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Journal articles on the topic 'Afterglow (Physics) – Mathematical models'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

Fox, Derek B., and Peter W. A. Roming. "Observations of short gamma-ray bursts." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365, no. 1854 (February 9, 2007): 1293–305. http://dx.doi.org/10.1098/rsta.2006.1974.

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We review recent observations of short-hard gamma-ray bursts and their afterglows. The launch and successful ongoing operations of the Swift satellite, along with several localizations from the High-Energy Transient Explorer mission, have provoked a revolution in short-burst studies: first, by quickly providing high-quality positions to observers; and second, via rapid and sustained observations from the Swift satellite itself. We make a complete accounting of Swift-era short-burst localizations and proposed host galaxies, and discuss the implications of these observations for the distances, energetics and environments of short bursts, and the nature of their progenitors. We then review the physical modelling of short-burst afterglows: while the simplest afterglow models are inadequate to explain the observations, there have been several notable successes. Finally, we address the case of an unusual burst that threatens to upset the simple picture in which long bursts are due to the deaths of massive stars, and short bursts to compact-object merger events.

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2

Medvedev, Mikhail V. "Electron acceleration in relativistic GRB shocks." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365, no. 1854 (February 9, 2007): 1177–78. http://dx.doi.org/10.1098/rsta.2006.1983.

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The shock model of gamma-ray bursts (GRBs) has two free parameters: ϵ B and ϵ e . It has been shown that ϵ B should range between few×10 −3 and few×10 −4 . However, how to calculate the value of ϵ e has remained an outstanding theoretical problem for over a decade. Here, we demonstrate that the Weibel theory inevitably predicts that . The GRB afterglow data fully agree with this theoretical prediction. Our result explains why the electrons are close to equipartition in GRBs. This ϵ e – ϵ B relation can potentially be used to reduce the number of free parameters in afterglow models.

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3

Ciolfi, Riccardo. "Short gamma-ray burst central engines." International Journal of Modern Physics D 27, no. 13 (October 2018): 1842004. http://dx.doi.org/10.1142/s021827181842004x.

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Growing evidence connects the progenitor systems of the short-hard subclass of gamma-ray bursts (GRBs) to the merger of compact object binaries composed of two neutron stars (NSs) or of an NS and a black hole (BH). The recent observation of the binary NS (BNS) merger event GW170817 associated with GRB 170817A brought a great deal of additional information and provided further support to the above connection, even though the identification of this burst as a canonical short GRB (SGRB) remains uncertain. Decades of observational constraints and theoretical models consolidated the idea of a jet origin for the GRB prompt emission, which can also explain the multiwavelength afterglow radiation observed in most of the events. However, the mechanisms through which a BNS or NS–BH merger remnant would power a collimated outflow are much less constrained. Understanding the properties of the remnant systems and whether they can provide the right conditions for jet production has been a main driver of the great effort devoted to study BNS and NS–BH mergers, and still represents a real challenge from both the physical and the computational points of view. One fundamental open question concerns the nature of the central engine itself. While the leading candidate system is a BH surrounded by a massive accretion disk, the recent observation of plateau-shaped X-ray afterglows in some SGRBs would suggest a longer-lived engine, i.e. a metastable (or even stable) massive NS, which would also exclude NS–BH progenitors. Here we elaborate on this key aspect, considering three different scenarios to explain the SGRB phenomenology based on different hypotheses on the nature of the merger remnant. Then, we discuss the basic properties of GRB 170817A and how this event would fit within the different frameworks of the above scenarios, under the assumption that it was or was not a canonical SGRB.

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CHANG, PHILIP, ANATOLY SPITKOVSKY, and JONATHAN ARONS. "LONG TERM EVOLUTION OF MAGNETIC TURBULENCE IN RELATIVISTIC COLLISIONLESS SHOCKS." International Journal of Modern Physics D 17, no. 10 (September 2008): 1769–75. http://dx.doi.org/10.1142/s021827180801339x.

