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40ed8e5b630b00826712ff7ff5085390b1642fe7 | subsection | 18 | 20 | Mass duality as T-duality | But detailed analysis of the correspondence between
our and a string theory of the black hole goes over basic
intention of this work. | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
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2db5ef005d1f28ed1a52d83c5d6b068cc91eb0ad | subsection | 19 | 20 | Conclusion | In conclusion we can shortly repeat and point out the following.
In this work we showed that single horizon black hole behaves as a
"laser". It is in many aspects conceptually analogous to
remarkable Corley and Jacobson work on the two horizon black hole
laser. We started by proposition that circumference of the black
... | {
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} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
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"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
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06a005abab3a5ba7455ac5c991aef2f7ca7cdf09 | abstract | 0 | 11 | Abstract | Focussing on null fields as simple models of laser beams we discuss the
classical relativistic motion of charges in strong electromagnetic fields. We
suggest a universal, Lorentz and gauge invariant measure of laser intensity and
explicitly calculate and interpret it for crossed field, plane wave and vortex
models. | {
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d57c91fa659c19545b0a35ae214a427dea436e91 | subsection | 1 | 11 | Introduction | Since the advent of chirped pulse amplification optical lasers have reached
unprecedented intensities, the present state of the art being
about 10^{22} W/cm^2. The associated electric fields of
approximately 10^{13} V/m have renewed interest in the idea
of particle acceleration using laser fields. Using plasma wake
fie... | {
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e13244e31ad4dfa5ae8b776536c701f197239b84 | subsection | 2 | 11 | Introduction | This is defined by m_e c^2=eE_\mathrm {crit}\protect {\mathchoice{\displaystyle \lambda }{\hbox{}}{\hspace{0.0pt}\vrule width.7height.1pt depth.1pt\hss }{\box }0}\textstyle \lambda \scriptstyle \lambda \scriptscriptstyle \lambda C, where \displaystyle \lambda \textstyle \lambda \scriptstyle \lambda \scriptscriptstyle \... | {
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b1d70d7e15e88f7f01b9c7ea30e6baa930c86527 | subsection | 3 | 11 | Motion in plane wave fields | Since we know from (REF ) that a_0 is a classical quantity, it will suffice to consider the classical motion of a charged particle (chosen to be an electron throughout), as governed by the Lorentz equation\dot{p}_\mu (\tau ) = \frac{e}{m_e} \, F_{\mu \nu } \big ( x(\tau )\big ) \, p^\nu (\tau ) \; ,maintaining both gau... | {
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e34d6c63363e40cd6011c96c0701cb48214140c4 | subsection | 4 | 11 | Motion in plane wave fields | Thus, quite remarkably, we can trade the x dependence of F^{\mu \nu } for a dependence solely on proper time \tau
whereupon the equation of motion becomes linear. Consequently it
can be solved analytically by exponentiation. As the time dependence resides in
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b76d293738f5d2ea1af9773a20cb0dc5aca1ac1b | subsection | 5 | 11 | Motion in plane wave fields | As a result of this discussion the exponential series
(REF ) is truncated to second order, giving the simple expressionp_\mu (\tau ) = \left[ g_{\mu \nu } +
G_{\mu \nu }(\tau ) + {\textstyle \frac{1}{2}} G^2_{\mu \nu } (\tau ) \right]
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6d98f5860e1645f82cfdb515adf0b4785a3cb6cc | subsection | 6 | 11 | Motion in plane wave fields | Altogether, a_0 is a ratio of two energies as described below ().Taking p as the momentum of a probe photon
a_0 also determines the amount of vacuum birefringence in
ultra-strong laser fields due to an induced effective metric,
h_{\mu \nu } = g_{\mu \nu } - \kappa \, T_{\mu \nu } with \kappa =
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db44f72fb3b70320aec3f135f069b2a2e444e54a | subsection | 7 | 11 | Motion in plane wave fields | In this case b_1 is as above, b_2=-C(\Omega \tau -\sin \Omega \tau ) and b_3 = -2C b_2 i.e. there is no frequency doubling.
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508d6c9a7bbb0f4542b2e8ba677531c436ecfeb1 | subsection | 8 | 11 | Electromagnetic vortices | A more sophisticated null field model of a laser beam is the electromagnetic vortex found by Białynicki-Birula . The author states that this
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369934cfb669b2ed6e7382bea8999acaa65d8209 | subsection | 9 | 11 | Electromagnetic vortices | Its solution amounts, in our notation, to introducing a co-moving basis \epsilon _\tau via\zeta (\tau ) \equiv e^{-i\Omega \tau /2} \, Z(\tau ) \equiv \epsilon _\tau \cdot x \; .In terms of \zeta the transverse equation of motion is\ddot{\zeta } + i \Omega \dot{\zeta } - \frac{\Omega ^2}{4} \zeta - \omega _c \Omega \, ... | {
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0.01068115234375,
0.0994873046875,
0.01123046875,
0.1260986328125,
0.150146484375,
0.08795166015625,
0.1868896484375,
0.0098876953125,
0.10388183593... |
eebe479834f4c60eee4841273f53f249e8ff667e | subsection | 10 | 11 | Discussion and conclusion | We have given a thorough discussion of particle motion in null
fields from a particle physics perspective. The fields in
question are regarded as simple models of laser fields, for which the usual Lorentz and gauge
invariants quadratic in electromagnetic fields vanish. It
follows that the laser fields can only be chara... | {
"cite_spans": []
} | 10.1016/j.optcom.2009.01.051 | 0807.1841 | A Lorentz and gauge invariant measure of laser intensity | [
"Thomas Heinzl",
"Anton Ilderton"
] | [
"physics.class-ph",
"hep-ph"
] | 2,008 | en | Physics | [
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9fdf809602f91944acdb5a26258a6d373ebbf6d1 | abstract | 0 | 22 | Abstract | A fully implicit finite difference scheme has been developed to solve the
hydrodynamic equations coupled with radiation transport. Solution of the time
dependent radiation transport equation is obtained using the discrete ordinates
method and the energy flow into the Lagrangian meshes as a result of radiation
interacti... | {
"cite_spans": []
} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
Hydrodynamics coupled to Radiation Transport Equation | [
"Karabi Ghosh",
"S. V. G. Menon"
