Song
2007-01-01
It was verified that if Goldbach conjecture was a fault, the number of simultaneous independent linear equations educed from this assumption would be unreasonably at least one more than the number of unknowns involved in these equations.
The Spin-2 Equation on Minkowski Background
Beyer, Florian; Frauendiener, Jörg; Whale, Ben
2014-01-01
The linearised general conformal field equations in their first and second order form are used to study the behaviour of the spin-2 zero-rest-mass equation on Minkowski background in the vicinity of space-like infinity.
Hydrodynamic Equations for Microscopic Phase Densities
Gerasimenko, V. I.; Shtyk, V. O.; Zagorodny, A. G.
2009-01-01
The evolution equations for the generalized microscopic phase densities are introduced. The evolution equations of average values of microscopic phase densities are derived and a solution of the initial-value problem of the obtained hydrodynamic type hierarchy is constructed.
On oscillatory solutions of certain difference equations
Directory of Open Access Journals (Sweden)
Grzegorz Grzegorczyk
2006-01-01
Full Text Available Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed.
On oscillatory solutions of certain difference equations
Grzegorz Grzegorczyk; Jarosław Werbowski
2006-01-01
Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed.
OSCILLATION CRITERIA FOR FORCED SUPERLINEAR DIFFERENCE EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Using Riccati transformation techniques,some oscillation criteria for the forced second-order superlinear difference equations are established.These criteria are dis- crete analogues of the criteria for differential equations proposed by Yan.
Fractional Complex Transform for Fractional Differential Equations
Li, Zheng-Biao; He, Ji-Huan
2010-01-01
Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Two examples are given.
A new class of variational equation problems
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
Applying an analysis method to a group of multivariable equations, a new class of variational equations are proved. This method is more concise and more direct than the others. This result can be applied to some stochastic control models.
Exact Solutions to Short Pulse Equation
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; ZHENG Ming-Hua; LIU Shi-Kuo
2009-01-01
In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.
Exact Solutions to Degasperis-Procesi Equation
Institute of Scientific and Technical Information of China (English)
ZHANG Lin-Na; FU Zun-Tao; LIU Shi-Kuo
2008-01-01
In this paper,dependent and independent variable transformations are introduced to solve the Degasperis-Procesi equation.It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.
Introduction to linear algebra and differential equations
Dettman, John W
1986-01-01
Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
An axiomatic approach to Maxwell's equations
Heras, José A
2016-01-01
This paper suggests an axiomatic approach to Maxwell's equations. The basis of this approach is a theorem formulated for two sets of functions localized in space and time. If each set satisfies a continuity equation then the theorem provides an integral representation for each function. A corollary of this theorem yields Maxwell's equations with magnetic monopoles. It is pointed out that the causality principle and the conservation of electric and magnetic charges are the most fundamental physical axioms underlying these equations. Another application of the corollary yields Maxwell's equations in material media. The theorem is also formulated in the Minkowski space-time and applied to obtain the covariant form of Maxwell's equations with magnetic monopoles and the covariant form of Maxwell's equations in material media. The approach makes use of the infinite-space Green function of the wave equation and is therefore suitable for an advanced course in electrodynamics.
Constructing Academic Inadequacy: African American Athletes' Stories of Schooling.
Benson, Kirsten F.
2000-01-01
This qualitative study interviewed eight academically "at risk" African American athletes at a southeastern university with a major revenue-producing football program. Analysis suggested that the athletes' marginal academic performance was constructed in a system of interrelated practices engaged in by all the significant members of the academic…
Inadequacy of von Neumann entropy for characterizing extractable work
International Nuclear Information System (INIS)
The lack of knowledge that an observer has about a system limits the amount of work it can extract. This lack of knowledge is normally quantified using the Gibbs/von Neumann entropy. We show that this standard approach is, surprisingly, only correct in very specific circumstances. In general, one should use the recently developed smooth entropy approach. For many common physical situations, including large but internally correlated systems, the resulting values for the extractable work can deviate arbitrarily from those suggested by the standard approach.
Institutional Biosafety Committees and the Inadequacies of Risk Regulation.
Bereano, Philip L.
1984-01-01
Discusses institutional biosafety committees (IBC) which provide quasi-independent reviews of recombinant DNA work done at an institution. Considers the nature of IBC operation, the National Institutes of Health "Guidelines for Recombinant DNA Research," the composition of IBCs, the intellectual content of IBC deliberations, and issues related to…
Correctional Recreation Today, A Pitiful Reflection of Our Past Inadequacy.
Calloway, Jim
1981-01-01
A brief history of correctional recreation from the late 1800s through the time of the Attica riots is presented. Since the prison population is 60 to 70 percent Black, there is an obvious need for greater minority participation in the development of correctional recreation. (JN)
The inadequacies of absolute prohibition of reproductive cloning.
Lee, Martin Lishexian
2004-02-01
This study reviews debates on human cloning and its benefits, considers international and domestic laws, and argues that the choice of reproductive means is a human right. In exercise of this right, a balanced approach should be adopted, in order to benefit human society while protecting human dignity adequately. The immaturity of cloning techniques indicates that at the present time human reproductive cloning is too risky. Thus a temporary ban on such cloning is appropriate, but the ban on relevant scientific research and animal experimentation is inappropriate as it denies the spirit of freedom of scientific inquiry, and hinders making the benefits of scientific advancement available to human society as a whole. PMID:15018212
Filling transitions on rough surfaces: inadequacy of Gaussian surface models
Dufour, Renaud; Herminghaus, Stephan
2015-01-01
We present numerical studies of wetting on various topographic substrates, including random topographies. We find good agreement with recent predictions based on an analytical interface-displacement-type theory \\cite{Herminghaus2012, Herminghaus2012a}. The phase diagrams are qualitatively as predicted, but differently in this study the critical points are found to lie within the physical parameter range (i.e., at positive contact angle) in all cases studied. Notably, it is corroborated that Gaussian random surfaces behave qualitatively different from all non-Gaussian topographies investigated, exhibiting a qualitatively different phase diagram. This shows that Gaussian random surfaces must be used with great care in the context of wetting phenomena.