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We study the long term evolution of magnetic fields generated by an initially unmagnetized collisionless relativistic e+e- shock. Our 2D particle-in-cell numerical simulations show that downstream of such a Weibel-mediated shock, particle distributions are approximately isotropic, relativistic Maxwellians, and the magnetic turbulence is highly intermittent spatially, non-propagating, and decaying. Using linear kinetic theory, we find a simple analytic form for these damping rates. Our theory predicts that the overall magnetic energy decays as (ωp t)-q with q ~ 1, which compares favorably with simulations, but predicts overly rapid damping of short-wavelength modes. The magnetic trapping of particles within the magnetic structures may be the origin of this discrepancy. We conclude that initially unmagnetized relativistic shocks in electron-positron plasmas are unable to form persistent downstream magnetic fields. These results put interesting constraints on synchrotron models for the prompt and afterglow emission from GRBs.

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5

Zhang, Xuefeng, Zhoujian Cao, and He Gao. "Long-term postmerger simulations of relativistic star coalescence: Formation of toroidal remnants and gravitational wave afterglow." International Journal of Modern Physics D 28, no. 01 (January 2019): 1950026. http://dx.doi.org/10.1142/s0218271819500263.

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It has been estimated that a significant proportion of binary neutron star merger events produce long-lived massive remnants supported by differential rotation and subject to rotational instabilities. To examine formation and oscillation of rapidly rotating neutron stars (NS) after merger, we present an exploratory study of fully general-relativistic hydrodynamic simulations using the public code Einstein Toolkit. The attention is focused on qualitative aspects of long-term postmerger evolution under [Formula: see text]-rotational symmetry. As simplified test models, we use a moderately stiff [Formula: see text] ideal-fluid equation-of-state and unmagnetized irrotational equal-mass binaries with three masses well below the threshold for prompt collapse. Our high resolution simulations generate postmerger “ringdown” gravitational wave (GW) signals of 170 ms, sustained by rotating massive NS remnants without collapsing to black holes. We observe that the high-density double-core structure inside the remnants gradually turns into a quasi-axisymmetric toroidal shape. It oscillates in a quasi-periodic manner and shrinks in size due to gravitational radiation. In the GW spectrograms, dominant double peaks persist throughout the postmerger simulations and slowly drift to higher frequencies. A new low-frequency peak emerges at about 100 ms after merger, owing to the growth of GW-driven unstable oscillation modes. The long-term effect of grid resolution is also investigated using the same initial model. Moreover, we comment on physical conditions that are favorable for the transient toroidal configuration to form, and discuss implication of our findings on future GW observation targeting rapidly rotating NS remnants.

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6

Lamb, Gavin P., Joseph J. Fernández, Fergus Hayes, Albert K. H. Kong, En-Tzu Lin, Nial R. Tanvir, Martin Hendry, Ik Siong Heng, Surojit Saha, and John Veitch. "Inclination Estimates from Off-Axis GRB Afterglow Modelling." Universe 7, no. 9 (September 5, 2021): 329. http://dx.doi.org/10.3390/universe7090329.

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For gravitational wave (GW) detected neutron star mergers, one of the leading candidates for electromagnetic (EM) counterparts is the afterglow from an ultra-relativistic jet. Where this afterglow is observed, it will likely be viewed off-axis, such as the afterglow following GW170817/GRB 170817A. The temporal behaviour of an off-axis observed GRB afterglow can be used to reveal the lateral jet structure, and statistical model fits can put constraints on the various model free-parameters. Amongst these parameters is the inclination of the system to the line of sight. Along with the GW detection, the afterglow modelling provides the best constraint on the inclination to the line-of-sight and can improve the estimates of cosmological parameters, for example, the Hubble constant, from GW-EM events. However, modelling of the afterglow depends on the assumed jet structure and—often overlooked—the effects of lateral spreading. Here we show how the inclusion of lateral spreading in the afterglow models can affect the estimated inclination of GW-EM events.

