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Exploring extragalactic particle accelerators with gamma-rays and neutrinos

Prof. Felix Aharonian

The main objective of the proposed topic of the thesis will be related to theoretical studies of spectral and temporal features of gamma-rays and neutrinos produced in small (sub parsec) and large (multi-kpc) scale jets of active galactic nuclei - the potential sites of acceleration of highest energy cosmic rays with spectra extending to 10^20 eV. 

Cosmological implication of the Amati relation

Prof. Lorenzo Amati and Prof. Massimo della Valle

Because of their huge luninosities and a redshift distribution extending up to more than z = 8, Gamma-Ray Bursts (GRBs) are in principle very powerful cosmological probes. The possibility of extracting information on cosmological parameters from GRBs by means of the Amati relation, linking the isotropic-equivalent radiated energy (Eiso) to thespectral peak photon energy (Ep), has already been demonstrated by us since 2008. The aim of this project is to make a significant step forward in this investigation by: a) exploiting the substantially enriched sample of GRBs with measured redshift and spectral parameters; b) exploiting the expected observations of Cepheids up to z = 0.5 - 1 by next generation telescopes (e.g., EELT) to calibrate thr Amati relation; join the cosmological information derivable from GRBs with next generation probes like high-redshift Cepheids, high-redshift novae, improved results from Galaxy clusters.  

The study of transparency of electron-positron-baryon plasma

Prof. Alexey Aksenov, Remo Ruffini and Gregory Vereshchagin

The sources of Gamma Ray Bursts are optically thick and contain electron-positron plasma which plays important dynamical role providing self accelerated expansion of plasma. As result of expansion eventually it becomes transparent to photons, trapped during optically thick phase. Such photons are emitted as a strong flash of radiation identified as P-GRB within the fireshell model of GRBs. The thesis is expected to focus on the study of transparency of electron-positron plasma with baryonic loading by combining relativistic hydrodynamic and kinetic treatments 

Equilibrium configurations of 2 charged spinning masses in General Relativity

Prof. V. Belinski

Detailed study of the possibility of realization and of physical properties of all of the three types of such configurations: Black Hole – Naked Singularity, Black Hole – Black Hole and Naked Singularity – Naked Singularity.References:
1. G.Alekseev and V.Belinski  "”Equilibrium configurations of two charged masses in General Relativity”, Phys. Rev. D76, 021501(R),  (2007).
2. G.Alekseev and V.Belinski  "”On the equilibrium state of two rotating charged masses in General Relativity”, arXiv:1211.3964 [gr-qc].
3. V. Belinski and E. Verdaguer “Gravitational Solitons”, Cambridge University Press, Cambridge Monographs on Mathematical Physics, 2001.

Gravitational waves of solitonic nature

Prof. V. Belinski

The study of propagation and collision of gravitational solitonic waves in vacuum and in material medium. Physical parameters of the waves (energy, mass, velocities). Cosmological applications.
1. V.Belinski “Gravitational breather and topological properties of gravisolitons”, Phys. Rev., D44, 3109, (1991).
2. V. Belinski “One-soliton cosmological waves”,  Sov. Phys. JETP, 50, 623, (1979).
3. V. Belinski and E. Verdaguer “Gravitational Solitons”, Cambridge University Press, Cambridge Monographs on Mathematical Physics, 2001.

"Disguised" short Gamma-Ray Bursts: correlations between peak luminosity, circumburst medium density and offset from the center of their host galaxies

Prof. Carlo Bianco

Within the fireshell model Gamma-Ray Bursts are divided into three classes: "genuine" short GRBs, characterized by a low value of the fireshell baryon loading; "long" GRBs, characterized by a high value of the fireshell baryon loading and an average CircumBurst Medium (CBM) density of the order of 1 particle/cm^3; and "disguised" short GRBs, characterized by a high value of the fireshell baryon loading, like the "long" ones, but with an average CBM density of the order of 10^{-3} particles/cm^3, typical of a galactic halo environment. We have therefore proposed that the progenitors of this new class of "disguised" short GRBs are merging binary systems, formed by neutron stars and/or white dwarfs in all possible combinations, which spiraled out from their birth place into the halo. "Disguised" short GRBs present an occasional softer extended emission after an initial spike-like emission. Since the soft tail peak luminosity is directly related to the CBM density, GRBs displaying a more luminous prolonged soft tail should have a systematically smaller offset from the center of their host galaxy. Some observational evidence was found in this sense. However, this point is still debated, mostly due to the paucity of the sample of identified "disguised" short GRBs. The thesis work will involve the analysis of different GRBs within the fireshell model, using data from Swift and Fermi satellites, to identify the "disguised" short ones so to increase their sample. In particular, the properties of the corresponding host galaxies will be studied in details, to establish a correlation between the soft tail peak luminosity of the "disguised" short GRBs and their offset from the center of the host galaxies. 

