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* [48] D. Petz. _An Invitation to the Algebra of Canonical Commutation Relations_. Leuven University Press, Leuven, 1990.
* [49] I. Segal. Tensor algebras over Hilbert spaces. _I. Trans. Am. Math. Soc._, 81:106-134, 1956.
* [50] M. Takesaki. Conditional expectations in von Neumann algebras. _J. Funct. Anal._, 9:306-321, 1971.
* [51] R. Van Handel and H. Mabuchi. Quantum projection filter for a highly nonlinear model in cavity QED. to appear: J. Opt. B, arXiv:quant-ph/0503222, California Institute of Technology, 2005.
* [52] R. Van Handel, J. K. Stockton, and H. Mabuchi. Feedback control of quantum state reduction. _IEEE Transactions on Automatic Control_, 50:768-780, 2005.
* [53] R. Van Handel, J. K. Stockton, and H. Mabuchi. Modelling and feedback control design for quantum state preparation. to appear J. Opt. B, California Institute of Technology, 2005.
* [54] D. Williams. _Probability with Martingales_. Cambridge University Press, Cambridge, 1991.
* [55] H. M. Wiseman and G. J. Milburn. Quantum theory of field-quadrature measurements. _Phys. Rev. A_, 47:642-662, 1993.
* [56] M. Zakai. On the optimal filtering of diffusion processes. _Z. Wahrsch. th. verw. Geb._, 11:230-243, 1969. | [] |
|
* [31] Yuan Rong, The existence of almost periodic solutions of retarded differential equations with piecewise argument, _Nonlinear analysis, Theory, methods and Applications,_**48** 1013-1032 (2002).
* [32] Yuan Rong, On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument, _J. Math. Anal. Appl.,_**303** 103-118 (2005).
**e-mail:** [email protected] | [] |
|
cosmological tests pertaining neoclassical test, luminosity distance, angular diameter distance and look back time for the model are discussed in Sections \(5-8\). As a concluding part some discussions are made in Section \(9\).
## 2 The Metric and Field Equations
We consider the Kaluza-Klein type Robertson Walker (RW) space time
\[ds^{2}=dt^{2}-a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2} \theta d\phi^{2})+(1-kr^{2})d\psi^{2}\right],\] (1)
where \(a(t)\) is the scale factor, \(k=0,\;\pm 1\) is the curvature parameter.
The usual energy-momentum tensor is modified by addition of a term
\[T^{vac}_{ij}=-\Lambda(t)g_{ij},\] (2)
where \(\Lambda(t)\) is the cosmological term and \(g_{ij}\) is the metric tensor.
Einstein's field equations (in gravitational units \(c=1,\;\;G=1\)) read as
\[R_{ij}-\frac{1}{2}Rg_{ij}=-8\pi T_{ij}-\Lambda(t)g_{ij}.\] (3)
The energy-momentum tensor \(T_{ij}\) in the presence of a perfect fluid has the form
\[T_{ij}=(p+\rho)u_{i}u_{j}-pg_{ij},\] (4)
where \(p\) and \(\rho\) are, respectively, the energy and pressure of the cosmic fluid, and \(u_{i}\) is the fluid five-velocity such that \(u^{i}u_{i}=1\).
The Einstein filed Eqs. (3) and (4) for the metric (1) take the form
\[6\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right)=8\pi\rho+\Lambda(t),\] (5)
\[\frac{3\ddot{a}}{a}+3\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right)=-8 \pi p+\Lambda(t).\] (6)
An over dot indicates a derivative with respect to time \(t\). The energy conservation law can be written as
\[\dot{\rho}+4(\rho+p)H=-\frac{\dot{\Lambda}}{8\pi}.\] (7)
where \(H=\frac{\dot{a}}{a}\), is the Hubble parameter.
For complete determinacy of the system, we consider the barotropic equation of state
\[p=\omega\rho,\] (8) | [] |
|
* [12] Pitjeva, E.V., High-Precision Ephemrides of Planets-EPM and Determinations of Some Astronomical Constants, _Sol. Sys. Res._, **39**, 176-186, 2005a.