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7

Fil’chenkov, M. L., and Yu P. Laptev. "MATHEMATICAL MODELS IN THEORETICAL PHYSICS." Metafizika, no. 3 (December 15, 2020): 64–68. http://dx.doi.org/10.22363/2224-7580-2020-3-64-68.

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Quantum theory and relativity theory as well as possible reconciliation have been analyzed from the viewpoint of mathematical models being used in them, experimental affirmation, interpretations and their association with dualistic paradigms.

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8

MIMICA, PETAR, DIMITRIOS GIANNIOS, and MIGUEL ANGEL ALOY. "SIMULATIONS OF DYNAMICS AND EMISSION FROM MAGNETIZED GRB AFTERGLOWS." International Journal of Modern Physics D 19, no. 06 (June 2010): 985–90. http://dx.doi.org/10.1142/s0218271810017007.

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The role of magnetic fields in the GRB flow is still controversial. The afterglow emission, particularly the early phases, may provide a probe into the magnetization of the outflow. Using ultrahigh resolution relativistic MHD simulations, the interaction between radially expanding magnetized ejecta with the interstellar medium is studied. We explore the effect of the magnetic field strength of the ejecta on the afterglow structure, particularly regarding the presence and strength of a reverse shock. We compute synthetic afterglow light curves to quantify the effect of the magnetization of the flow on observed radiation.

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9

Voon, Lok C. Lew Yan, Roderick Melnik, and Morten Willatzen. "Physics-Based Mathematical Models for Nanotechnology." Journal of Physics: Conference Series 107 (March 1, 2008): 011001. http://dx.doi.org/10.1088/1742-6596/107/1/011001.

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10

Kleiner, Johannes. "Mathematical Models of Consciousness." Entropy 22, no. 6 (May 30, 2020): 609. http://dx.doi.org/10.3390/e22060609.

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In recent years, promising mathematical models have been proposed that aim to describe conscious experience and its relation to the physical domain. Whereas the axioms and metaphysical ideas of these theories have been carefully motivated, their mathematical formalism has not. In this article, we aim to remedy this situation. We give an account of what warrants mathematical representation of phenomenal experience, derive a general mathematical framework that takes into account consciousness’ epistemic context, and study which mathematical structures some of the key characteristics of conscious experience imply, showing precisely where mathematical approaches allow to go beyond what the standard methodology can do. The result is a general mathematical framework for models of consciousness that can be employed in the theory-building process.

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11

Kraenkel, R. A., and M. Senthilvelan. "Mathematical Models of Generalized Diffusion." Physica Scripta 63, no. 5 (May 1, 2001): 353–56. http://dx.doi.org/10.1238/physica.regular.063a00353.

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12

Banasiak, J. "Kinetic models – mathematical models of everything?" Physics of Life Reviews 16 (March 2016): 140–41. http://dx.doi.org/10.1016/j.plrev.2016.01.005.

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13

Ben Abdallah, Naoufel, Pierre Degond, and Florian Méhats. "Mathematical models of magnetic insulation." Physics of Plasmas 5, no. 5 (May 1998): 1522–34. http://dx.doi.org/10.1063/1.872810.

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14

Gavaghan, David, Alan Garny, Philip K. Maini, and Peter Kohl. "Mathematical models in physiology." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364, no. 1842 (March 22, 2006): 1099–106. http://dx.doi.org/10.1098/rsta.2006.1757.