GRB luminosity function and redshift distribution

Prof. Carlo Bianco

The aim of the thesis is to use all the available GRB data (from the historical observations by BATSE and BeppoSAX all the way to the latest observations by Swift, Fermi and Agile) to study the luminosity unction and redshift distribution of GRBs pertaining to different classes ("canonical long", "genuine short", "disguised short", "IGC-GRBs", etc.). References: -) M. Schmidt, ApJ, 700 (2009) 633. -) L.J. Rangel Lemos, C.L. Bianco, H.J. Mosquera Cuesta, J.A. Rueda, R. Ruffini, PoS(Texas2010), 204. 

Relativistic kinematics and dynamics of GRBs

Prof. Carlo Bianco

The aim of the thesis is to study the relativistic effects in the kinematics and dynamics of GRBs, in particular the ones related to the shape of the surfaces of equal arrival time (EQTS) of the emitted radiation, in view of the detailed temporal structure of the observed GRB light curves and spectra.
-) C.L. Bianco, F.A. Massucci, R. Ruffini; International Journal of Modern Physics D, 20 (2011) 1919.
-) C.L. Bianco, R. Ruffini; ApJ, 633 (2005) L13.  

Study of the gravitationnal and electromagnetic signature from the gamma-ray burst sources

Prof. Michel Boer

Gravitationnal wave detectors VIRGO and LIGO reached their nominal sensitivity after a decade of construction and commisionning. This opens a new window on the Universe, and enables joint searches of objects emitting also electromagnetic radiation and neutrinos. First generation interferometers have already given constrains on the Crab ellipticity or on the gravitationnal wave stochastic background. With the advent of the second generation of detectors, within the next years, we expect to see at least close-by binary sources. Third generation interferometers, such as the Einstein Telescope, will allow the detection and study of a large variety of phenomena, from compact objects detection to the implications in fundamental physics and cosmology. The identification of an optical signature in coincidence with a gravitational signal would lead to a significant increase of the detectability distance, the horizon of the detector, hence of the event rate by a factor of about 3. Gamma-ray bursts progenitors are among the best promising candidates for such pioneering detections of events in gravitational waves, since they are thought to be caused by the collapse of very massive stars, or the coalescence of compact binary systems. The goal of the thesis will be to prepare and to optimise these new experiments. Using different classes of GRB progenitors the candidate will look for the signal in gravitational waves, as well as from electromagnetic detectors, and for possible coincidences. He will model the temporal evolution of the fireball, derive the characteristic times, and the lags between gravitationnal and optical radiations. The candidate will participate in the follow-up of GRB sources using the TAROT and Zador telescopes, detected by the SWIFT satellite. This work will be also made in collaboration with the VIRGO and LIGO teams.   

Chemical Evolution of Rotating, Collapsing, Protostellar Clouds

Prof. Sandip Chakrabarti 

Hydrodynamic and Radiative properties of Black Hole Accretion Flows and their Observational Consequences