* [13] Pitjeva, E.V., Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft, _Astron. Lett._, **31**, 340-349, 2005b.
* [14] Standish, E.M., The Astronomical Unit now, in: Kurtz, D.W. (ed.), _Transits of Venus: New Views of the Solar System and Galaxy, Proceedings IAU Colloquium No. 196_, (Cambridge University Press, Cambridge), pp. 163-179, 2005. | [] |
|
# On the Role of Hadamard Gates
in Quantum Circuits
D. J. Shepherd1
_University of Bristol, Department of Computer Science_
Footnote 1: [email protected], [email protected]
(March 23, 2006)
###### Abstract
We study a reduced quantum circuit computation paradigm in which the only allowable gates either permute the computational basis states or else apply a "global Hadamard operation", _i.e._ apply a Hadamard operation to every qubit simultaneously. In this model, we discuss complexity bounds (lower-bounding the number of global Hadamard operations) for common quantum algorithms : we illustrate upper bounds for Shor's Algorithm, and prove lower bounds for Grover's Algorithm. We also use our formalism to display a gate that is neither quantum-universal nor classically simulable, on the assumption that Integer Factoring is not in **BPP**.
## 1 Introduction
A Quantum Circuit (or Quantum Logic Network) is usually presented as being composed both of _wires_ that carry qubits and _gates_ that tap those wires to modify the qubits they carry, [5]. In section 2 we specify the notation used to describe computation with quantum circuits, and specify exactly which features we shall be allowing within the kinds of circuits we wish to consider.
The main focus is to enquire about the difference it makes if we limit to using 'classical' gates (ones which preserve the set of computational basis states) and 'global Hadamard transforms' (where a Hadamard gate is applied once to each qubit.) We will show in section 3 that our imposed limitations do not limit computational power in any real sense, and in subsequent sections will discuss what algorithms and algorithm-primitives tend to | [] |
|
## References
* [1] M.F. Atiyah: _"K-Theory"_, Benjamin 1967. D.G. Quillen: _"Higher Algebraic K-Theory"_, Graduate Lecture Course, Oxford, Michaelmas 1994 (personal notes).
* [2]M.F. Atiyah: _Collected Works (Vol 5: Gauge Theory)_, Oxford University Press (1990). M.F. Atiyah: _"The Geometry and Physics of Knots"_, Accademia Nazionale dei Lincei, Cambridge University Press, 1990.
* [3] A. Connes: _"Noncommutative Geometry"_, Academic Press, (1994).
* [4] D. J. Saunders: _The Geometry of Jet Bundles_, London Mathematical Society Lecture Note Series 142, Cambridge University Press, (1989).
* [5] S. Majid: _"Riemannian geometry of quantum groups and finite groups with nonuniversal differentials"_, Commun. Math. Phys. 225 (2002) 131-170.
* [6] A. Connes and D. Kreimer: _"Hopf Algebras, Renormalisation and Noncommutative Geometry"_, Commun. Math. Phys. 199, (1998), 203-242. A. Connes, D. Kreimer: _"Renormalization in quantum field theoy and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem"_, Commun. Math. Phys. 210.1 (2000), 249-273. A. Connes, D. Kreimer: _"Renormalization in quantum field theoy and the Riemann-Hilbert problem. II. The \(\beta\)-function, diffeomorphisms and the renormalisation group"_, Commun. Math. Phys. 216.1 (2001), 215-241.
* | [] |
|
rature must be more than 215 MeV because the thermal photon spectra is a superposition of emission rates for all the temperatures from initial to freeze-out. This indicates that the temperature of the system formed after the collisions is higher than the transition temperature for deconfinement. The lower limit in the initial temperature can be used to put a conservative upper bound on the thermalization time, which in the present case is \(\sim 1.3\) fm. The invariant mass distribution of lepton pairs in the similar framework will be reported shortly [71].