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Computational modelling of biological processes and systems has witnessed a remarkable development in recent years. The search-term ( modelling OR modeling ) yields over 58 000 entries in PubMed, with more than 34 000 since the year 2000: thus, almost two-thirds of papers appeared in the last 5–6 years, compared to only about one-third in the preceding 5–6 decades. The development is fuelled both by the continuously improving tools and techniques available for bio-mathematical modelling and by the increasing demand in quantitative assessment of element inter-relations in complex biological systems. This has given rise to a worldwide public domain effort to build a computational framework that provides a comprehensive theoretical representation of integrated biological function—the Physiome. The current and next issues of this journal are devoted to a small sub-set of this initiative and address biocomputation and modelling in physiology, illustrating the breadth and depth of experimental data-based model development in biological research from sub-cellular events to whole organ simulations.

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15

Zefreh, K. Z., F. M. Welford, and Jan Sijbers. "Investigation on the effect of exposure time on scintillator afterglow for ultra-fast tomography acquisition." Journal of Instrumentation 11, no. 12 (December 7, 2016): C12014. http://dx.doi.org/10.1088/1748-0221/11/12/c12014.

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16

VERGANI, S. D., and C. GUIDORZI. "GRB 070311: A COMMON ORIGIN FOR THE PROMPT AND AFTERGLOW EMISSION." International Journal of Modern Physics D 17, no. 09 (September 2008): 1359–62. http://dx.doi.org/10.1142/s0218271808012917.

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GRB 070311 was a long burst that triggered INTEGRAL. We present prompt γ-ray, early NIR/optical, late optical and X-ray data on this burst and its afterglow. Interestingly, the H-band light curve acquired with REM exhibits two pulses at 80 and 140 s after the peak of the γ-ray burst, with possible evidence for a contemporaneous faint γ-ray tail. The late optical and X-ray afterglow underwent a rebrightening between 3 × 104 and 2 × 105 s after the burst with energy comparable with that of the prompt emission extrapolated in the X-ray band. After fitting the early γ-ray and optical light curves, we modelled the time profile of the late rebrightening as the time-rescaled version of the prompt γ-ray pulse over an underlying power law. This result supports a common origin for both prompt and late X-ray/optical afterglow rebrightening of GRB 070311 within the external shock scenario.

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17

Schloemann, E. F. "Mathematical Models of Microwave Ferrites." Le Journal de Physique IV 07, no. C1 (March 1997): C1–433—C1–436. http://dx.doi.org/10.1051/jp4:19971176.

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18

Fagnola, Franco, John E. Gough, Hendra I. Nurdin, and Lorenza Viola. "Mathematical models of Markovian dephasing." Journal of Physics A: Mathematical and Theoretical 52, no. 38 (August 27, 2019): 385301. http://dx.doi.org/10.1088/1751-8121/ab38ec.

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19

STAFF, J., B. NIEBERGAL, and R. OUYED. "A THREE-STAGE MODEL FOR THE INNER ENGINE OF GRBs: PROMPT EMISSION AND EARLY AFTERGLOW." International Journal of Modern Physics D 17, no. 09 (September 2008): 1383–89. http://dx.doi.org/10.1142/s0218271808012954.

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We describe a model within the "quark-nova" scenario to interpret the recent observations of early X-ray afterglows of long gamma-ray bursts (GRBs) with the Swift satellite. This is a three-stage model within the context of a core-collapse supernova. STAGE 1 is an accreting (proto-) neutron star leading to a possible delay between the core collapse and the GRB. STAGE 2 is accretion onto a quark star, launching an ultrarelativistic jet generating the prompt GRB. This jet also creates the afterglow as the jet interacts with the surrounding medium creating an external shock. Slower shells ejected from the quark star (during accretion), can re-energize the external shock leading to a flatter segment in the X-ray afterglow. STAGE 3, which occurs only if the quark star collapses to form a black hole, consists of an accreting black hole. The jet launched in this accretion process interacts with the preceding quark star jet, and could generate the flaring activity frequently seen in early X-ray afterglows. Alternatively, a STAGE 2b can occur in our model if the quark star does not collapse to a black hole. The quark star in this case can then spin down due to magnetic braking, and the spin down energy may lead to flattening in the X-ray afterglow as well. This model seems to account for both the energies and the timescales of GRBs, in addition to the newly discovered early X-ray afterglow features.