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1a61e3b04dfcc777293cfc4a7c8654ca552a5be0 | subsection | 1 | 22 | Introduction | Radiation transport and its interaction with matter via emission,
absorption and scattering of radiation have a substantial effect on
both the state and the motion of materials in high temperature
hydrodynamic flows occurring in inertial confinement fusion (ICF),
strong explosions and astrophysical systems . For
many a... | {
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f4e8609e2cc77dbe514fd9e761d2d021c934ee16 | subsection | 2 | 22 | Grid structure | For hydrodynamics calculations, the medium is divided into
a number of cells as shown in Fig. REF . The coordinate
of the i th vertex is denoted by r_i and the region between
the (i-1) and i th vertices is the i th cell. The density
of the i th grid is \rho _i and its mass is given bym_i=\acute{c} \times \rho _i \times... | {
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} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
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d18491ee1dd12bb980375006131c8ae90fe408c7 | subsection | 3 | 22 | Lagrangian step | During a time interval \Delta t\ the vertexes r_i
of the cells move as\tilde{r}_i = r_i+u_i^\star \Delta t\\
u_i^\star =(1/2)(u_i+\tilde{u}_i)where u_i^\star is the average of velocity values at the
beginning and end of the Lagrangian step, u_i and
\tilde{u}_i , respectively. | {
"cite_spans": []
} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
Hydrodynamics coupled to Radiation Transport Equation | [
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491b9aac6191ccf9f4d65ef06f39fd6a6d66b56a | subsection | 4 | 22 | Discretized form of the hydrodynamic equations | In the Lagrangian formulation of hydrodynamics, the mass of each
cell remains constant thereby enforcing mass conservation.The Lagrangian differential equation for the conservation of
momentum is\rho \frac{d\vec{u}}{dt}=-\vec{\nabla }PHere, the total pressure is the sum of the electron, ion and
radiation pressures i.e.... | {
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8e17c9767d5a10100f327cebde9610612daac92c | subsection | 5 | 22 | Discretized form of the hydrodynamic equations | This equation can be discretized to obtain the change in total
pressure along a Lagrangian trajectory in terms of the velocity
\tilde{u}_i at the end of the time step :P_{i}^{1/2}=P_{i}+q_{i}-
\rho _{i} v_{i}^2 \frac{1}{r_{i-1/2}^\alpha }
\times [\frac{r_{i}^\alpha \tilde{u}_{i} - r_{i-1}^
\alpha \tilde{u}_{i-1}}{r_{i}... | {
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23944bf370335a742ad59f4b432e570693a79564 | subsection | 6 | 22 | Discretized form of the hydrodynamic equations | \sigma _R(T_{elec}) is the
Rosseland opacity, E_R(r,T_{elec}) is the radiation energy flux
and \sigma _R(T_{elec})\ B(T_{elec}) is radiation emission rate.
P_{ie} is the ion-electron energy exchange term given byP_{ie}(Tergs/cm^3/\mu s)=2.704 \times 10^{-40} n_{elec}\ n_{ion}\\
\times \frac{T_{ion}-T_{elec}}{T_{elec}^{... | {
"cite_spans": []
} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
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79f202c65f2ae34b388fca12c1b635608f10fd08 | subsection | 7 | 22 | Discretized form of the hydrodynamic equations | Here the Coulomb logarithm for ion-electron collision
is \ln \Lambda =\mbox{max}\lbrace 1, \ (23 - \ln [(n_{elec})^{0.5} Z\ T_{elec}^{-1.5}]) \rbracewith T_{elec} expressed in eV.The discrete form of the energy equations for ions and electrons areT_{ion,i}^{n,k}=T_{ion,i}^{n-1}-(P_{ion,i}^{n,k-1}\ \Delta V_i^{n,k}+
\fr... | {
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4fa512335b20b256eb46396fb9b47ce02ebc85ba | subsection | 8 | 22 | Discrete ordinates method for solving
the radiation transport equation. | In the Gray approximation, or one group model, the
time dependent radiation transport equation in a
stationary medium is\frac{1}{c}\frac{\partial I}{\partial t}+
\vec{\Omega }.\vec{\nabla }I+(\sigma _R(T)+\sigma _s )
I(\vec{r},\vec{\Omega },t )=\sigma _R(T)B(T)\\
+\frac{\sigma _s}{4\pi }\int I(\vec{r},\vec{\Omega }\acu... | {
"cite_spans": []
} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
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"Karabi Ghosh",
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c32a4a510a949fb3d6cebe78137cbfa7880cf3fe | subsection | 9 | 22 | Discrete ordinates method for solving
the radiation transport equation. | This iteration arises because
the opacity \sigma _R(T) and the radiation emission rate
\sigma _R(T)B(T) are functions of the local temperature T. The
converged spatial temperature distribution is assumed to be known
for the hydrodynamic cycle for the previous time step. Starting
with the corresponding values of \sigma ... | {
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27c424f8b98bc185ca5f0db5094a2d81dfebb0b6 | subsection | 10 | 22 | Discrete ordinates method for solving
the radiation transport equation. | However this modification is not necessary
in the implicit method as the iterations are performed for
converging the temperature distribution.To solve Eq. [REF ], it is written in the discrete angle
variable as \frac{\mu _m}{r^2}\frac{\partial }{\partial r}(r^2 I_m)+
\frac{2}{r\omega _m}(\alpha _{m+1/2}I_{m+1/2}-
\alph... | {
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d67ddb7e00e377da56f63e16d14e19704e7049f1 | subsection | 11 | 22 | Discrete ordinates method for solving
the radiation transport equation. | As mentioned earlier, planar geometry equations are obtained if the
terms involving \alpha _{m\pm 1/2} are omitted and the replacements
V_i=r_{i+1/2}-r_{i-1/2} and A_{i+1/2}=1 are made. Thus, both
geometries can be treated on the same lines using this approach.
The difference scheme is completed by assuming that the fl... | {
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d9a597375ddacf7bfdca67c78300471ff053b3de | subsection | 12 | 22 | Discrete ordinates method for solving
the radiation transport equation. | The mesh-angle
sweeps are repeated until the scattering source distribution
converges to a specified accuracy. The rate of radiation energy
absorbed by unit mass of the material in the i th mesh is\varepsilon _i=\sigma _{Ri}^{n,k-1}[E_{Ri}^{n,k}-B_i^{n,k-1}]/
\rho ^{n,k}_iwhich determines the coupling between radiation... | {
"cite_spans": []
} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
Hydrodynamics coupled to Radiation Transport Equation | [
"Karabi Ghosh",
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8d22c47ad3353a93997ba7db253ba713830f8e95 | subsection | 13 | 22 | Implicit radiation hydrodynamics solution method | The sample volume is divided into 'L' meshes of equal width.