In spite of the encouraging situation described above it is worthwhile to mention the following. The experimental data [8] for real photon spectrum in \(Au+Au\) collision has been obtained from the analysis of the low mass and high \(p_{T}\) lepton pairs by assuming
Figure 6: Photon emission from QGP phase for two values of strong coupling constant \(\alpha_{s}\). Solid (dotted) line indicates results for \(\alpha_{s}=0.3\) (temperature dependent coupling). Here \(T_{i}=400\) MeV, \(\tau_{i}=0.2\) fm and \(T_{f}=120\) MeV. Type I EOS has been used here. | [
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* [24] McClintock, J. E. & Remillard, R. A. Black Hole Binaries. _Compact Stellar X-Ray Sources_, Eds. M. van der Klis & W. Lewin, Cambridge University Press, Cambridge (in the press)
* [25] Fender, R. P., Pooley, G. G., Durouchoux, P., Tilanus, R. P. J. & Brocksopp, C. The very flat radio-millimetre spectrum of Cygnus X-1. _Mon. Not. Roy. Astron. Soc._**312**, 853-858 (2000)
* [26] Di Salvo, T., Done, C., Zycki, P. T., Burderi, L. & Robba, N. R. Probing the Inner Region of Cygnus X-1 in the Low/Hard State through Its X-Ray Broadband Spectrum. _Astrophys. J._**547**, 1024-1033 (2001)
* [27] Gallo, E., Fender, R. P. & Pooley, G. G. A universal radio/X-ray correlation in low/hard state black hole binaries. _Mon. Not. Roy. Astron. Soc._**344**, 60-72 (2003)
* [28] Livio, M., Pringle, J. E. & King, A. R. The Disk-Jet Connection in Microquasars and AGN. _Astrophys. J._**593**, 184-188 (2003)
* [29] Malzac, J., Merloni, A. & Fabian, A. C., Jet-disc coupling through a common energy reservoir in the black hole XTE J1118+480. _Mon. Not. Roy. Astron. Soc._**351**, 253-264 (2004)
* [30] Fender, R. P., Gallo, E. & Jonker, P. G. Jet-dominated states: an alternative to advection across black hole event horizons in 'quiescent' X-ray binaries. _Mon. Not. Roy. Astron. Soc._**343**, L99-L103 (2003)
Correspondence and requests for materials should be addressed to E.G. ([email protected])
We are grateful to Dimitris Mislis and Romano Corradi for the H\(\alpha\) observation presented in this | [] |
|
* (25) E. L. Lehmann, _Nonparametrics: Statistical methods based on Ranks_, San Francisco: Holden-Day Inc (1975).
* (26) G. S. Watson, _Statistics on Spheres_, New York, Wiley (1983). | [] |
|
* [15] J. M. Link _et al._ [FOCUS Collaboration], Phys. Lett. B **535**, 43 (2002) [arXiv:hep-ex/0203031].
* [16] J. G. Korner and G. A. Schuler, Z. Phys. C **46**, 93 (1990).
* [17] J. M. Link _et al._ [FOCUS Collaboration], Phys. Lett. B **544**, 89 (2002) [arXiv:hep-ex/0207049]; Phys. Lett. B **607**, 67 (2005) [arXiv:hep-ex/0410067].
* [18] L. Rosselet _et al._, Phys. Rev. D **15**, 574 (1977).
* [19] F. A. Berends, A. Donnachie and G. C. Oades, Phys. Rev. **171**, 1457 (1968).
* [20] L. Edera and M. R. Pennington, arXiv:hep-ph/0506117. | [] |
|
Given a vector field \(X\) we can lift it to a section of \(E\) in two ways: \(X^{+}\) a section of \(V\) and \(X^{-}\) a section of \(V^{\perp}\). We then have the following
**Theorem 2**: _Let \(X,Y\) be two vector fields, and \(V\subset E\) a generalized metric. Let \(g\) be the metric on \(M\) and \(H\) the curvature of the gerbe as defined in Proposition 1. Then, using the Courant bracket on sections of \(E\),_
\[[X^{-},Y^{+}]-[X,Y]^{-}=2g\nabla_{X}Y\in\Omega^{1}\]
_where \(\nabla\) is a connection which preserves the metric \(g\) and has skew torsion \(-H\)._
* **Remark:** Interchanging the roles of \(V\) and \(V^{\perp}\), we get a connection with torsion \(+H\).