Inadequacy of federal forum for resolution of oil spill damages
International Nuclear Information System (INIS)
This paper presents a barrister's observations of how the determination of liability and damage following an oil spill might be made more manageable. The author concludes that standard trial in Federal Court, with traditional right to a jury and existing rules of evidence, is not an appropriate forum for the adjudication of massive oil spill cases. Rather, a special multi-destruct panel should be created to deal specifically with oil spill damage cases
Textbook Sexual Inadequacy? A Review of Sexuality Texts.
Goettsch, Stephen L.
1987-01-01
Reviews eight current human sexuality textbooks for both their general organization and substantive content. Addresses specifically the content areas of sexual response cycle; sexual disfunction; acquaintance rape; AIDS and sexually transmitted diseases; extramarital sex; abortion; homosexuality; and pornography. Identifies as a recurring fault…
Solving Equations of Multibody Dynamics
Jain, Abhinandan; Lim, Christopher
2007-01-01
Darts++ is a computer program for solving the equations of motion of a multibody system or of a multibody model of a dynamic system. It is intended especially for use in dynamical simulations performed in designing and analyzing, and developing software for the control of, complex mechanical systems. Darts++ is based on the Spatial-Operator- Algebra formulation for multibody dynamics. This software reads a description of a multibody system from a model data file, then constructs and implements an efficient algorithm that solves the dynamical equations of the system. The efficiency and, hence, the computational speed is sufficient to make Darts++ suitable for use in realtime closed-loop simulations. Darts++ features an object-oriented software architecture that enables reconfiguration of system topology at run time; in contrast, in related prior software, system topology is fixed during initialization. Darts++ provides an interface to scripting languages, including Tcl and Python, that enable the user to configure and interact with simulation objects at run time.
Langevin Equation for DNA Dynamics
Grych, David; Copperman, Jeremy; Guenza, Marina
Under physiological conditions, DNA oligomers can contain well-ordered helical regions and also flexible single-stranded regions. We describe the site-specific motion of DNA with a modified Rouse-Zimm Langevin equation formalism that describes DNA as a coarse-grained polymeric chain with global structure and local flexibility. The approach has successfully described the protein dynamics in solution and has been extended to nucleic acids. Our approach provides diffusive mode analytical solutions for the dynamics of global rotational diffusion and internal motion. The internal DNA dynamics present a rich energy landscape that accounts for an interior where hydrogen bonds and base-stacking determine structure and experience limited solvent exposure. We have implemented several models incorporating different coarse-grained sites with anisotropic rotation, energy barrier crossing, and local friction coefficients that include a unique internal viscosity and our models reproduce dynamics predicted by atomistic simulations. The models reproduce bond autocorrelation along the sequence as compared to that directly calculated from atomistic molecular dynamics simulations. The Langevin equation approach captures the essence of DNA dynamics without a cumbersome atomistic representation.
Quantization of Equations of Motion
Directory of Open Access Journals (Sweden)
D. Kochan
2007-01-01
Full Text Available The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential. A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail.
Solutions manual to accompany 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
Homogenization of ordinary and linear transport equations
Peirone, Roberto
1996-01-01
The homogenization of first order ordinary differential equations in $\\mathbb{R}^N$ and associated linear transport equations are studied. We prove the equivalence between $G$-convergence and strong $G$-convergence for the ordinary equations. We give a sufficient condition, which is also necessary in the autonomous case, for the weak homogenization of the linear transport equations. This condition is satisfied when div$_x f=0$.
Partial differential equations of mathematical physics
Sobolev, S L
1964-01-01
Partial Differential Equations of Mathematical Physics emphasizes the study of second-order partial differential equations of mathematical physics, which is deemed as the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. The book discusses in detail a wide spectrum of topics related to partial differential equations, such as the theories of sets and of Lebesgue integration, integral equations, Green's function, and the proof of the Fourier method. Theoretical physicists, experimental physicists, mathematicians engaged in pure and applied math
Some Aspects of Extended Kinetic Equation
Directory of Open Access Journals (Sweden)
Dilip Kumar
2015-09-01
Full Text Available Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.
Loewner equations in multiply connected domains
Böhm, Christoph
2016-01-01
The first goal of this thesis is to generalize Loewner's famous differential equation to multiply connected domains. The resulting differential equations are known as Komatu--Loewner differential equations. We discuss Komatu--Loewner equations for canonical domains (circular slit disks, circular slit annuli and parallel slit half-planes). Additionally, we give a generalisation to several slits and discuss parametrisations that lead to constant coefficients. Moreover, we compare Komatu--Loewne...
Perturbative linearization of reaction-diffusion equations
International Nuclear Information System (INIS)
We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, namely the Fisher equation and the time-dependent Ginzburg-Landau equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution
Jarosław Mikołajski; Ewa Schmeidel
2006-01-01
Third order linear homogeneous differential and recurrence equations with constant coefficients are considered. We take the both equations with the same characteristic equation. We show that these equations (differential and recurrence) can have solutions with different properties concerning oscillation and boundedness. Especially the numbers of suitable types of solutions taken out from fundamental sets are presented. We give conditions under which the asymptotic properties considered are th...