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20

Lin, En-Tzu, Fergus Hayes, Gavin P. Lamb, Ik Siong Heng, Albert K. H. Kong, Michael J. Williams, Surojit Saha, and John Veitch. "A Bayesian Inference Framework for Gamma-ray Burst Afterglow Properties." Universe 7, no. 9 (September 17, 2021): 349. http://dx.doi.org/10.3390/universe7090349.

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In the field of multi-messenger astronomy, Bayesian inference is commonly adopted to compare the compatibility of models given the observed data. However, to describe a physical system like neutron star mergers and their associated gamma-ray burst (GRB) events, usually more than ten physical parameters are incorporated in the model. With such a complex model, likelihood evaluation for each Monte Carlo sampling point becomes a massive task and requires a significant amount of computational power. In this work, we perform quick parameter estimation on simulated GRB X-ray light curves using an interpolated physical GRB model. This is achieved by generating a grid of GRB afterglow light curves across the parameter space and replacing the likelihood with a simple interpolation function in the high-dimensional grid that stores all light curves. This framework, compared to the original method, leads to a ∼90× speedup per likelihood estimation. It will allow us to explore different jet models and enable fast model comparison in the future.

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21

Korepanov, I. G. "Fundamental mathematical structures of integrable models." Theoretical and Mathematical Physics 118, no. 3 (March 1999): 319–24. http://dx.doi.org/10.1007/bf02557328.

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22

Aksulu, M. D., R. A. M. J. Wijers, H. J. van Eerten, and A. J. van der Horst. "A new approach to modelling gamma-ray burst afterglows: using Gaussian processes to account for the systematics." Monthly Notices of the Royal Astronomical Society 497, no. 4 (August 5, 2020): 4672–83. http://dx.doi.org/10.1093/mnras/staa2297.

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ABSTRACT The afterglow emission from gamma-ray bursts (GRBs) is a valuable source of information to understand the physics of these energetic explosions. The fireball model has become the standard to describe the evolution of the afterglow emission over time and frequency. Because of recent developments in the theory of afterglows and numerical simulations of relativistic outflows, we are able to model the afterglow emission with realistic dynamics and radiative processes. Although the models agree with observations remarkably well, the afterglow emission still contains additional physics, instrumental systematics, and propagation effects that make the modelling of these events challenging. In this work, we present a new approach to modelling GRB afterglows, using Gaussian processes (GPs) to take into account systematics in the afterglow data. We show that, using this new approach, it is possible to obtain more reliable estimates of the explosion and microphysical parameters of GRBs. We present fit results for five long GRBs and find a preliminary correlation between the isotropic energetics and opening angles of GRBs, which confirms the idea of a common energy reservoir for the kinetic energy of long GRBs.

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23

GEHRELS, N., and J. K. CANNIZZO. "GAMMA-RAY BURSTS — OBSERVATIONS." International Journal of Modern Physics D 19, no. 06 (June 2010): 977–84. http://dx.doi.org/10.1142/s021827181001710x.

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We are in an exciting period of discovery for gamma-ray bursts. The Swift observatory is detecting 100 bursts per year, providing arcsecond localizations and sensitive observations of the prompt and afterglow emission. The Fermi observatory is observing 250 bursts per year with its medium-energy GRB instrument and about 10 bursts per year with its high-energy LAT instrument. In addition, rapid-response telescopes on the ground are providing new capabilities to study optical emission during the prompt phase and spectral signatures of the host galaxies. The combined data set is enabling great advances in our understanding of GRBs including afterglow physics, short burst origin, and high-energy emission.

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24

Tayler, A. B., and J. B. Keller. "Mathematical Models in Applied Mechanics." Journal of Applied Mechanics 55, no. 1 (March 1, 1988): 252. http://dx.doi.org/10.1115/1.3173653.