* **Proof:** Since \(\pi[A,B]=[\pi A,\pi B]\), it is clear that \(\pi([X^{-},Y^{+}]-[X,Y]^{-})=0\) and so \([X^{-},Y^{+}]-[X,Y]^{-}\) is a one-form.
Put \(\Delta_{X}Y=[X^{-},Y^{+}]-[X,Y]^{-}\). Then
\[\Delta_{fX}Y = [fX^{-},Y^{+}]-[fX,Y]^{-}\] \[= f[X^{-},Y^{+}]-(Yf)X^{-}-f[X,Y]^{-}+(Yf)X^{-}\] \[= f\Delta_{X}Y\]
where we have used (3) together with \((X^{-},Y^{+})=0\).
Similarly
\[\Delta_{X}fY = [X^{-},fY^{+}]-[X,fY]^{-}\] \[= (Xf)Y^{+}+f[X^{-},Y^{+}]-(Xf)Y^{-}-f[X,Y]^{-}\] \[= f\Delta_{X}Y+Xf(Y^{+}-Y^{-})\] \[= f\Delta_{X}Y+(Xf)2gY\]
where we have used half the difference of the two splittings to define the metric as in Proposition 1.
These two expressions show that \(\Delta_{X}Y=2g\nabla_{X}Y\) for some connection \(\nabla\) on \(T\). We next show it preserves the metric.
If \(\xi\) is a \(1\)-form then \(i_{Z}\xi=2(\xi,Z^{+})\). We therefore have
\[g(\nabla_{X}Y,Z)+g(Y,\nabla_{X}Z)=(\Delta_{X}Y,Z^{+})+(Y^{+},\Delta_{X}Z).\] | [] |
|
BI-TP 2005/30
**The impact of QCD plasma instabilities**
**on bottom-up thermalization**
Dietrich Bodeker 1
Footnote 1: e-mail: [email protected]
_Fakultat fur Physik, Universitat Bielefeld, 33615 Bielefeld, Germany_
**Abstract**
QCD plasma instabilities, caused by an anisotropic momentum distributions of the particles in the plasma, are likely to play an important role in thermalization in heavy ion collisions. We consider plasmas with two different components of particles, one strongly anisotropic and one isotropic or nearly isotropic. The isotropic component does not eliminate instabilities but it decreases their growth rates. We investigate the impact of plasma instabilities on the first stage of the "bottom-up" thermalization scenario in which such a two-component plasma emerges, and find that even in the case of non-abelian saturation instabilities qualitatively change the bottom-up picture. | [] |
|
**The Conformal Penrose Limit and the Resolution of the pp-curvature Singularities**
R. Guven1
Footnote 1: Present address: Department of Physics, Iş\(\imath\)k University, Kumbaba Mevkii, Şile, Istanbul 34980, Turkey
_Department of Mathematics, Bogazici University, Bebek, Istanbul 34342, Turkey._
_ABSTRACT___
We consider the exact solutions of the supergravity theories in various dimensions in which the space-time has the form \(M_{d}\times S^{D-d}\) where \(M_{d}\) is an Einstein space admitting a conformal Killing vector and \(S^{D-d}\) is a sphere of an appropriate dimension. We show that, if the cosmological constant of \(M_{d}\) is negative and the conformal Killing vector is space-like, then such solutions will have a conformal Penrose limit: \(M^{(0)}_{d}\times S^{D-d}\) where \(M^{(0)}_{d}\) is a generalized \(d\)-dimensional AdS plane wave. We study the properties of the limiting solutions and find that \(M^{(0)}_{d}\) has 1/4 supersymmetry as well as a Virasoro symmetry. We also describe how the pp-curvature singularity of \(M^{(0)}_{d}\) is resolved in the particular case of the \(D6\)-branes of \(D=10\) type IIA supergravity theory. This distinguished case provides an interesting generalization of the plane waves in \(D=11\) supergravity theory and suggests a duality between the \(SU(2)\) gauged \(d=8\) supergravity theory of Salam and Sezgin on \(M^{(0)}_{8}\) and the \(d=7\) ungauged supergravity theory on its pp-wave boundary.