Nonlinear SCHRÖDINGER-PAULI Equations
Ng, Wei Khim; Parwani, Rajesh R.
2011-11-01
We obtain novel nonlinear Schrüdinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
Weierstrass solutions for dissipative BBM equation
Mancas, Stefan C.; Spradlin, Greg; Khanal, Harihar
2013-01-01
In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified BBM equation. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant make the equation integrable in terms of Weiers...
Notes on the infinity Laplace equation
Lindqvist, Peter
2016-01-01
This BCAM SpringerBriefs is a treaty of the Infinity-Laplace Equation, which has inherited many features from the ordinary Laplace Equation, and is based on lectures by the author. The Infinity.Laplace Equation has delightful counterparts to the Dirichlet integral, the mean value property, the Brownian motion, Harnack's inequality, and so on. This "fully non-linear" equation has applications to image processing and to mass transfer problems, and it provides optimal Lipschitz extensions of boundary values.
BOUNDARY CONTROL OF MKDV-BURGERS EQUATION
Institute of Scientific and Technical Information of China (English)
TIAN Li-xin; ZHAO Zhi-feng; WANG Jing-feng
2006-01-01
The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1]. The existence of the solution of MKdV-Burgers equation with the feedback control law was proved. On the base, priori estimates for the solution was given. At last, the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given.
Soliton Equations with Self-Consistent Sources
Myrzakulov, Ratbay
2014-01-01
We consider some soliton equations with self-consistent sources. A brief review of main SESCS is presented. In particular we construct the Heisenberg ferromagneic equation with self-consistent sources (HFESCS) which is integrable. The corresponding Lax representation is presented. Some properties of HFESCS are analyzed. The relation between soliton equations with self-consistent potentials and soliton equations with self-consistent sources is studied.
Kinetic equation for a dense soliton gas
El, G. A.; Kamchatnov, A. M.
2005-01-01
We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with t...
Onsager equations and time dependent neutron transport
International Nuclear Information System (INIS)
The diffusion of neutrons following an abrupt, localized temperature fluctuation can be conducted in the framework of Onsager-type transport equations. Considering Onsager equations as a generalized Fick's law, time-dependent particle and energy 'generalized diffusion equations' can be obtained. Aim of the present paper is to obtain the time-dependent diffusion Onsager-type equations for the diffusion of neutrons and to apply them to simple trial cases to gain a feeling for their behaviour. (author)
Some Aspects of Extended Kinetic Equation
Dilip Kumar
2015-01-01
Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.
Some constant solutions to Zamolodchikov's tetrahedron equations
Hietarinta, Jarmo
1992-01-01
In this letter we present constant solutions to the tetrahedron equations proposed by Zamolodchikov. In general, from a given solution of the Yang-Baxter equation there are two ways to construct solutions to the tetrahedron equation. There are also other kinds of solutions. We present some two-dimensional solutions that were obtained by directly solving the equations using either an upper triangular or Zamolodchikov's ansatz.
Introduction to differential equations with dynamical systems
Campbell, Stephen L
2011-01-01
Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Cam
Symmetry Breaking for Black-Scholes Equations
Institute of Scientific and Technical Information of China (English)
YANG Xuan-Liu; ZHANG Shun-Li; QU Chang-Zheng
2007-01-01
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.
Symmetry Breaking for Black-Scholes Equations
Yang, Xuan-Liu; Zhang, Shun-Li; Qu, Chang-Zheng
2007-06-01
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.
Symmetry Breaking for Black-Scholes Equations
International Nuclear Information System (INIS)
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.
The Modified Enskog Equation for Mixtures
Beijeren, H. van; Ernst, M.H.
1973-01-01
In a previous paper it was shown that a modified form of the Enskog equation, applied to mixtures of hard spheres, should be considered as the correct extension of the usual Enskog equation to the case of mixtures. The main argument was that the modified Enskog equation leads to linear transport coe
Errors in equations for galaxy rotation speeds
Nicholson, Kenneth F.
2003-01-01
Shown are the errors and difficulties of the equations used for galaxy rotation speeds in the book "Galactic Dynamics" (Binney and Tremaine). A usable and accurate set of equations is then presented. The new equations allow easy determination of galaxy mass distribution from the rotation profile with no need for dark matter or any knowledge of galaxy surface light.
TWO PROBLEMS OF HERMITE ELLIPTIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Huaug Feirain
2009-01-01
In this article, the author investigates some Hermite elliptic equations in a modified Sobolev space introduced by X. Ding [2]. First, the author shows the existence of a ground state solution of semilinear Hermite elliptic equation. Second, the author studies the eigenvalue problem of linear Hermite elliptic equation in a bounded or unbounded domain.
Some Functional Equations Originating from Number Theory
Indian Academy of Sciences (India)
Soon-Mo Jung; Jae-Hyeong Bae
2003-05-01
We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.
Multi component equations of state for electrolytes
DEFF Research Database (Denmark)
Lin, Yi; Thomsen, Kaj; de Hemptinne, Jean-Charles
2007-01-01
Four equations of state have been implemented and evaluated for multi component electrolyte solutions at 298.15K and 1 bar. The equations contain terms accounting for short-range and long-range interactions in electrolyte solutions. Short range interactions are described by one of the three equat...