Prof. Sandip Chakrabarti 

Probing the Early Universe with Cosmic Large Scale Structure

Prof. Jaan Einasto

Our theoretical understanding of the formation and evolution of cosmic large-scale struc- ture (LSS) has proven to be a powerful tool (along with cosmic microwave background (CMB) experiments and type Ia supernova surveys) in helping to establish “concordance cosmological model” – the standard model of cosmology. On one hand the standard cos- mological model is remarkably simple: in its most basic form a model with only 5-6 free parameters is able to accommodate a large body of diverse observational data. On the other hand we should be worried since we have good physical theories available for only ∼ 5% of the stuff out there, while the rest ∼ 95% is provided by poorly understood dark components – dark energy (DE) and dark matter (DM). Mostly from CMB mea- surements we have been able to learn a great deal about the initial fluctuations: we know that the initial fluctuations are compatible with the predictions of the simplest inflationary models, i.e. Gaussian adiabatic perturbations with approximately scale- free Harrison-Zeldovich spectrum. Any detectable deviation from Gaussianity would certainly be extremely interesting as it would tell us something about the nature of the physical processes setting up these primordial perturbations. So far the strongest upper bounds (there has been no convincing detection so far) on allowed level of non- Gaussianities has been obtained from CMB measurements. However, it has been shown that future LSS surveys will be at least as competitive as upcoming CMB experiments in constraining the possible level of primordial non-Gaussianity. Indeed, from the anal- ysis of the current LSS data there are already some hints that might argue in favor of non-Gaussian models: • unexpectedly massive galaxy clusters at high redshifts; • surprisingly large number of very rich superclusters; • excess power on the largest scales as hinted by several recent power spectrum measurements from the Sloan Digital Sky Survey (SDSS) galaxy data. In this PhD project the student will work on available LSS datasets, particularly with the data from the SDSS. The main focus of the study will be: • search for primordial non-Gaussianities; • constraining the properties of dark components (DE, DM) throughout cosmic evo- lution. As the statistical analysis (in particular, testing of statistical methods and error analy- sis) of the LSS data often requires comparison with simulation outputs, the student will most probably run some cosmological N-body simulations or use already available sim- ulation data. These tasks will understandably require some knowledge of programming languages, and thus it would be beneficial if the student has already some experience with computer programming.   

Study of new missions devoted to Gamma Ray Bursts Astronomy

Prof. Filippo Frontera

The origin of GRB, their emission mechanisms and their use as rulers for cosmology reuire further experimental efforts. New missions are requested. Among the proposed, there is MIRAX. The PhD student will be associated to the MIRAX development, if this mission will be approved.  

Feasible science with deep gamma-ray observations from 100 keV to 600 kev and beyond

Prof. Filippo Frontera

The development of Laue lenses for gamma-rays is at an advanced stage of development in our institute. With Laue lenses a sensitivity 2 orders of magnitude better than that achieved thus far is expected. The PhD thesis would concern the science cases that could be faced with astronomical observations in the Laue lens passband. The expectations should be quantitatively investigated. 

A study of the contamination of CMB fluctuation maps from foreground sources

Prof. Paolo Giommi 

Systematic studies of the erratic time variability of the prompt emission and early afterglow of GRBs

Prof. Cristiano Guidorzi   

Multiwavelength observations and analysis of gamma ray bursts - supernovae events

Prof. Luca Izzo

The current possibility to analyze gamma ray burst emission has recently allowed the formulation of a new model for the GRB-SN association (Rueda & Ruffini 2012). The induced gravitational collapse scenario acting in GRBs-SNe has been applied with success to low-z GRBs, as 090618 and 110709B (Izzo et al. 2012; Penacchioni et al. 2012). The candidate will work in a competitive group and will be involved in the search for new GRB-SN candidates, starting from the analysis of observations in almost "real-time" of new GRBs using data from satellites as Swift and Fermi. The physical characteristics of the IGC scenario are able to estimate the distance of the GRB and then predicts the occurrence of a SN associated to the GRB. Moreover, part of the time will be dedicated to the theoretical development of the same model, which needs a more complete treatment that involves the study of the physics of Neutron stars and Black Holes originating GRBs, as well the late phases of very massive stars.

On the Difference between Euler-Heisenberg-Schwinger and Unruh Effect of Particle Creation

Prof. Hagen Kleinert 

Rapidly rotating neutron stars with realistic equations of states: analysis of the multipole moments

Prof.ssa Jutta Kunz

Rapidly rotating neutron stars are calculated with realistic equations of state in GR and for generalized Einstein equations. The multipole moments are extracted and analyzed. Possible observational signatures are identified.

Gravitomagnetic effects on clocks and pulsar timing

Prof. Claus Laemmerzahl

The influence of the gravitomagnetic field on clocks on satellites orbiting the Earth as well as on pulsars orbiting a Kerr black hole shall be investigated. This has importance for positioning and pulsar timing.