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25

Penenko, V. V., and E. A. Tsvetova. "Mathematical models of environmental forecasting." Journal of Applied Mechanics and Technical Physics 48, no. 3 (May 2007): 428–36. http://dx.doi.org/10.1007/s10808-007-0053-4.

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26

Zachary, Deborah. "Teaching, learning and using mathematical models in physics." Physics Education 24, no. 6 (November 1, 1989): 339–43. http://dx.doi.org/10.1088/0031-9120/24/6/004.

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27

Walstad, Allan. "On the merits of mathematical models." Physics Today 59, no. 6 (June 2006): 10. http://dx.doi.org/10.1063/1.2218526.

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28

Serdyuk, V. S., A. M. Dobrenko, O. A. Tsorina, E. V. Bakiko, and S. V. Yanchij. "Mathematical models for estimating production risks." Journal of Physics: Conference Series 1050 (July 2018): 012077. http://dx.doi.org/10.1088/1742-6596/1050/1/012077.

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29

Kudryavtsev, A. A., and A. I. Ledyankin. "On the electron and vibrational temperatures in a nitrogen afterglow plasma." Physica Scripta 53, no. 5 (May 1, 1996): 597–602. http://dx.doi.org/10.1088/0031-8949/53/5/017.

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30

Ball, J. M. "Mathematical models of martensitic microstructure." Materials Science and Engineering: A 378, no. 1-2 (July 2004): 61–69. http://dx.doi.org/10.1016/j.msea.2003.11.055.

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31

Blackmore, Denis, Roman Samulyak, and Anthony Rosato. "New Mathematical Models for Particle Flow Dynamics." Journal of Nonlinear Mathematical Physics 6, no. 2 (January 1999): 198–221. http://dx.doi.org/10.2991/jnmp.1999.6.2.6.

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32

Brushlinskii, K. V. "Mathematical Models of Plasma in Morozov’s Projects." Plasma Physics Reports 45, no. 1 (January 2019): 33–45. http://dx.doi.org/10.1134/s1063780x19010021.

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33

Bilge, Ayse Humeyra, Arif Selcuk Ogrenci, and Onder Pekcan. "Mathematical models for phase transitions in biogels." Modern Physics Letters B 33, no. 09 (March 30, 2019): 1950111. http://dx.doi.org/10.1142/s0217984919501112.

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It has been shown that reversible and irreversible phase transitions of biogels can be represented by epidemic models. The irreversible chemical sol–gel transitions are modeled by the Susceptible-Exposed-Infected-Removed (SEIR) or Susceptible-Infected-Removed (SIR) epidemic systems whereas reversible physical gels are modeled by a modification of the Susceptible-Infected-Susceptible (SIS) system. Measured sol–gel and gel–sol transition data have been fitted to the solutions of the epidemic models, either by solving the differential equations directly (SIR and SEIR models) or by nonlinear regression (SIS model). The gel point is represented as the “critical point of sigmoid,” defined as the limit point of the locations of the extreme values of its derivatives. Then, the parameters of the sigmoidal curve representing the gelation process are used to predict the gel point and its relative position with respect to the transition point, that is, the maximum of the first derivative with respect to time. For chemical gels, the gel point is always located before the maximum of the first derivative and moves backward in time as the strength of the activation increases. For physical gels, the critical point for the sol–gel transition occurs before the maximum of the first derivative with respect to time, that is, it is located at the right of this maximum with respect to temperature. For gel–sol transitions, the critical point is close to the transition point; the critical point occurs after the maximum of the first derivative for low concentrations whereas the critical point occurs after the maximum of the first derivative for higher concentrations.

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34

VERGANI, S. D., D. MALESANI, and E. MOLINARI. "THE INTERPLAY OF PROMPT AND AFTERGLOW EMISSION IN GRB 060418." International Journal of Modern Physics D 17, no. 09 (September 2008): 1343–49. http://dx.doi.org/10.1142/s0218271808012899.