Equations of state for light water
International Nuclear Information System (INIS)
The equations of state for light water were developed, based on the tables of Keenan and Keyes. Equations are presented, describing the specific volume, internal energy, enthalpy and entropy of saturated steam, superheated vapor and subcooled liquid as a function of pressure and temperature. For each property, several equations are shown, with different precisions and different degress of complexity. (Author)
A new analogue of Gauss' functional equation
Directory of Open Access Journals (Sweden)
Themistocles M. Rassias
1995-12-01
Full Text Available Gauss established a theory on the functional equation (Gauss' functional equation f(a+b2,ab=f(a,bÃ¢Â€Â‰Ã¢Â€Â‰(a,b>0, where f:R+ÃƒÂ—R+Ã¢Â†Â’R is an unknown function of the above equation.
Solving Absolute Value Equations Algebraically and Geometrically
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
The Schrodinger equation and negative energies
Bruce, S
2008-01-01
We present a nonrelativistic wave equation for the electron in (3+1)-dimensions which includes negative-energy eigenstates. We solve this equation for three well-known instances, reobtaining the corresponding Pauli equation (but including negative-energy eigenstates) in each case.
Some new modular equations and their applications
Yi, Jinhee; Sim, Hyo Seob
2006-07-01
Ramanujan derived 23 beautiful eta-function identities, which are certain types of modular equations. We found more than 70 of certain types of modular equations by using Garvan's Maple q-series package. In this paper, we prove some new modular equations which we found by employing the theory of modular form and we give some applications for them.
Multicomponent nonisothermal nucleation. 1. Kinetic equation
Kurasov, Victor
1999-01-01
The first part of the theory for the multicomponent nonisothermal nucleation is presented. On the base of analysis of the elementary acts of interation between an embryo and environment the kinetic equation is derived. This equation will be solved later and gives the nucleation rate for nonisothermal nucleation in the system with many condensating components. The orders of operators in this equation are estimated.
Differential Galois Theory of Linear Difference Equations
Hardouin, Charlotte; Singer, Michael F.
2008-01-01
We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions.
Difference Galois theory of linear differential equations
Di Vizio, Lucia; Hardouin, Charlotte; Wibmer, Michael
2013-01-01
We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.
Non RG logarithms via RG equations
Malyshev, Dmitry
2004-01-01
We compute complete leading logarithms in $\\Phi^4$ theory with the help of Connes and Kreimer RG equations. These equations are defined in the Lie algebra dual to the Hopf algebra of graphs. The results are compared with calculations in parquet approximation. An interpretation of the new RG equations is discussed.
Invalidity of the spectral Fokker-Planck equation forCauchy noise driven Langevin equation
DEFF Research Database (Denmark)
Ditlevsen, Ove Dalager
2004-01-01
The standard Langevin equation is a first order stochastic differential equation where the driving noise term is a Brownian motion. The marginal probability density is a solution to a linear partial differential equation called the Fokker-Planck equation. If the Brownian motion is replaced by so......-called alpha-stable noise (or Levy noise) the Fokker-Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. In stead it has been attempted to formulate an equation for the characteristic function (the Fourier transform......) corresponding to the density function. This equation is frequently called the spectral Fokker-Planck equation. This paper raises doubt about the validity of the spectral Fokker/Planck equation in its standard formulation. The equation can be solved with respect to stationary solutions in the particular case...
Dual Isomonodromic Problems and Whitham Equations
Takasaki, K
1997-01-01
The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter $\\epsilon$. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the Seiberg- Witten solutions of low energy supersymmetric gauge theories.
Binomial moment equations for stochastic reaction systems.
Barzel, Baruch; Biham, Ofer
2011-04-15
A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations. PMID:21568538
The Dirac equation and its solutions
International Nuclear Information System (INIS)
The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.
On physical equation of porous material
International Nuclear Information System (INIS)
The authors offer simple-to-use physical equations of porous body in the framework of the associated law of flow. The equations meet the conditions of cylindrical type flow known in the theory of porous body plasticity. The equations offered are compared with the physical equations meeting the conditions of elliptical type flow. They were applied in modeling the process of press titanium sponge. Together with the physical equations offered, the mathematical tools and the experimental procedure to model the processes of pressing the non-compacted raw material blanks are proposed too
The Dirac equation and its solutions
Bagrov, Vladislav G
2014-01-01
Dirac equations are of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly.In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.
Quadratic field equations on the brane
International Nuclear Information System (INIS)
It is shown that four-dimensional vacuum Einstein solutions simply embedded in five dimensions obey the Gauss-Bonnet-Einstein field equations: Gab+ αGBab + δ55abαexp[-2χ/√α]GB4 = 0 and the Pauli-Einstein equations Gab - 3αPab/5 = 0, and the Bach-Einstein equations Bab = 0. General equations are calculated for which these and similar results follow. It is briefly argued that such field equations could be significant on large distance scales.
First integrals of difference Hamiltonian equations
International Nuclear Information System (INIS)
In the present paper, the well-known Noether's identity, which represents the connection between symmetries and first integrals of Euler-Lagrange equations, is rewritten in terms of the Hamiltonian function. This approach, based on the Hamiltonian identity, provides a simple and clear way to find first integrals of canonical Hamiltonian equations without integration. A discrete analog of the Hamiltonian identity is developed. It leads to a connection between symmetries and first integrals of difference Hamiltonian equations that can be used to conserve the structural properties of Hamiltonian equations under discretizaton. The results are illustrated by a number of examples for both continuous and difference Hamiltonian equations.