TeV gamma-ray extragalactic sources with HESS experiment

Prof. Giovanni Lamanna

Major astroparticle physics research observatories on ground and in space provide outstanding data for the understanding of the origin and propagation of cosmic radiation. High-energy gamma rays and antimatter cosmic-rays are also key messengers for the indirect search of dark matter in our Galaxy. The PhD thesis would concern the investigation of inclusive search with these messengers through the analysis of data from H.E.S.S., Fermi and AMS and the phenomenological implications in the field of particle physics and astrophysics.  

Hidden symmetries of gravity and BKL cosmology

Prof. Hermann Nicolai 

Bouncing Cosmology

Prof. Mario Novello 

Non-gravitational Black Hole: effective metric in non-linear Electrodynamics

Prof. Mario Novello 

Gravitational lensing of rotating black holes

Prof. Volker Perlick

An exact lens equation, based on the lightlike geodesic equation without approximations, is to be worked out for stationary axisymmetric spacetimes. This is then to be used for investigating the multiple-imaging properties, the caustics and the shadow of rotating black holes and other compact objects.

Differential interferometry of the Broad Line Region in the Active Galactic Nuclei

Prof. Romain Petrov and Prof. Alessandro Marconi

The « Broad Line Region » (BLR) in the active galactic nuclei of Seyfert I galaxies or Quasars corresponds to emission lines of width several thousands of kilometers per second. These lines are generally attributed to the accretion disk of gas and plasma around a supermassive central black hole. Few constraints, especially on their sizes are provided by time­resolved multi­wavelength photometry. The largest ones are 200 micro­arcseconds wide, in principle too small to be resolved, even with the largest facilities available like stellar interferometers. However, an exciting new technique called “differential interferometry” enables one, given the interferometer is coupled with a spectrograph, to access spatial scales as small as the extent of the largest BLRs. This subject is linked to the exploitation of the Astronomical Multi­BEams Recombiner (AMBER) located in Chile at the Very Large Telescope Interferometer (VLTI). New developments are in progress to upgrade the limiting magnitude of AMBER, giving access to the brightest AGNs. In particular, an observing campaign will take place in 2011 to resolve the innermost regions of few of these extragalactic objects. The student will be involved in the observations, the data extraction and the interpretation and modelling of these objects. He will be part of a well established team of interferometrists (F. Millour, B. Lopez, P. Berio) and will collaborate with the extragalactic team at the Côte d’Azur Observatory (E. Slezak). Reference: Marconi, A., Maiolino, R., Petrov, R. G., “Extragalactic Astronomy with the VLTI: a new window on the Universe”, Astrophysics and Space Science, v. 286, Issue 1, p. 245­254, (2003).  

Massive stars: from numerical simulations to observations

Prof. Mikhail Popov

The investigation of hydrodynamics instabilities and nucleosynthesis in 100-300 solar masses stars which could be progenitors of pair-instability supernova explosions. The used hydrodynamic code is based on piece-wise parabolic approximation of quantities, which are constructed by a grid of characteristic lines with the minimum utilization of interpolation operators. The outcome of nuclear elements is computed by tracer particle method, which is a post-processing step of integration of a reaction network over the recorded temperature and density time histories along the paths the tracer particles took in a hydrodynamic simulation. The investigation includes the study of influence of magnetic fields, rotational effects and other parameters of initial configuration, estimation of light curves and a spectrum.   

Discrete inhomogeneous relativistic cosmology

Prof. Kjell Rosquist

(a) In a new approach to cosmology, we employ a mean field philosophy to study cosmological evolution and to estimate observational quantities. One advantage, compared to traditional schemes, is the close connection to the observed discrete matter distribution which is used as an input in the evolution equations. Another advantage is that we use only a minimum of assumptions. In particular, no symmetries, such as isotropy or homogeneity, need to be imposed at the outset. (b) Cosmological observations depend on light beams from distant sources. The beams are focused by the spacetime curvature. In the standard Friedmann cosmology, the curvature is determined by the Ricci tensor which represents the part of the curvature which is determined locally by the matter. In the real universe, however, the curvature is dominated by the Weyl curvature which represents the vacuum part of the curvature. The aim of this project is to investigate the influence of the Weyl curvature on the focusing of light beams in order to obtain a better basis for the interpretation of cosmological observations. Working on the above projects can include the development of the theory and/or to study luminosity-redshift relations which can be compared with observational data such as those coming from supernova measurements 

Fundamental interactions and structure properties of rotating neutron stars

Prof. Jorge A. Rueda

Construction of equilibrium configurations of neutron stars taking into account the strong, weak, electromagnetic, and gravitational interactions within the frameqwork of the general theory of relativity. The effects of uniform rotation are also considered. 