The initial position and velocity of all the vertices are
defined according to the problem under consideration. Also
the initial pressure, temperature and internal energy of
all the meshes are entered as input.For any time step, the temperature of the inciden... | {
"cite_spans": []
} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
Hydrodynamics coupled to Radiation Transport Equation | [
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65164e3ae058f52162690a05b8d477ea5105f332 | subsection | 14 | 22 | Implicit radiation hydrodynamics solution method | The stability analysis of Von Neumann introduces additional
reduction in time step due to the material compressibility
.The above procedure is repeated up to the time we are interested
in following the evolution of the system.
The solution method described above is clearly depicted in the
flowchart given in Fig. REF . ... | {
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8005a1df7311c5328ce038cf2d05c5a64fee3b45 | subsection | 15 | 22 | Semi-implicit method | In the semi-implicit scheme, Eq. [REF ] is retained
and P_i^{1/2} is expressed as P_i^{1/2} = (P_i+\tilde{P}_i)/2
wherein \tilde{P}_i is the pressure at the end of the time step.
Starting with the previous time step values for \tilde{P}_i, the
position and velocity of each mesh is obtained and \tilde{P}_i
is iterativel... | {
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791ad9261417881ae250ac143d22702418e3a315 | subsection | 16 | 22 | Shock propagation in Aluminium | In the indirect drive inertial confinement fusion, high power
laser beams are focused on the inner walls of high Z cavities
or hohlraums, converting the driver energy to x-rays which implode
the capsule. If the x-ray from the hohlraum is allowed to fall
on an aluminium foil over a hole in the cavity, the low Z material... | {
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d3cc5391acbf020ea35b151b116907caa7b88eb8 | subsection | 17 | 22 | Shock propagation in Aluminium | The shock velocity changes from 3.54 to
5.46 cm/\mu s at 1.5 ns when the incident radiation
temperature increases to 200 eV.The performance of the implicit and semi-implicit schemes are
compared by studying the convergence properties and the CPU cost
for the problem of shock wave propagation in aluminium. The
convergen... | {
"cite_spans": []
} | 0807.1842 | Convergence of Implicit Difference Scheme for 1D Lagrangian
Hydrodynamics coupled to Radiation Transport Equation | [
"Karabi Ghosh",
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df836f1bd4618a0047f6cb30e86704ec4695551c | subsection | 18 | 22 | Point explosion problem | The self similar problem of a strong point explosion was formulated
and solved by Sedov . The problem considers a perfect
gas with constant specific heats and density \rho _0 in which a
large amount of energy E is liberated at a point instantaneously.
The shock wave propagates through the gas starting from the point
wh... | {
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e2cac38f692306cfc5a0204247033092de761a2c | subsection | 19 | 22 | Point explosion problem | The dimensionless parameter
\xi _0, which depends on the specific heat ratio \gamma is
obtained from the condition of conservation of energy evaluated
with the solution obtained.Also, the distributions of velocity, pressure, density and
temperature behind the shock front are generated numerically
using the hydrodynamic... | {
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da4d833e581b39be532aaffa2de5087cb446d0e8 | subsection | 20 | 22 | Point explosion problem | In the implicit method, faster convergence is
attained at the cost of slightly higher CPU time as shown
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19b49932db0bbf0d0e51d66b2960c34ddec10d45 | subsection | 21 | 22 | Conclusions | In this paper we have developed and studied the performance of
fully implicit radiation hydrodynamics scheme as compared to the
semi-implicit scheme. The time dependent radiation transport
equation is solved and energy transfer to the medium is
accounted exactly without invoking approximation methods.
To validate the c... | {
"cite_spans": []
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1fa21e58038517c57edc0d48463a7f9e243abce5 | abstract | 0 | 32 | Abstract | Despite the importance of urban traffic flows, there are only a few
theoretical approaches to determine fundamental relationships between
macroscopic traffic variables such as the traffic density, the utilization, the
average velocity, and the travel time. In the past, empirical measurements have
primarily been describ... | {
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ca15d61904690909f4128e303327364650d786e8 | subsection | 1 | 32 | Introduction | The study of urban traffic flows has a long history (see Ref. for an overview).
For more than a decade now, physicists have contributed various interesting models, ranging from cellular automata , , to fluid-dynamic approaches , . Complementary, one should mention, for example, Refs. , as representatives of publication... | {
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e42b9286cecae17a4791658bb558ba6f48baacfc | subsection | 2 | 32 | Introduction | This is common in queueing theory and transportation planning, where formulas such as the capacity constraint function (REF ) are used.After discussing elementary relationships for cyclically signalized intersections of urban road networks in Sec. , we will start in Sec. with the discussion of undersaturated traffic co... | {
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dc1684f9bf8905802b80c789f10ce9b95c7989aa | subsection | 3 | 32 | Elementary Relationships for Cyclically Operated Intersections | Let us study a single intersection with a periodically operated traffic light. We shall have green phases j of duration \Delta T_j, during which one or several of the traffic streams i are served.
\beta _{ij} shall be 1, if traffic stream i is served by green phase j, otherwise \beta _{ij} = 0.