Some Recent Advances in Partial Difference Equations
Petropoulou, Eugenia N
2010-01-01
Lately there is an increasing interest in partial difference equations demonstrated by the enormous amount of research papers devoted to them. The initial reason for this increasing interest was the development of computers and the area of numerical analysis, where partial difference equations arise naturally when discretizing a partial differential equation. The aim of this e-book is to provide some recent advances in the field of partial difference equations. Applications of partial difference equations in numerical analysis and systems theory are also presented. This e-book will be of use t
The Numerical Approximation of Functional Differential Equations
Venturi, Daniele
2016-01-01
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equations), quantum field theory (Schwinger-Dyson equations) and statistical physics (equations for generating functionals and effective action methods). However, no effective numerical method has yet been developed to compute their solution. The purpose of this manuscript is to fill this gap, and provide a new perspective on the problem of numerical approximation of nonlinear functionals and functional differential equations. The proposed methods will be described and demonstrated in various examples.
Equation of state of beryllium
International Nuclear Information System (INIS)
A new, wide-range equation of state (EOS) has been constructed for Be. The composite theoretical model incorporates ionization equilibrium and condensed-matter and multiphase physics. It also satisfies all thermodynamic equilibrium constraints. The theoretical EOS has been compared with all available high-pressure and high-temperature Be data, and satisfactory agreement is generally achieved. The most interesting feature is the theoretical prediction of melting at just below 220 GPa (2 Mb), indicating an extremely wide pressure range for solid Be. A striking feature is the appearance of shell-structure effects in physical-process paths: 2 large loops appear on the principal Hugoniot and the behavior of release isentropes from rho = rho0 is significantly affected
Model Equations: "Black Box" Reconstruction
Bezruchko, Boris P.; Smirnov, Dmitry A.
Black box reconstruction is both the most difficult and the most tempting modelling problem when any prior information about an appropriate model structure is lacking. An intriguing thing is that a model capable of reproducing an observed behaviour or predicting further evolution should be obtained only from an observed time series, i.e. "from nothing" at first sight. Chances for a success are not large. Even more so, a "good" model would become a valuable tool to characterise an object and understand its dynamics. Lack of prior information causes one to utilise universal model structures, e.g. artificial neural networks, radial basis functions and algebraic polynomials are included in the right-hand sides of dynamical model equations. Such models are often multi-dimensional and involve quite many free parameters.
The equations of medieval cosmology
Buonanno, Roberto
2008-01-01
In Dantean cosmography the Universe is described as a series of concentric spheres with all the known planets embedded in their rotation motion, the Earth located at the centre and Lucifer at the centre of the Earth. Beyond these "celestial spheres", Dante represents the "angelic choirs" as other nine spheres surrounding God. The rotation velocity increases with decreasing distance from God, that is with increasing Power (Virtu'). We show that, adding Power as an additional fourth dimension to space, the modern equations governing the expansion of a closed Universe (i. e. with the density parameter \\Omega_0>1) in the space-time, can be applied to the medieval Universe as imaged by Dante in his Divine Comedy. In this representation the Cosmos acquires a unique description and Lucifer is not located at the centre of the hyperspheres.
The equations of medieval cosmology
Buonanno, Roberto; Quercellini, Claudia
2009-04-01
In Dantean cosmography the Universe is described as a series of concentric spheres with all the known planets embedded in their rotation motion, the Earth located at the centre and Lucifer at the centre of the Earth. Beyond these "celestial spheres", Dante represents the "angelic choirs" as other nine spheres surrounding God. The rotation velocity increases with decreasing distance from God, that is with increasing Power (Virtù). We show that, adding Power as an additional fourth dimension to space, the modern equations governing the expansion of a closed Universe (i.e. with the density parameter Ω0 > 1) in the space-time, can be applied to the medieval Universe as imaged by Dante in his Divine Comedy. In this representation, the Cosmos acquires a unique description and Lucifer is not located at the centre of the hyperspheres.
An introduction to differential equations
Ladde, Anil G
2012-01-01
This is a twenty-first century book designed to meet the challenges of understanding and solving interdisciplinary problems. The book creatively incorporates "cutting-edge" research ideas and techniques at the undergraduate level. The book also is a unique research resource for undergraduate/graduate students and interdisciplinary researchers. It emphasizes and exhibits the importance of conceptual understandings and its symbiotic relationship in the problem solving process. The book is proactive in preparing for the modeling of dynamic processes in various disciplines. It introduces a "break-down-the problem" type of approach in a way that creates "fun" and "excitement". The book presents many learning tools like "step-by-step procedures (critical thinking)", the concept of "math" being a language, applied examples from diverse fields, frequent recaps, flowcharts and exercises. Uniquely, this book introduces an innovative and unified method of solving nonlinear scalar differential equations. This is called ...
Stochastic integration and differential equations
Protter, Philip E
2003-01-01
It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...
Structural equation modeling in epidemiology
Directory of Open Access Journals (Sweden)
Leila Denise Alves Ferreira Amorim
2010-12-01
Full Text Available Structural equation modeling (SEM is an important statistical tool for evaluating complex relations in several research areas. In epidemiology, the use and discussion of SEM have been limited thus far. This article presents basic principles and concepts in SEM, including an application using epidemiological data analysis from a study on the determinants of cognitive development in young children, considering constructs related to organization of the child's home environment, parenting style, and the child's health status. The relations between the constructs and cognitive development were measured. The results showed a positive association between psychosocial stimulus at home and cognitive development in young children. The article presents the contributions by SEM to epidemiology, highlighting the need for an a priori theoretical model for improving the study of epidemiological questions from a new perspective.