Probing compact stars theory from cooling

Prof. Jorge A. Rueda

The luminosity coming from the cooling of a compact star depends on the details of its internal structure, e.g. pressure, density, and gravitational potential inside the star. Thus, the observation of the surface temperature of compact stars ovser the time constraints the theory of compact stars. We solve the radiative transport equations for different cooling  mechanisms. The aim is to compare and contrast the cooling luminosity  of different compact stars e.g. globally and locally neutral neutron stars, hybrid stars, strange stars

Probing compact stars theory from cooling
Prof. Jorge A. Rueda

Neutron star theory and gamma-ray bursts
Prof. Jorge A. Rueda

The gravitational collapse of neutron stars is one of the possible progenitors of  gamma-ray bursts (GRBs). In particular, short GRBs are believed to be the outcome of the merger of binary neutron stars. It has been also recently shown in (Rueda & Ruffini 2012)  that the GRBs accompanied by a supernova explosion (GRB-SN) can be the result of the gravitational collapse of a neutron star to a black hole induced by the accretion of the material expelled in the SN explosion of a close companion evolved star. This scenario has been called induced gravitational collapse (IGC).  Thus, short GRBs and GRB-SN systems can unveil crucial information on the theory of neutron stars and probe their internal structure. The aim is to study the details of these processes in light of a self-consistent theory of neutron stars that includes all the fundamental interactions within the framework of general relativity.

Physics and Astrophysics of massive, rotating, highly magnetized white dwarfs
Prof. Jorge A. Rueda

We construct equilibrium configurations of white dwarfs within general relativity in the non-rotating and rotating cases. The finite temperatures, nuclear, quantum, and electromagnetic effects are all accounted for. The physics and astrophysics application of the magnetosphere of highly magnetized white dwarfs is also studied. We apply in addition the results to the study of astrophysical systems where massive white dwarfs are relevant, e.g. massive white dwarf binary mergers, massive accreting white dwarfs, novae; type Ia supernovae, as well as soft gamma-ray repeaters and anomalous X-ray pulsars.

Dark matter distribution in galactic cores and halos

Prof. Remo Ruffini

Structure of the Universe and Dark Matter

Prof. Remo Ruffini  

Supernovae and GRBs

Prof. Remo Ruffini

The aim of the thesis is to study the association between GRBs and supernovae, particularly in view of the Induced Gravitational Collapse (IGC) scenario proposed within the Fireshell model.References:-) L. Izzo, R. Ruffini, A.V. Penacchioni,  et al.; Astronomy & Astrophysics, 543 (2012) A10.-) A.V. Penacchioni, R. Ruffini, L. Izzo, et al.; Astronomy & Astrophysics, 538 (2012) A58. 

The mass formula of Black Holes and its application for Gamma Ray Bursts

Prof. Remo Ruffini (Nice) and Prof. Donato Bini (Rome)

The rotational and electromagnetic energy of Black Holes are linked to energy source of Gamma Ray Bursts via study of exact analytic and numerical solution of collapsing shell in General Relativity. The definition of the Blackholic energy is analyzed also in connection to the electrodynamic properties of the Fermi solution in Special Relativity. Recent work by R. Jantzen is generalized. 

URCA process in a collapsing core endowed with electromagnetic structure

Prof. Remo Ruffini and Prof. She-Sheng Xue

The case of collapsing core fulfilling general relativistic, electromagnetic, weak and strong interactions is being studied with special attention to the URCA process. The number fluxes and energy spectra of neutrino and anti-neutrino emission are studied. 

The role of 17 keV neutrinos in cosmological structure formation

Prof. Remo Ruffini and Prof. Gregory Vereshchagin

A unified approach to halos in globular clusters, dwarf galaxies and galaxies as well as nuclear condensations in galactic cores are analyzed within General Relativity and quantum statistics. 