The setup time after pha... | {
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5424735a4d3c350f85113a604ecbd20476c5968b | subsection | 4 | 32 | Elementary Relationships for Cyclically Operated Intersections | (REF ) impliesf_i(u_i,\delta _i) = (1+\delta _i) \frac{A_i}{\widehat{Q}_i} = (1+\delta _i) u_i \, ,whereu_i = \frac{A_i}{\widehat{Q}_i}is called the utilization of the service or outflow capacity \widehat{Q}_i (see Fig. REF ). \delta _i \ge 0 is a safety factor to cope with variations in the arrival flow (inflow) A_i. ... | {
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d0e5766f3c59b69462f531858ffc23a07dd1836c | subsection | 5 | 32 | Elementary Relationships for Cyclically Operated Intersections | If Q_{\rm out} represents the characteristic outflow from congested traffic per lane into an area of free flow, the overall service capacity by all service lanes is given by the minimum of the number of lanes I_i used by vehicle stream i upstream the intersection, and the number I^{\prime }_i of lanes downstream of it:... | {
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e8b6acbeed64d0e6eda131dda7a668286b202c5e | subsection | 6 | 32 | Elementary Relationships for Cyclically Operated Intersections | Then, the next green phase for this traffic stream starts at time t^{\prime }_0 = t_0 + (1-f_i)T_{\rm cyc},
as f_iT_{\rm cyc} is the green time period and (1-f_i)T_{\rm cyc} amounts to the sum of the amber and red time periods. Due to O_i(t) \ge A_i(t) and O_i(t^{\prime }_0) > A_i(t^{\prime }_0), t^{\prime }_0 - t_0 = ... | {
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817d0da2f28dc946535718e72125b660382d06b6 | subsection | 7 | 32 | Elementary Relationships for Cyclically Operated Intersections | Considering formulas (REF ) and (REF ), the excess green time is\Delta T_i - T_i = f_i T_{\rm cyc} - \frac{u_i(1-f_i)T_{\rm cyc}}{1-u_i} = \frac{f_i - u_i}{1-u_i} T_{\rm cyc} \, .Hence, the percentage of delayed vehicles is\frac{A_i[T_{\rm cyc} - (\Delta T_i - T_i)]}{A_iT_{\rm cyc}} = 1 - \frac{f_i-u_i}{1-u_i} = \frac{... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
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1a9ac477b7c4107ffd9d2a2e0577d84208b6e02c | subsection | 8 | 32 | Elementary Relationships for Cyclically Operated Intersections | As the average number of delayed vehicles is
(\Delta N_i^{\rm max}+0)/2 and a fraction (1-f_i)/(1-u_i)\le 1 of vehicles is delayed, together with Eqs. (REF ) and (REF ) we find\Delta N_i^{\rm av}(u_i,\lbrace f_j\rbrace ) &=& \frac{(1-f_i)}{(1-u_i)} \frac{\Delta N_i^{\rm max}(u_i,\lbrace f_j\rbrace )}{2} \\
&=& u_i \wid... | {
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346c099acc5dd3eee9892ae70c63aff8fd9a4948 | subsection | 9 | 32 | Efficiency of Traffic Operation | In reality, the average delay time will depend on the time-dependence of the inflow A_i(t), and on how well the traffic light is coordinated with the arrival of vehicle platoons. In particular, this implies a dependence on the signal offsets. In the best case, the average delay is zero, but in the worst case, it may al... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
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3fc8539e541ab189cca4b2f6182cdaa83c9ceddc | subsection | 10 | 32 | Efficiency of Traffic Operation | Nevertheless, we will demonstrate the general dependence on the utilization u_i in the following.For this, we will study the case of excess green times (\delta _i > 0), which are usually chosen to cope with the stochasticity of vehicle arrivals, i.e. the fact that the number of vehicles arriving during one cycle time i... | {
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c815deadfea5d4ecaaf4448f645218c5cd8e41c6 | subsection | 11 | 32 | Fundamental Relationships for Undersaturated Traffic | The travel time is generally given by the sum of the free travel time {\cal T}_i^0 = L_i/V_i^0 and the average delay time {\cal T}_i^{\rm av}, where L_i denotes the length of the road section used by vehicle stream i and V_i^0 the free speed (or speed limit). With Eq. (REF ), we get{\cal T}_i(\lbrace u_j\rbrace ,\epsil... | {
"cite_spans": []
} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
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07ed22308e3070368ad7a2852181c9a5bb8a9c6e | subsection | 12 | 32 | Fundamental Relationships for Undersaturated Traffic | (REF ), we obtain the equation\rho _i^{\rm av}(\lbrace u_j\rbrace ,\epsilon _i,L_i) = \frac{u_i\widehat{Q}_i}{L_i} (1-\epsilon _i) \frac{(1-u_i) T_{\rm los}}{2(1-\sum _j u_j)} \, ,which can be numerically inverted to give the utilization u_i as a function of the scaled densities \rho _j^{\rm av}L_j/(1-\epsilon _j). The... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
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4289e94a1169aa1a08c2ccf3773fdc3c8fe42094 | subsection | 13 | 32 | Transition to Congested Traffic | The utilizations u_i increase proportionally to the arrival flows A_i, i.e. they go up during the rush hour. Eventually,\sum _j f_j = \sum _j (1+\delta _j) u_j \rightarrow 1 \, ,which means that the intersection capacity is reached. Sooner or later, there will be no excess capacities anymore, which implies \delta _i \r... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
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3e6274df94a77813e4d0f875690a98e55bd999f5 | subsection | 14 | 32 | Transition to Congested Traffic | Therefore, the related maximum delay time is (1-u_i)T_{\rm cyc} (N_i^{\rm max} - 1)/N_i^{\rm max}, which reduces the average delay time {\cal T}_i^{\rm av} by (1-u_i)T_{\rm cyc} /(2N_i^{\rm max}).\Delta N_i^{\rm av} = \frac{\Delta N_i^{\rm max}}{2} \, .As a consequence, we have{\cal T}_i(\lbrace u_j\rbrace ) = \frac{L_... | {
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bdd779c4baa48ce8dee131355b0d918b5b7ddb13 | subsection | 15 | 32 | Transition to Congested Traffic | For this reason, the cycle time is limited to a finite valueT_{\rm cyc}^{\rm max}(\lbrace u_j^0\rbrace ) = \frac{T_{\rm los}}{1-\sum _j u_j^0} \, ,where typically u_j^0 \le u_j.