Sobolev gradients and differential equations
Neuberger, John William
1997-01-01
A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling.
Inferring Mathematical Equations Using Crowdsourcing.
Directory of Open Access Journals (Sweden)
Szymon Wasik
Full Text Available Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game-so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical equations that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an equation that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players.
Inferring Mathematical Equations Using Crowdsourcing.
Wasik, Szymon; Fratczak, Filip; Krzyskow, Jakub; Wulnikowski, Jaroslaw
2015-01-01
Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game-so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical equations that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an equation that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players. PMID:26713846
Exact solution to fractional logistic equation
West, Bruce J.
2015-07-01
The logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences. Herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time. The solution to the fractional logistic equation (FLE) is obtained using the Carleman embedding technique that allows the nonlinear equation to be replaced by an infinite-order set of linear equations, which we then solve exactly. The formal series expansion for the initial value solution of the FLE is shown to be expressed in terms of a series of weighted Mittag-Leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity. The numerical integration to the FLE provides an excellent fit to the analytic solution. We propose this approach as a general technique for solving a class of nonlinear fractional differential equations.
10. Exploring the Conformal Constraint Equations
Butscher, Adrian
One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the rescaled Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity. This problem can be avoided by means of a technique of H. Friedrich, which replaces the Einstein equations in the unphysical spacetime by an equivalent system of equations which is regular at the boundary. The initial value problem for these equations produces a system of constraint equations known as the conformal constraint equations. This work describes some of the properties of the conformal constraint equations and develops a perturbative method of generating solutions near Euclidean space under certain simplifying assumptions.
A New Route to the Majorana Equation
Directory of Open Access Journals (Sweden)
Eckart Marsch
2013-09-01
Full Text Available In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group.
Christianto V.; Smarandache F.
2008-01-01
In the present article we argue that it is possible to write down Schrodinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R.M.Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this...
Difference equations and cluster algebras I: Poisson bracket for integrable difference equations
Inoue, Rei
2010-01-01
We introduce the cluster algebraic formulation of the integrable difference equations, the discrete Lotka-Volterra equation and the discrete Liouville equation, from the view point of the general T-system and Y-system. We also study the Poisson structure for the cluster algebra, and give the associated Poisson bracket for the two difference equations.
Indian Academy of Sciences (India)
Yusuf Gurefe; Abdullah Sonmezoglu; Emine Misirli
2011-12-01
In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the (, ) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.
The properties of the first equation of the Vlasov chain of equations
Perepelkin, E. E.; Sadovnikov, B. I.; Inozemtseva, N. G.
2015-05-01
A derivation of the first Vlasov equation as a well-known Schrödinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred. A physical meaning of the phase of the wave function which is a scalar potential of the probabilistic flow velocity is demonstrated. Occurrence of the velocity potential vortex component leads to the Pauli equation for one of the spinar components. A scheme for the construction of the Schrödinger equation solution from the Vlasov equation solution and vice-versa is shown. A process of introduction of the potential to the Schrödinger equation and its interpretation are given. The analysis of the potential properties gives us the Maxwell equation, the equation of the kinematic point movement, and the equation for movement of the medium within electromagnetic fields.
Dust levitation about Itokawa's equator
Hartzell, C.; Zimmerman, M.; Takahashi, Y.
2014-07-01
levitation about Itokawa, we must include accurate plasma and gravity models. We use a 2D PIC code (described in [8]) to model the plasma environment about Itokawa's equator. The plasma model includes photoemission and shadowing. Thus, we model the plasma environment for various solar incidence angles. The plasma model gives us the 2D electric field components and the plasma potential. We model the gravity field around the equatorial cross-section using an Interior Gravity model [9]. The gravity model is based on the shape model acquired by the Hayabusa mission team and, unlike other models, is quick and accurate close to the surface of the body. Due to the nonspherical shape of Itokawa, the electrostatic force and the gravity may not be collinear. Given our accurate plasma and gravity environments, we are able to simulate the trajectories of dust grains about the equator of Itokawa. When modeling the trajectories of the grains, the current to the grains is calculated using Nitter et al.'s formulation [10] with the plasma sheath parameters provided by our PIC model (i.e., the potential minimum, the potential at the surface, and the sheath type). Additionally, we are able to numerically locate the equilibria about which dust grains may levitate. Interestingly, we observe that equilibria exist for grains up to 20 microns in radius about Itokawa's equator when the Sun is illuminating Itokawa's 'otter tail'. This grain size is significantly larger than the stably levitating grains we observed using our 1D plasma and gravity models. Conclusions and Future Work: The possibility of dust levitation above asteroids has implications both for our understanding of their evolution and for the design of future missions to these bodies. Using detailed gravity and plasma models, we are above to propagate the trajectories of dust particles about Itokawa's equator and identify the equilibria about which these grains will levitate. Using these simulations, we see that grains up to 20 microns
Lakrib, Mustapha; Sari, Tewfik
2006-01-01
We prove averaging theorems for ordinary differential equations and retarded functional differential equations. Our assumptions are weaker than those required in the results of the existing literature. Usually, we require that the nonautonomous differential equation and the autonomous averaged equation are locally Lipschitz and that the solutions of both equations exist on some interval. We extend this result to the case of vector fields which are continuous in the spatial variable uniformly ...