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We present observations of the early afterglow emission of GRB 060418. Thanks to the simultaneous coverage at optical, X-ray and gamma-ray wavelengths, we can detect and separate the external shock emission (visible in the optical and late X-ray data) and the central engine activity (early X and gamma rays). The two components are clearly distinguished based on temporal and spectral properties. The detection of the afterglow onset (in the optical) allows the determination of the fundamental fireball properties, namely its bulk Lorentz factor and total energy. The early time X-ray flare closely resembles the prompt emission gamma-ray pulses in its temporal profile, being wider at low energies and showing lags between the hard and soft bands. This provides a strong suggestion that X-ray flares are a continuation of the prompt emission.

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35

MASTICHIADIS, A., and D. KAZANAS. "THE SUPERCRITICAL PILE MODEL FOR GRBs: THE PROMPT TO EARLY AFTERGLOW STAGE." International Journal of Modern Physics D 17, no. 09 (September 2008): 1641–50. http://dx.doi.org/10.1142/s021827180801325x.

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The "supercritical pile" is a very economical GRB model that provides for the efficient conversion of the energy stored in the protons of a relativistic blast wave (RBW) into radiation and at the same time produces — in the prompt GRB phase, even in the absence of any particle acceleration — a spectral peak at an energy ~ 1 MeV. We extend this model to include also the evolution of the RBW Lorentz factor Γ and thus follow the spectral and temporal features of this model into the GRB early afterglow stage. One of the novel features of the present treatment is the inclusion of the feedback of the GRB produced radiation on the evolution of Γ with radius. This way one can obtain afterglow light curves with steep decays followed by a relatively flatter flux stage, as observed in a large number of bursts.

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36

Abdallah, N. Ben, S. Mas-Gallic, and P. A. Raviart. "A Mathematical analysis of electric probe models." Transport Theory and Statistical Physics 25, no. 3-5 (April 1996): 263–81. http://dx.doi.org/10.1080/00411459608220701.

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37

Wang, Qi, and Tianyu Zhang. "Review of mathematical models for biofilms." Solid State Communications 150, no. 21-22 (June 2010): 1009–22. http://dx.doi.org/10.1016/j.ssc.2010.01.021.

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38

Filippov, A., M. Kidera, V. Mironov, T. Nakagawa, and G. Shirkov. "Numerical Simulation of the Bremsstrahlung Emission from ECR Source in Afterglow Mode." Physica Scripta T92, no. 1 (2001): 218–21. http://dx.doi.org/10.1238/physica.topical.092a00218.

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39

Roose, Tiina, and Andrea Schnepf. "Mathematical models of plant–soil interaction." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1885 (September 25, 2008): 4597–611. http://dx.doi.org/10.1098/rsta.2008.0198.

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In this paper, we set out to illustrate and discuss how mathematical modelling could and should be applied to aid our understanding of plants and, in particular, plant–soil interactions. Our aim is to persuade members of both the biological and mathematical communities of the need to collaborate in developing quantitative mechanistic models. We believe that such models will lead to a more profound understanding of the fundamental science of plants and may help us with managing real-world problems such as food shortages and global warming. We start the paper by reviewing mathematical models that have been developed to describe nutrient and water uptake by a single root. We discuss briefly the mathematical techniques involved in analysing these models and present some of the analytical results of these models. Then, we describe how the information gained from the single-root scale models can be translated to root system and field scales. We discuss the advantages and disadvantages of different mathematical approaches and make a case that mechanistic rather than phenomenological models will in the end be more trustworthy. We also discuss the need for a considerable amount of effort on the fundamental mathematics of upscaling and homogenization methods specialized for branched networks such as roots. Finally, we discuss different future avenues of research and how we believe these should be approached so that in the long term it will be possible to develop a valid, quantitative whole-plant model.