This implies that the sum of utilizations must fulfill\sum _j u_j \le \sum _j u_j^0 = 1 - \frac{T_{\rm los}}{T_{\rm cyc}^{\rm max}} \, .As soo... | {
"cite_spans": []
} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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81547765cfbbad89a7f2e9386ed6a58fc1b45f7e | subsection | 16 | 32 | Fundamental Relationships for Congested Traffic Conditions | In the congested regime, the number of delayed vehicles does not reach zero anymore, and platoons cannot be served without delay. Vehicles will usually have to wait several cycle times until they can finally pass the traffic light. This increases the average delay time enormously. It also implies that there are no exce... | {
"cite_spans": []
} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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c4ad9497cd78ab68f8dc3b143fa5ff3eaf57f253 | subsection | 17 | 32 | Fundamental Relationships for Congested Traffic Conditions | (REF ), the maximum number of delayed vehicles is\Delta N_i^{\rm max}(u_i,k) = \Delta N_i^{\rm min} (u_i,k) + u_i (1-u_i^0)\widehat{Q}_iT_{\rm cyc}^{\rm max} \, .Because of \Delta N_i^{\rm av} = (\Delta N_i^{\rm min} +\Delta N_i^{\rm max} )/2 and A_i = u_i \widehat{Q}_i, the average number of delayed vehicles is\Delta ... | {
"cite_spans": []
} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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6fd5464339dc9b10daa7c9e43e6f45e7ed00435c | subsection | 18 | 32 | Fundamental Relationships for Congested Traffic Conditions | The quotient of the number kA_iT_{\rm cyc}^{\rm max} of vehicles arriving in k cycles and the
number \widehat{Q}_i u_i^0T_{\rm cyc}^{\rm max} of vehicles served during one green time period, when rounded down, corresponds to the number n_{\rm s} of additional stops needed by newly arriving vehicles. Therefore, with A_i... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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08a36d20bf7a754ef69cc503a6e80e18f15ca4d7 | subsection | 19 | 32 | Fundamental Relationships for Congested Traffic Conditions | In summary, the delay time of a newly arriving vehicle at time t (when averaging over the triangular part for the sake of simplicity), is{\cal T}_i^{\rm av}(u_i,t) &=& (1-u_i^0)\frac{T_{\rm cyc}^{\rm max}}{2} + \left\lfloor \frac{u_i(t-t_{i0})}{u_i^0T_{\rm cyc}^{\rm max}} \right\rfloor (1-u_i^0)T_{\rm cyc}^{\rm max} \\... | {
"cite_spans": []
} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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193a33836da180cf89a112ff65fc4ea55411b452 | subsection | 20 | 32 | Fundamental Relationships for Congested Traffic Conditions | In this way, we obtaink+\frac{1}{2} = \frac{\Delta N_i^{\rm av}(u_i,\lbrace u_j^0\rbrace ,k)}{(u_i-u_i^0)\widehat{Q}_iT_{\rm cyc}^{\rm max}(\lbrace u_j^0\rbrace )} - \frac{u_i^0(1-u_i)}{2(u_i - u_i^0)} \, .Therefore, while in Sec. we could express the average travel time and the average velocity either in dependence of... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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cd33735259dd58aecb42147bc20e027f8abc29d7 | subsection | 21 | 32 | Fundamental Relationships for Oversaturated Traffic Conditions | We have seen that, under congested conditions, the number of delayed vehicles is growing on average. Hence, the vehicle queue will eventually fill the road section reserved for vehicle stream i completely. Its maximum storage capacity per lane for delayed vehicles is\Delta N_i^{\rm jam}(L_i) = L_i \rho _i^{\rm jam} \, ... | {
"cite_spans": []
} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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f686e7a18dbf72aafdd92a46bd53678dbcce93f6 | subsection | 22 | 32 | Fundamental Relationships for Oversaturated Traffic Conditions | Therefore, it would make sense to reduce the cycle time to a value T_{\rm cyc} < T_{\rm cyc}^{\rm max}
in the oversaturated regime.Note, however, that the travel times on the road section reserved for stream i are not growing anymore, because the road section is limited to a length L_i. This allows us to determine the ... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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04e19011f36ace33d863fb19f2ca581b42963c72 | subsection | 23 | 32 | Transition from Oversaturated to Undersaturated Traffic Conditions | If the arrival flow A_i after the rush hour drops below the value of \sigma _i u_i^0 T_{\rm cyc}^{\rm max}, the vehicle queue will eventually shrink, and the road section used by vehicle stream i enters from the oversaturated into the congested regime. The formulas for the evolution of the number of delayed vehicles ar... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
"Dirk Helbing"
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5eb3252579b1edbbf5c77302b6458efe37437a1e | subsection | 24 | 32 | Summary and Outlook | Based on a few elementary assumptions, we were able to derive fundamental relationships for the average travel time {\cal T}_i^{\rm av} and average velocity V_i^{\rm av}. These relationships are functions of the utilization u_i of the service capacity of a cyclically signalized intersection and/or the average number \D... | {
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} | 10.1140/epjb/e2009-00093-7 | 0807.1843 | Derivation of a Fundamental Diagram for Urban Traffic Flow | [
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25672b20abc81d607a183335f7d48cb68581215d | subsection | 25 | 32 | Summary and Outlook | Therefore, the average travel time does not only depend on the utilization u_i, but also on the average vehicle queue \Delta N_i^{\rm av} (or the average density \rho _i^{\rm av}). Although the traffic light control can still improve the average travel times by synchronizing with the arrival of vehicles, the related ef... | {
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c1b8a7d406928e3721d96153378fab709238b014 | subsection | 26 | 32 | Transferring the Link-Based Urban Fundamental Diagrams to an Area-Based One | We may finally ask ourselves, whether the above formulas would also allow one to make predictions about the average travel times and speeds for a whole area of an urban traffic network, rather than for single road sections (“links”) only. This would correspond to averaging over the link-based fundamental diagrams of th... | {
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87ad48dbc67b965237fbc44b357cd65f61a9da1a | subsection | 27 | 32 | Transferring the Link-Based Urban Fundamental Diagrams to an Area-Based One | However, as the link-based fundamental diagram between the flow Q(\rho ) = \rho V^{\rm av}(\rho ) and the density \rho is convex, evaluating the flow at some average density overestimates the average flow.When averaging over speed values, they have to be weighted by the number of vehicles concerned, i.e. by the density... | {
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6ceeb0aebb7b8aaab1a943fd961d9efd4f350260 | subsection | 28 | 32 | Transferring the Link-Based Urban Fundamental Diagrams to an Area-Based One | 6, see Fig. 7 in Ref. . He extracted these from original data of GPS-equipped taxis by Prof. Masao Kuwahara from the University of Tokyo. The fit of the theoretically predicted relationship to the empirical data was carried out by Anders Johansson. Furthermore, the author is grateful for partial support by the Daimler-... | {