Chang, Yi-Fang
2009-01-01
We proposed a new type of soliton equation, whose solutions may describe some statistical distributions, for example, Cauchy distribution, normal distribution and student distribution, etc. The equation possesses two characters. Further, from an extension of this type of equation we may obtain the exponential distribution, and the Fermi-Dirac distribution in quantum statistics. Moreover, by using the method of the soliton-solution, the nonlinear Klein-Gordon equation and nonlinear Dirac equat...
The telegraph equation in charged particle transport
Gombosi, T. I.; Jokipii, J. R.; Kota, J.; Lorencz, K.; Williams, L. L.
1993-01-01
We present a new derivation of the telegraph equation which modifies its coefficients. First, an infinite order partial differential equation is obtained for the velocity space solid angle-averaged phase-space distribution of particles which underwent at least a few collisions. It is shown that, in the lowest order asymptotic expansion, this equation simplifies to the well-known diffusion equation. The second-order asymptotic expansion for isotropic small-angle scattering results in a modified telegraph equation with a signal propagation speed of v(5/11) exp 1/2 instead of the usual v/3 exp 1/2. Our derivation of a modified telegraph equation follows from an expansion of the Boltzmann equation in the relevant smallness parameters and not from a truncation of an eigenfunction expansion. This equation is consistent with causality. It is shown that, under steady state conditions in a convecting plasma, the telegraph equation may be regarded as a diffusion equation with a modified transport coefficient, which describes a combination of diffusion and cosmic-ray inertia.
Mathematical aspects of classical nonlinear field equations
International Nuclear Information System (INIS)
In these notes we review some important advances in the mathematical analysis of classical nonlinear field equations. Our particular interest will be devoted to the nonlinear Schroedinger equation and the nonlinear Klein-Gordon equation. Since specifically nonlinear phenomena only become evident in the long-time behavior we are mainly concerned with global results rather than with solutions of these equations for a short time. We start with a thorough discussion of the nonlinear Schroedinger equation and its application to the theory of lasers and its bound states. In the following we study the general question of the existence of solutions of the nonlinear Klein-Gordon equation and other relativistic wave equations. The next section outlines the importance of the conservation laws which follow from the invariance properties of the equations. Special consideration is given to the Yang-Mills equations. The final section exposes a discussion of scattering theory mainly in the context of the nonlinear Klein-Gordon equation and summarizes the exciting development in recent years which has taken place in the nonlinear inverse scattering problem. (HJ)
Institute of Scientific and Technical Information of China (English)
苏颖; 刘宏岩
2001-01-01
Through the studies of Exploration to Wide and Complete Chapteron Normal and Abnormal Chnages in Six Climates and the basic theory of doctrine on five elements' motion and six kinds of natural factor,the regularity and the easy way of inquiry have been found about abnormal changes in climate,five seasons and generation of the excessiveness and inadequacy. And corrections have been made to the teaching materials of the Yellow Emperor's Internal Classic (textbook in the sixth edition) as well as some mistakes in some books found in the process of inquiry. Simultaneously,correction have also been made to some wrong expositions handed down from one generation to generation for hundreds of years. That's of great value for further study of the theory,doctrine on five elements' motion and six kinds of natural factors and is of some realistic significance in prevention,diagnosis and treatment of diseases in clinical practice.
Stability analysis of ecomorphodynamic equations
Bärenbold, F.; Crouzy, B.; Perona, P.
2016-02-01
In order to shed light on the influence of riverbed vegetation on river morphodynamics, we perform a linear stability analysis on a minimal model of vegetation dynamics coupled with classical one- and two-dimensional Saint-Venant-Exner equations of morphodynamics. Vegetation is modeled as a density field of rigid, nonsubmerged cylinders and affects flow via a roughness change. Furthermore, vegetation is assumed to develop following a logistic dependence and may be uprooted by flow. First, we perform the stability analysis of the reduced one-dimensional framework. As a result of the competitive interaction between vegetation growth and removal through uprooting, we find a domain in the parameter space where originally straight rivers are unstable toward periodic longitudinal patterns. For realistic values of the sediment transport parameter, the dominant longitudinal wavelength is determined by the parameters of the vegetation model. Bed topography is found to adjust to the spatial pattern fixed by vegetation. Subsequently, the stability analysis is repeated for the two-dimensional framework, where the system may evolve toward alternate or multiple bars. On a fixed bed, we find instability toward alternate bars due to flow-vegetation interaction, but no multiple bars. Both alternate and multiple bars are present on a movable, vegetated bed. Finally, we find that the addition of vegetation to a previously unvegetated riverbed favors instability toward alternate bars and thus the development of a single course rather than braiding.
Numerical methods for ordinary differential equations
Butcher, John C
2008-01-01
In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author''s pioneering text is fully revised and updated to acknowledge many of these developments. It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding. Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler. Features of the book includeIntroductory work on differential and difference equations.A comprehensive introduction to the theory and practice of solving ordinary differential equations numeri...
Stochastic differential equation model to Prendiville processes
Energy Technology Data Exchange (ETDEWEB)
Granita, E-mail: granitafc@gmail.com [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); Bahar, Arifah [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); UTM Center for Industrial & Applied Mathematics (UTM-CIAM) (Malaysia)
2015-10-22
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
A state estimation of Liu equations
Ananyev, B. I.
2015-11-01
This paper is concerned with state estimation problems for so-called Liu equations. These equations are counterparts of well-known Ito ones and they were introduced by B. Liu under elaboration of his uncertain theory. The Liu equations may be solved backward and they represent a more convenient object for the state estimation problem solution especially for the case when distributions of disturbances are unknown. Using the dynamic programming principle, we derive an equation for the informational set consisting of all states that are compatible with measuring data. Special cases of Liu equations and constraints for disturbances are examined. Among them the linear equations with quadratic constraints are considered in most details. Some examples are also given.