## 1 | [] |
|
HI 022103.9+270204: marginal ALFA detection, marginal LBW confirmation.
HI 022224.8+262552: previously measured opt cz=11179.
HI 022335.8+271851: marginal detection corroborated by previous HI det at cz=10645.
HI 022340.2+270927: previous HI det at cz=10647.
HI 022348.9+272848: previous HI det at cz=10706.
HI 022355.8+270618: previous HI det at cz=5470.
HI 022405.6+263900: sev. small LSB features in field, most notably: (a) 022402.0+263958 and (b) 022408.9+263952; optical id ambiguous.
HI 022459.4+260314: marginal detection corroborated by previous HI det at cz=10080.
HI 022533.4+264458: previous HI det at cz=10348.
HI 022538.8+271709: previous HI det at cz=8984.
HI 022558.5+271607: 2MASS galaxy with measured opt. cz=10493; in group with NGC 916 at 022547.6+271432, cz=9614. Marginal HI parameters.
HI 022609.4+273549: previous HI det at cz=9980.
HI 022617.1+260750: peculiar (double) gal (CGCG 483-047) at opt. cz=10128. Marginal HI det.
HI 022620.2+271315: Very marginal ALFA detection, corroborated by previous HI det at cz=10357.
HI 022629.9+273937: previous HI det at cz=9710.
HI 022632.1+274941: previous HI det at cz=9621.
HI 022741.1+271328: marginal detection corroborated by previous HI det at cz=5123.
HI 022742.7+261406: previous HI det at cz= 9504.
HI 022745.2+263507: previous HI det at cz=9789.
HI 022745.2+263507: merging system, tidal features.
HI 022751.5+275429: previously measured opt cz=10468.
HI 022816.3+261854: previous HI det at cz=5237.
HI 023052.0+261047: weak det, corroborated by previous HI det at cz=13289.
HI 023103.8+274053: previous HI det at cz=4587.
HI 023137.5+261010: previous HI det at cz=10869.
HI 023851.0+275109: previous HI det at cz=4587.
HI 024416.4+260648: opt id is galaxy in compact group HCG 020, at anomalous opt cz=10561 (other galaxies in group near 14500 km s\({}^{-1}\)). Marginal det.
HI 024600.5+280145: previous HI det at cz=7954.
HI 024609.7+270247: previous HI det at cz=5728. | [] |
|
scenario without any phase transition) has not been considered here, because we feel this scenario is not realistic at RHIC collision at \(\sqrt{s_{NN}}=200\) GeV. The other EOS, thermal quasi-particle model considered in [73] gives \(T_{i}\) more than 300 MeV. Therefore, \(T_{i}=300\) MeV obtained here for type II EOS is the conservative lower limit. Photon productions from thermal QCD and \({\cal{N}}=4\) SYM have been compared to the data. In both the cases similar values of the initial temperatures are obtained.
The extracted average temperature (\(T_{av}\)) from the slope of the photon \((p_{T})\) spectra is found to be \(\sim\) 265 MeV for the \(p_{T}\) range between 1.25 to 2.25 GeV where thermal contributions dominate. When the effects of flow is ignored (by putting \(v_{r}=0\) for all time) the 'true' average temperature is found to be 215 MeV. The initial tempe
Figure 5: Direct photon spectra at RHIC energies measured by PHENIX Collaboration. Solid (dotted) line depicts the pQCD + thermal (thermal) photon yield. Thermal photons from QGP phase is obtained from SYM theory [22]. Here \(T_{i}\) = 300 MeV and \(\tau_{i}=0.5\) fm. Type II EOS is used to obtain the thermal contributions. | [
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# Simulation of guiding of multiply charged projectiles through insulating capillaries
K. Schiessl
Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
W. Palfinger
Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
K. Tokesi
Institute of Nuclear Research of the Hungarian Academy of Sciences, (ATOMKI), H-4001 Debrecen, P.O.Box 51, Hungary Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
H. Nowotny
Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
C. Lemell
[email protected] Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
J. Burgdorfer
Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
(October 12, 2023)
###### Abstract
Recent experiments have demonstrated that highly charged ions can be guided through insulating nanocapillaries along the direction of the capillary axis for a surprisingly wide range of injection angles. Even more surprisingly, the transmitted particles remain predominantly in their initial charge state, thus opening the pathway to the construction of novel ion-optical elements without electric feedthroughs. We present a theoretical treatment of this self-organized guiding process. We develop a classical trajectory transport theory that relates the microscopic charge-up with macroscopic material properties. Transmission coefficients, angular spread of transmitted particles, and discharge characteristics of the target are investigated. Partial agreement with experiment is found.