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6544bccb484d5391cc2f5fb736ab6c0f4e539b55 | subsection | 29 | 32 | Determination of Average Travel Times and Velocities | Let f(x) be a function and w(x) a weight function. Then, the average of the function between x=x_0 and x=x_1 is defined as\frac{\int \limits _{x_0}^{x_1} dx^{\prime } \; w(x^{\prime })f(x^{\prime })}{ \int \limits _{x_0}^{x_1} dx^{\prime } \; w(x^{\prime })} \, .In case of uniform arrivals of vehicles, we have a functi... | {
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809f04f94427346cb9baea63160d48cc287eee00 | subsection | 30 | 32 | Determination of Average Travel Times and Velocities | This formula corrects the naive formulaV_i^{\rm av} \approx \frac{L_i}{{\cal T}_i} = \frac{L_i}{{\cal T}_i^0 + \frac{\Delta N_1^{\rm max}}{2u_i\widehat{Q}_i}}\approx \frac{L_i}{{\cal T}_i^0}\left( 1 - \frac{\Delta N_1^{\rm max}}{2u_i\widehat{Q}_i{\cal T}_i^0}\right)\, ,where we have used 1/(1+x) \approx 1 - x. Therefor... | {
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30ceba4693bba17c7a1f6b5e5413dbb238efff02 | subsection | 31 | 32 | Determination of Average Travel Times and Velocities | (REF ) and the percentage of delayed vehicles, which is given by Eq. (REF ). The same result follows fromV_i^{\rm av} = \frac{L_i}{{\cal T}_i^0 + {\cal T}_i^{\rm av}} \approx V_i^0 \left( 1 - \frac{{\cal T}_i^{\rm av}}{{\cal T}_i^0} \right)together with Eq. (REF ). | {
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1935ba8832249daf85117ea06bfa247d3264bc3e | abstract | 0 | 9 | Abstract | We generalize the one-to one correspondence between quasi normal modes in 3-
dimensional anti deSitter black holes and the poles of the retarded correlators
in the boundary conformal field theory to include logarithmic operators in the
latter. This analysis is carried out explicitly for the logarithmic mode in
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6720497665f9552b794d5ccea621c93fb715b3bc | subsection | 1 | 9 | Introduction | It was observed in that there is a one to one correspondence between the quasi-normal frequencies of linear perturbations in a 3-dimensional BTZ black hole background and the poles, in momentum space, of the retarded propagator of the respective dual operators in the boundary conformal field theory. This correspondence... | {
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5f4b52a99966770ab57f909cd5621e0a1c6942cb | subsection | 2 | 9 | Algebraic structure | There is a simple algebraic structure relating the massive graviton solution for generic mass m>1 to the logarithmic solution at m=1. To describe it we consider the equation for motion for tensor linear perturbations h_{\mu \nu } in the transverse trace-less gauge(\nabla ^2+2)\left[\epsilon _\mu ^{\ \alpha \beta }\nabl... | {
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bbd1f758d330aaf4aeb2ba1bd690829b3a67937e | subsection | 3 | 9 | Logarithmic quasi-normal modes | It was shown in that in topologically massive gravity the quasi-normal modes for massive gravitons in the BTZ black hole background with metric (u=\tau +\phi , v=\tau -\phi )ds^2 = \frac{1}{4}\left(du^2-2\cosh (2\rho )dudv+dv^2\right)+d\rho ^2\;,are descendents of a "chiral highest weight" solution, h(m)_{\mu \nu }, to... | {
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928a1de15771c67ab090e2bbebc9306a1b998be6 | subsection | 4 | 9 | Logarithmic quasi-normal modes | Furthermore, since L_k and \bar{L}_k commute with the equation of motion ,\tilde{h}^{(n)}_{\mu \nu }=(\bar{L}_1 L_{-1})^{(n)}\tilde{h}_{\mu \nu }satisfies the third order equation (REF ).
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2677549a744ac54cff2519b10c55766443bc87e6 | subsection | 5 | 9 | Logarithmic quasi-normal modes | In view of a conformal field theory interpretation of \tilde{h}^{(1)} we should note that the vv-component of the metric is not dominant at large \rho which in turn leads to difficulties in identifying the dual operator in the CFT. The curvature perturbation induced by \tilde{h}^{(1)} is then obtained using (REF ) as\d... | {
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384349da1bfd909087d854e2d05c3b6f9efbc8c8 | subsection | 6 | 9 | Relation to Logarithmic CFT | The simplest version of a logarithmic conformal field theory (see e.g. for a review), which is sufficient for our purpose arises
in the presence of two operators C and D with degenerate eigenvalue of L_0 such thatL_0 |C>=h|C>\;,\qquad L_0|D>=h|D>+|C>\;.The 2-point functions of these operators are then given by<C(x) C(0... | {
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14391cffa901a8c5bc978006fd63d08e2bde950c | subsection | 7 | 9 | Relation to Logarithmic CFT | We then conclude that the momentum space representation of G^{DD}_R(t,\sigma ) has double poles while that of G^{CD}_R(t,\sigma ) has simple poles at the same location.We will argue below that it is precisely these double poles that are responsible for the linear time dependence of the corresponding quasi-normal mode. ... | {
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10093927f3a3ab9200574cc6bb7c9d7f330bddf7 | subsection | 8 | 9 | Acknowledgments: | This was work supported in parts by the Transregio TRR 33 `The Dark
Universe', the Excellence Cluster `Origin and Structure of the
Universe' of the DFG as well as the DFG grant Ma 2322/3-1. I would like to thank D. Grumiller for helpful correspondence and Sergey Solodukhin for helpful comments and for pointing out some... | {
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} | 10.1088/1126-6708/2008/09/073 | 0807.1844 | Quasi-Normal Modes for Logarithmic Conformal Field Theory | [
"Ivo Sachs"
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855dbb76d8387bb57e4bfc0900bb9d2d423d8eca | abstract | 0 | 7 | Abstract | The improvement in observational facilities requires refining the modelling
of the geometrical structures of astrophysical objects. Nevertheless, for
complex problems such as line overlap in molecules showing hyperfine structure,
a detailed analysis still requires a large amount of computing time and thus,
misinterpret... | {
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3a61eacd1130c28e4b2120578613bb11abd6955b | subsection | 1 | 7 | Introduction | In the nucleated instability (also called core
instability) hypothesis of giant planet
formation, a critical mass for static core envelope
protoplanets has been found. Mizuno () determined
the critical mass of the core to be about 12 \,M_\oplus
(M_\oplus =5.975 \times 10^{27}\,\mathrm {g} is the Earth mass), which
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5e247d2063d59a5fdb713f947a6611b9ae58ea0e | subsection | 2 | 7 | Baker's standard one-zone model | In this section the one-zone model of Baker (),
originally used to study the Cepheïd pulsation mechanism, will
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89b44fcc6fc79eac1891d5c923e3d8521e17cc6d | subsection | 3 | 7 | Baker's standard one-zone model | Substituting into Baker's criteria, using
thermodynamic identities and definitions of thermodynamic quantities,\Gamma _1 = \left( \frac{ \partial \ln P}{ \partial \ln \rho }
\right)_{S} \, , \;
\chi ^{}_\rho = \left( \frac{ \partial \ln P}{ \partial \ln \rho }
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f1edf2d5706c6cc597c3a943fbd94f98c90f71e0 | subsection | 4 | 7 | Baker's standard one-zone model | Once the microphysics, i.e. the thermodynamics
and opacities (see Table REF ), are specified (in practice
by specifying a chemical composition) the one-zone stability can
be inferred if the thermodynamic state is specified.