Diffusion phenomenon for linear dissipative wave equations
Said-Houari, Belkacem
2012-01-01
In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, we prove that for {equation presented} decays with the rate {equation presented} [0,1] faster than that of either u or v, where u is the solution of the linear wave equation with initial data {equation presented} [0,1], and v is the solution of the related heat equation with initial data v 0 = u 0 + u 1. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415-433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by t -γ/2. © European Mathematical Society.
Stochastic differential equation model to Prendiville processes
International Nuclear Information System (INIS)
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution
Large lattice fractional Fokker–Planck equation
International Nuclear Information System (INIS)
An equation of long-range particle drift and diffusion on a 3D physical lattice is suggested. This equation can be considered as a lattice analog of the space-fractional Fokker–Planck equation for continuum. The lattice approach gives a possible microstructural basis for anomalous diffusion in media that are characterized by the non-locality of power law type. In continuum limit the suggested 3D lattice Fokker–Planck equations give fractional Fokker–Planck equations for continuous media with power law non-locality that is described by derivatives of non-integer orders. The consistent derivation of the fractional Fokker–Planck equation is proposed as a new basis to describe space-fractional diffusion processes. (paper)
Fokker–Planck equation on fractal curves
International Nuclear Information System (INIS)
A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function
Multigrid method for nonlinear poroelasticity equations
Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, Cornelis
2016-01-01
In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretizati...
Introducing Equational Semantics for Argumentation Networks
Gabbay, Dov M.
2011-01-01
This paper provides equational semantics for Dung’s argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the “extensions” of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, etc, etc. ...
Equation of state of HE detonation products
Nadykto B.A.
2011-01-01
Computational analysis of steady-state HE detonation parameters is possible if one knows the equation of state of detonation products and thermal energy released at the Jouget point during detonation. There are a number of equations of state of HE detonation products that result from different assumptions concerning detonated material conditions. The paper considers one more version of the equation of state for HE detonation products.
Partial differential equations of parabolic type
Friedman, Avner
2008-01-01
This accessible and self-contained treatment provides even readers previously unacquainted with parabolic and elliptic equations with sufficient background to understand research literature. Author Avner Friedman - Director of the Mathematical Biosciences Institute at The Ohio State University - offers a systematic and thorough approach that begins with the main facts of the general theory of second order linear parabolic equations. Subsequent chapters explore asymptotic behavior of solutions, semi-linear equations and free boundary problems, and the extension of results concerning fundamenta
Some remarks on first order wave equations
International Nuclear Information System (INIS)
First order wave equations are presented. Formalizm based on spin projection operators is given and written are the conditions following from the Lorentz invariance and other requirements. Some recently proposed equations and comments on some principles are analysed. The following topics are covered: relativistic invariance, β0-matrix, Harish-Chandra's and Umezawa-Visconti conditions, new s=1/2 equations, space reflections, derivability from lagrangian, definiteness of energy and charge, multiple representations
Estimating Structural Change in Linear Simultaneous Equations
Huang Weihong; Zhang Yang
2004-01-01
Tests and estimation for changes in the coefficients of linear regression models, particularly the analysis of covariance and the Chow tests, are well known to econometricians and are widely used. This paper demonstrates that analogous estimation can also be constructed in simultaneous equation models when equations are estimated by common estimator like OLS, 2SLS and LIML. In the present paper, we discuss the problem of estimating structural changes in equations from a simultaneous structura...
Strong solutions for the Levi curvature equation
Citti, G.; Montanari, A.
2000-01-01
We consider the prescribed Levi curvature equation, a second order quasilinear equation whose associated operator can be represented as a sum of squares of nonlinear vector fields. For this equation we introduce a notion of derivatives modeled on the geometry of the associated operator and prove an a priori $L^2$ estimate for these second order intrinsic derivatives of a viscosity solution. We then show that viscosity solutions are strong solutions in a natural sense and satisf...
The Boltzmann equation in the difference formulation
Energy Technology Data Exchange (ETDEWEB)
Szoke, Abraham [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks III, Eugene D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2015-05-06
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
Involutive reductions and solutions of differential equations
Engelmann, Joachim
2003-01-01
This work introduces the so-called involutive reduction procedure to simplify and solve differential equations. The method is based on symmetry analysis, which was developed by the Norwegian mathematician Sophus Lie. The involutive reduction procedure uses the given differential equation itself and the invariant surface condition of this differential equation. The invariant surface condition incorporates the symmetries of the problem. This coupled system of partial differential equati...
Approximation of PDEs with Underlying Continuity Equations
Klebanov, Ilja
2016-01-01
We develop a numerical method for the solution of special partial differential equations. We use an approximation space, which automatically adapts in space and time to the function that has to be approximated. For that purpose, we use the corresponding probability density function, transport maps to its probability distribution and the underlying continuity equation. The theory and numerical examples will be presented using the Schrödinger equation as the showcase PDE.
The Boltzmann equation in the difference formulation
International Nuclear Information System (INIS)
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
Markov processes and parabolic partial differential equations
Bossy, Mireille; Champagnat, Nicolas
2010-01-01
In the first part of this article, we present the main tools and definitions of Markov processes' theory: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov's backward and forward equations and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward SDEs (BSDEs). The second part of this article is devoted to the links between Markov processes and parabolic partial differential equations (PDEs). In part...