pacs: 34.50.Dy
## I | [] |
|
# Role of Orbitals in the Physics of Correlated Electron Systems
D.I.Khomskii
II. Physikalisches Institut,
Universitat zu Koln,
Zulpicher Str. 77,
50937 Koln, Germany
Rich properties of systems with strongly correlated electrons, such as transition metal (TM) oxides, is largely connected with an interplay of different degrees of freedom in them: charge, spin, orbital ones as well as crystal lattice. Specific and often very important role is played by orbital degrees of freedom. They can lead to a formation of different superstructures (an orbital ordering) which are associated with particular types of structural phase transitions -- one of very few examples where the microscopic origin of these transitions is really known; they largely determine the character of magnetic exchange and the type of magnetic ordering; they can also strongly influence many other important phenomena such as insulator-metal transitions (IMT), etc.
In this comment I will try to shortly summarize the main concepts and discuss some of the well-known manifestations of orbital degrees of freedom, but will mostly concentrate on a more recent development in this field. More traditional material is covered in several review articles [1, 2, 3]. Although I tried to cover the main new development in this area, the choice of topics of course is influenced by my own interests; other people probably would have stressed other parts of this big field. | [] |
|
where the \(*\)-product is defined by,
\[f*g(x)=e^{\frac{i}{2}\theta^{ij}\partial_{i}^{y}\partial_{j}^{z} }f(y)g(z)\mid_{y=z=x}\] (7)
\(\hat{F}_{kl}\) is the noncommutative field strength, and is related to the ordinary field strength, \(F_{kl}\) by,
\[\hat{F}_{kl}=F_{kl}+\theta^{ij}(F_{ki}F_{lj}-A_{i}\partial_{j}F_{ kl})+{\cal O}(F^{3})\] (8)
and,
\[\hat{F}_{kl}=\partial_{k}\hat{A}_{l}-\partial_{l}\hat{A}_{k}-i \hat{A}_{k}*\hat{A}_{l}+i\hat{A}_{l}*\hat{A}_{k}\] (9)
We now include world-sheet fermions. The action for the fermions coupled to \(B\)-field is given by,
\[S_{F}=\frac{i}{4\pi\alpha^{{}^{\prime}}}\int_{\Sigma}g_{MN}\bar{ \psi}^{M}\rho^{\alpha}\partial_{\alpha}\psi^{N}-\frac{i}{4}\int_{\partial \Sigma}B_{MN}\bar{\psi}^{N}\rho^{0}\psi^{M}\] (10)
The full action including the bosons and the fermions with the bulk and the boundary terms are invariant under the following supersymmetry transformations,
\[\delta X^{M} = \bar{\epsilon}\psi^{M}\] \[\delta\psi^{M} = -i\rho^{\alpha}\partial_{\alpha}X^{M}\epsilon\] (11)
We now write down the boundary equations by varying (10) with the following constraints,
\[\delta\psi^{M}_{L}=\delta\psi^{M}_{R}\mid_{\sigma=\pi}\mbox{ \hskip 14.45377ptand\hskip 14.45377pt}\delta\psi^{M}_{L}=-(-1)^{a}\delta\psi^{ M}_{R}\mid_{\sigma=0}\] (12)
where, \(a=0,1\) gives the NS and the R sectors respectively This gives the following boundary equations, | [] |
|
* (30) Hida K and Affleck I 2005 _cond-mat/0501697_
* (31) White S R 1992 _Phys. Rev. Lett._**69** 2863 White S R 1993 _Phys. Rev. B_**48** 10345
* (32) Kitazawa A and Okamoto K 1999 _J. Phys.:Condens. Matter_**11** 9765 | [] |
|
# Braneworld Cosmological Perturbation Theory at Low Energy
Jiro Soda
[email protected]
Sugumi Kanno
[email protected] Department of Physics, Kyoto University, Kyoto 606-8501, Japan
(October 12, 2023)
###### Abstract
Homogeneous cosmology in the braneworld can be studied without solving bulk equations of motion explicitly. The reason is simply because the symmetry of the spacetime restricts possible corrections in the 4-dimensional effective equations of motion. It would be great if we could analyze cosmological perturbations without solving the bulk. For this purpose, we combine the geometrical approach and the low energy gradient expansion method to derive the 4-dimensional effective action. Given our effective action, the standard procedure to obtain the cosmological perturbation theory can be utilized and the temperature anisotropy of the cosmic background radiation can be computed without solving the bulk equations of motion explicitly.