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4d4df58178a5e946a559dc3a84eb3bf48312dd16 | subsection | 5 | 7 | Baker's standard one-zone model | Regions of secular instability are
listed in Table 1.
[Figure: Vibrational stability equation of stateS_{\mathrm {vib}}(\lg e, \lg \rho ).>0 means vibrational stability.] | {
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8255467d5bfab3d708e541cb715c19c920dfa5d1 | subsection | 6 | 7 | Conclusions | The conditions for the stability of static, radiative
layers in gas spheres, as described by Baker's ()
standard one-zone model, can be expressed as stability
equations of state. These stability equations of state depend
only on the local thermodynamic state of the layer.
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4593ef31a5bf71ad0caa2e64eb2f78aaf4ff8dd3 | abstract | 0 | 54 | Abstract | This paper deals with the problem of existence and uniqueness of a solution
for a backward stochastic differential equation (BSDE for short) with one
reflecting barrier in the case when the terminal value, the generator and the
obstacle process are Lp-integrable with p in ]1,2[. To construct the solution
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2235acea496a78551651c0988552a64159eb4146 | subsection | 1 | 54 | Introduction | The notion of nonlinear backward stochastic differential equation (BSDE for short) was introduced by Pardoux and Peng (1990, ). A solution of this equation, associated with a terminal value \xi and a generator or coefficient f(t,\omega ,y,z), is a couple of adapted stochastic processes (Y_t,Z_t)_{t\le T} such that:\for... | {
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31e0983ec31a4a5989ea4367336a155b1bdcb919 | subsection | 2 | 54 | Introduction | Actually there have been relatively few papers which deal with the problem of existence/uniqueness of the solution for BSDEs in the case when the coefficients are not square integrable. Nevertheless we should point out that El-Karoui et al. (1997, ) and Briand et al. (2003, ) have proved existence and uniqueness of a s... | {
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726ed34ba2f5345d3d5bc3fd3769a94c6f07d4e8 | subsection | 3 | 54 | Notations, setting of the problem and preliminary results | Let (\Omega ,\mathcal {F},P) be a fixed probability space on which is defined a standard d-dimensional Brownian motion B=(B_{t})_{t\le T} whose natural filtration is ({\cal F}_{t}^{0}:=\sigma \lbrace B_{s},s\le t\rbrace )_{t\le T}. We denote by ({\cal F}_{t})_{t\le T} the completed filtration of ({\cal F}_{t}^{0})_{t\l... | {
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59b347dd019470db3b0a40667d78943fa02d242f | subsection | 4 | 54 | Notations, setting of the problem and preliminary results | Let \xi be an an \mathbb {R}-valued and \mathcal {F}_T-measurable random variable and let us consider a random function f :[0,T] \times \Omega \times \mathbb {R}\times \mathbb {R}^d \rightarrow \mathbb {R} measurable with respect to {\cal P} \times \mathcal {B}(\mathbb {R})\times \mathcal {B}(\mathbb {R}^d) where \cal ... | {
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0.0368347167... | |
a6ead226d4e1fd11ff4eeb35c013730fbaaa8e06 | subsection | 5 | 54 | Notations, setting of the problem and preliminary results | K_0=0 and K_T \in L^p(\Omega );
\displaystyle Y_t = \xi + \int _t^T f(s,Y_s,Z_s) ds + K_T - K_t - \int _t^T Z_s dB_s, 0 \le t \le T a.s.;
Y_t \ge L_t, 0 \le t \le T;
\displaystyle \int _0^T (Y_s - L_s) dK_s =0, P-a.s..The following corollary whose proof is given in will be used several times later, therefore for the... | {
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"raw": "Briand, Ph., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L. (2003). L^{p} solutions of backward stochastic differential equations, Stochastic Process. Appl., 108, 109–129.",
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"Said Hamadene",
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ab6773fe6b76f70657553f28d00dc90689a92497 | subsection | 6 | 54 | Notations, setting of the problem and preliminary results | There exists a real constant C_{p,\kappa } depending only on p and \kappa such that:\mathbb {E}\left[ \left( \int _0^T |Z_s|^2 ds \right)^{p/2} \right] \le C_{p,\kappa } \mathbb {E}\left[ \sup _{t \in [0,T]} |Y_t|^p + \left(\int _0^T |f(s,0,0)| ds \right)^p \right].Proof. Let \alpha be a real constant and for each inte... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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c71b73d58feb670975c31577402864f23c4c96f0 | subsection | 7 | 54 | Notations, setting of the problem and preliminary results | Next using Itô's formula yields:&& |Y_0|^2 + \int _0^{\tau _k} e^{\alpha s}|Z_s|^2 ds = e^{\alpha \tau _k}|Y_{\tau _k}|^2 + \int _0^{\tau _k} e^{\alpha s}Y_s (2f(s,Y_s,Z_s)-\alpha Y_s) ds \\
&& \qquad + 2 \int _0^{\tau _k} e^{\alpha s} Y_s dK_s - 2 \int _0^{\tau _k} e^{\alpha s} Y_s Z_s dB_s \\