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40

Tidriri, M. D. "A novel class of multiscale models in mathematical physics." Nonlinear Analysis: Theory, Methods & Applications 47, no. 7 (August 2001): 4995–5008. http://dx.doi.org/10.1016/s0362-546x(01)00611-3.

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Al-Ghafri, K. S. "Soliton-type solutions for two models in mathematical physics." Waves in Random and Complex Media 28, no. 2 (June 19, 2017): 261–69. http://dx.doi.org/10.1080/17455030.2017.1341669.

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Exner, P., and P. Šeba. "Mathematical models for quantum point-contact spectroscopy." Czechoslovak Journal of Physics 38, no. 1 (January 1988): 1–11. http://dx.doi.org/10.1007/bf01596513.

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Shaumiwaty, S., M. A. Lubis, T. Lubis, Dardanila, A. Purba, T. Nasution, Ramlan, and S. Hasrul. "Teacher performance toward students’ mathematical literacy in teaching linear program mathematical models." Journal of Physics: Conference Series 1663 (October 2020): 012066. http://dx.doi.org/10.1088/1742-6596/1663/1/012066.

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RUFFINI, REMO, CARLO LUCIANO BIANCO, SHE-SHENG XUE, PASCAL CHARDONNET, FEDERICO FRASCHETTI, and VAHE GURZADYAN. "EMERGENCE OF A FILAMENTARY STRUCTURE IN THE FIREBALL FROM GRB SPECTRA." International Journal of Modern Physics D 14, no. 01 (January 2005): 97–105. http://dx.doi.org/10.1142/s0218271805006201.

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It is shown that the concept of a fireball with a definite filamentary structure naturally emerges from the analysis of the spectra of Gamma-Ray Bursts (GRBs). These results, made possible by the recently obtained analytic expressions of the equitemporal surfaces in the GRB afterglow, depend crucially on the single parameter ℛ describing the effective area of the fireball emitting the X-ray and gamma-ray radiation. The X-ray and gamma-ray components of the afterglow radiation are shown to have a thermal spectrum in the co-moving frame of the fireball and originate from a stable shock front described self-consistently by the Rankine–Hugoniot equations. Precise predictions are presented on a correlation between spectral changes and intensity variations in the prompt radiation verifiable, e.g., by the Swift and future missions. The highly variable optical and radio emission depends instead on the parameters of the surrounding medium. The GRB 991216 is used as a prototype for this model.

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Grannan, E. R., and G. Swindle. "Rigorous results on mathematical models of catalytic surfaces." Journal of Statistical Physics 61, no. 5-6 (December 1990): 1085–103. http://dx.doi.org/10.1007/bf01014366.

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Avellaneda, Marco, and Andrew J. Majda. "Mathematical models with exact renormalization for turbulent transport." Communications in Mathematical Physics 131, no. 2 (July 1990): 381–429. http://dx.doi.org/10.1007/bf02161420.

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Raichenko, A. I. "Comparative analysis of one-particle mathematical models in materials science and oncology. I. Mathematical models." Powder Metallurgy and Metal Ceramics 44, no. 11-12 (November 2005): 578–82. http://dx.doi.org/10.1007/s11106-006-0028-7.

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GIORGI, C. "Mathematical models of thin thermoviscoelastic plates." Quarterly Journal of Mechanics and Applied Mathematics 53, no. 3 (September 1, 2000): 363–74. http://dx.doi.org/10.1093/qjmam/53.3.363.

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Kluth, Tobias. "Mathematical models for magnetic particle imaging." Inverse Problems 34, no. 8 (June 12, 2018): 083001. http://dx.doi.org/10.1088/1361-6420/aac535.

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Giorgi, Claudio, and Maria Grazia Naso. "Mathematical Models of Reissner–Mindlin Thermoviscoelastic Plates." Journal of Thermal Stresses 29, no. 7 (June 2006): 699–716. http://dx.doi.org/10.1080/01495730500499183.

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