Improved relativistic moment of inertia of rotating Neutron Stars

Prof. Remo Ruffini (Rome) and Prof. Jorge Rueda (Rome)

A self consistent model of rotating Neutron Stars fulfilling General Relativistic treatment as well as electromagnetic, weak and strong interactions is explored, as well as its consequences for observations.   

Observational study of the prompt emission in gamma-ray bursts with the Fermi telescope: the role of the photosphere of the relativistic jet

Prof. Felix Ryde

Gamma-ray bursts are the largest known explosions in the universe. Due to their huge brightness we are able to detect them from very large distances, thereby viewing the very early universe. Most of the observable energy in a GRB is released in the gamma-rays. In spite of this fact we do not yet know how it arises. This problem is therefore one of the most fundamental in high-energy astrophysics today, and much attention has been devoted to it both observationally and theoretically. The aim of the proposed PhD project is to attack this problem to study and determine the radiation and particle acceleration processes involved in producing the GRBs. In particular, this includes detailed studies of the spectral and temporal data available from the Fermi Gamma-ray Space Telescope, combined with in-depth theoretical considerations in order to understand these observations. The expected result will be better and deeper understanding of the radiation processes of GRBs, in particular of the broad-band spectra, up to 300 GeV and beyond will be understood. There are strong indications that the jet photosphere is important in shaping the spectrum, which is why significant effort will be devoted to understanding its behaviour. It is of great importance to solve this problem since (i) it is vital in giving us more clues to what the central engine is and how it behaves (ii) it will let us understand the shock physics and the ultra-relativistic environments in which the gamma-rays are created (iii) it will enable us to use bursts for cosmological studies.  

Bulk motion Comptonization in black hole/neutron star binary systems: prediction vs. observations

Prof. Lev Titarchuk 

Theory of Cosmological correlations

Prof. Gregory Vereshchagin

Presently observed Large Scale Structure of the Universe was formed during cosmological expansion. Initially small density fluctuations grown due to gravitational instability. Generally two regimes are considered: linear growth of perturbations and nonlinear structure formation. While linear regime is well studied analytically, nonlinear structure formation is mainly relied on numerical N-body simulations. Unlike linear theory, where the key object is the density contrast, in nonlinear theory the key objects are density correlations. The thesis should be focused on theory of cosmological density correlations at strongly nonlinear stage of cosmological structure formation.  

Emission mechanisms of Gamma Ray Bursts

Prof. Gregory Vereshchagin

Gamma Ray Bursts are the most powerful explosions of stellar objects in the Universe. Various models of GRBs consider different microphysical mechanisms of emission, including synchrotron, synchrotron - self Compton, inverse Compton, thermal ones. Unlike many astrophysical sources, the interpretation of data from GRBs requires the knowledge of dynamical history of these events, since they involve strong accelerations. Current observational data narrow down the list of possible emission mechanisms, and will eventually allow to rule many of them out. The thesis should be focused on microphysical theory of emission, including relativistic effects relevant for the observer.  

Electromagnetic fields space-time evolution in gravitationally collapsing compact stars

Prof. She-Sheng Xue

The main objective of the proposed topics for Ph.D. thesis's will be related to theoretical studies of both spatial and temporal evolution of electromagnetic fields and electron-positron pair-productions in compact stars, when they undergo perturbations or gravitational collapses. This will give us a chance to understand energetic sources accounting for gamma-ray bursts and soft gamma-ray repeaters. Approach will bases on both analytical and numerical studies of the Einstein-Maxwell equations, energy-momentum, particle-number conservations, and rate of electron-positron production, as well as equations of states.

The study of nonperturbative  electromagnetic dynamics and role of L'evy  tails
Prof. She-Sheng Xue and Prof. Hagen Kleinert

The study of nonperturbative  electromagnetic dynamics (QED) for strong electromagnetic fields and role of non-Gaussian distributions with L'evy  tails in strong laser fields of self-focusing Materials. Applications to structure formation in the Universe and Gamma-Ray Bursts.

(1)    H. Kleinert. ``Particles and quantum fields'', World Scientific
 http://users.physik.fu-berlin.de/kleinert/ /psles/.
(2)    R. Ruffini, G. Vereshchagin, and S.S. Xue. ``Electron-positron pairs in physics and astrophysics: From heavy nuclei to black holes''. Physics Reports, 487(1):1-140 (2010).
(3) H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, Singapore, 2009 (chapter 20)

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