&& \le e^{\alpha \tau _k... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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2c313a1eebfa4a9bcdfd9bfe0e48488ff23fdd91 | subsection | 8 | 54 | Notations, setting of the problem and preliminary results | Therefore&&|Y_0|^2 + (1-\varepsilon \kappa ) \int _0^{\tau _k} e^{\alpha s}|Z_s|^2 ds \le (e^{\alpha \tau _k}|Y_{\tau _k}|^2 +(1+\frac{1}{\varepsilon }) \sup _{s\le \tau _k}e^{2\alpha s}|Y_s|^2) \\
&& \qquad +\left( \int _0^{\tau _k} |f(s,0,0)| ds \right)^2 + (2\kappa +\kappa \varepsilon ^{-1}-\alpha ) \int _0^{\tau _k... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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6eb0975768a13802a4e797bfa22008cbbe9194d6 | subsection | 9 | 54 | Notations, setting of the problem and preliminary results | \\
& & \left. + \int _0^{\tau _k} |Z_s|^2 ds + \left| \int _0^{\tau _k} Z_s dB_s \right|^2 \right).Plugging this last inequality in the previous one to get:&&(1-\varepsilon C_\kappa )|Y_0|^2 + (1-\varepsilon \kappa )\int _0^{\tau _k} e^{\alpha s}|Z_s|^2 ds -\varepsilon C_\kappa \int _0^{\tau _k} |Z_s|^2 ds \\
&& \qquad... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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fdc92e0951af21dee8aeee6ebdeadff41ceaa0c4 | subsection | 10 | 54 | Notations, setting of the problem and preliminary results | \squareWe will now establish an estimate for the process Y. Actually we have:Lemma 2
We keep the notations of Lemma REF and we assume moreover that P-a.s. \int _0^T (Y_s-K_s)^+dK_s = 0. Then there exists a constant C_{p,\kappa } such that:\mathbb {E}\sup _{t \in [0,T]} |Y_t|^p \le C_{\kappa ,p} \left[ \mathbb {E}|\xi ... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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11a030b091644635329f65da05475fc51f10c4b3 | subsection | 11 | 54 | Notations, setting of the problem and preliminary results | But since f is Lipschitz then we have:&& e^{\alpha p t} | Y_t|^p + c(p) \int _t^u e^{\alpha ps } |Y_s|^{p-2} \mathbf {1}_{Y_s \ne 0} |Z_s|^2 ds \\
&& \quad \le e^{\alpha p u} |Y_u|^p + p(\kappa - \alpha ) \int _t^u e^{\alpha p s} |Y_s|^p ds + p \int _t^u e^{\alpha p s} |Y_s|^{p-1} |f(s,0,0)| ds \\
&& \qquad + p\kappa \... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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b4fd79470a8f2648cc33b820a23a85f15e455a01 | subsection | 12 | 54 | Notations, setting of the problem and preliminary results | Then plug the two last inequalities in the previous ones to obtain:&& e^{\alpha p t} | Y_t|^p + \frac{c(p)}{2} \int _t^u e^{\alpha ps } |Y_s|^{p-2} \mathbf {1}_{Y_s \ne 0} |Z_s|^2 ds \\
&& \quad \le e^{\alpha p u} |Y_u|^p + (p-1)\gamma ^{\frac{p}{p-1}}(\sup _{t\le s\le u}|Y_s|^{p})+ \gamma ^{-p} \left( \int _t^ue^{\alp... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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6087eb5dc3560e44f2414a6d5d394a8a0caf1ed4 | subsection | 13 | 54 | Notations, setting of the problem and preliminary results | It follows that:&&\int _t^u e^{\alpha p s} |Y_s|^{p-1} \mbox{sgn}(Y_s) dK_s \le \int _t^u e^{\alpha p s} |L_s|^{p-1} \mbox{sgn}(L_s) dK_s\\
&& \quad \le \int _t^u e^{\alpha p s} (L_s^+)^{p-1} dK_s \le \left( \sup _{t \in [0,T]} (L_s)^+ \right)^{p-1} \int _t^u e^{\alpha p s} dK_s \\
&& \quad \le \frac{(p-1)}{p} \frac{1}... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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288b44f804c683be7ed1c84788b9a966eee087d1 | subsection | 14 | 54 | Notations, setting of the problem and preliminary results | So using the predictable dual projection property (see e.g. ) we have: \forall t\le T,\mathbb {E}\left[ (K_T-K_t)^p \right] & = & \mathbb {E}\left[ \int _t^T p(K_T-K_s)^{p-1} dK_s \right] = p \mathbb {E}\int _t^T \mathbb {E}\left[ (K_T-K_s)^{p-1} | \mathcal {F}_s \right] dK_s \\
& \le & p \mathbb {E}\int _t^T \left[ \m... | {
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"raw": "Dellacherie, C. and Meyer, P.A. (1980). Probabilités et Potentiel V-VIII, Hermann, Paris.",
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} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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e34eb9300fdc776d3fdbb069ada38767d430a574 | subsection | 15 | 54 | Notations, setting of the problem and preliminary results | Thus using now Doob's maximal inequality to obtain:\frac{1}{2}\mathbb {E}\left[ (K_T-K_t)^p \right] & \le & C_p \sup _{s \in [t,T]} \mathbb {E}\left[ \mathbb {E}\left( 2\sup _{u \in [t,T]} |Y_u| + \int _t^T |f(u,Y_u,Z_u)| du \bigg | \mathcal {F}_s \right) \right]^{p} \\
& \le & \tilde{C}_p \mathbb {E}\left[ \sup _{u \i... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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63da40f48f336171fdd3701ee06f63bc34392976 | subsection | 16 | 54 | Notations, setting of the problem and preliminary results | Now the local martingale (\int _0^t e^{\alpha p s} |Y_t|^{p-1} \mbox{sgn}(Y_s) Z_s dB_s)_{t\le T} is actually a martingale, therefore taking expectation in (REF ) and taking into account of (REF ) to obtain:&& \frac{c(p)}{2}\mathbb {E}\int _t^T e^{\alpha ps } |Y_s|^{p-2} \mathbf {1}_{Y_s \ne 0} |Z_s|^2 ds \le e^{\alpha... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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