pacs: 04.50.+h, 98.80.Cq, 98.80.Hw
## I | [] |
|
J.L. is supported by NNSF of China (10474008). B.W. is supported by the "BaiRen" program of the Chinese Academy of Sciences and the 973 project (2005CB724508).
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|
\) GeV the maximum cross section is obtained for \(\theta\leq 3\pi/4\) and \(\xi\in[\pi/10,\pi/5]\), and the minimum at around \(3\pi/4\leq\theta\leq 7\pi/4\) and \(\pi/4\leq\xi\leq\pi/2\). The CP-odd phase effects are slightly smaller for constructive interference: around \(10\%(30\%)\) for the constructive (destructive) case. At \(\sqrt{s}=1000\) GeV, there is almost no destructive interference in the entire plane. The cross section can get enhanced at most by \(20\%\) in three different intervals shown in Fig. 5. The \(\chi^{+}_{1}\chi^{-}_{4}\) case is similar to the \(\chi^{+}_{1}\chi^{-}_{1}\) or \(\chi^{+}_{2}\chi^{-}_{2}\) and the \(\theta\) values for getting a maximum (minimum) value for the cross section are the same (around \(\pi/2\) for constructive and \(3\pi/2\) for destructive) each at different center of mass energies, 500 and 1000 GeV. The enhancement can be more than \(50\%\) for both energies. The destructive effects cannot exceed \(20\%\) at \(\sqrt{s}=1000\) GeV, but become more than \(60\%\) at \(\sqrt{s}=500\) GeV.
Figure 3: The CP phase dependence of the cross section of the process \(e^{+}e^{-}\to\chi^{+}_{1}\chi^{-}_{1}\). The left (right) figure is at \(\sqrt{s}=500(1000)\) GeV. All the other parameters of the model are chosen as stated in Section III. Also, \(M_{Z_{R}}=500\) GeV, \(\Gamma_{Z_{R}}=20\) GeV, \(M_{\tilde{\nu}_{L}}=150\) GeV and \(M_{\tilde{\nu}_{R}}=600\) GeV. The values for the contours are given in pb. | [
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be replaced with a term of the form \(\rho/\rho_{0}\), where \(\rho\) is the adjusted density, and \(\rho_{0}\) is the reference density. Alternatively, the original approximation in Equation 13 may be used with the altitude \(H\) being the density altitude.
Table 1 shows how the top 100 meter performances are re-ordered according to the given approximation. These are compared with the original back-of-the-envelope approximations given in Reference [14], based exclusively on altitude. The weather conditions have been taken from the NC DC database [24]. The relative humidity has been calculated from the mean dew-point. The cited temperature is the maximum temperature recorded on that particular date, which is assumed to be reflective of the conditions near the surface of the track (since the surface reflection and re-emission of heat from the rubberized material usually increases the temperature from the recorded mean). Typical trackside temperatures reported in [25] support this argument. For example, the surface temperature during the 100 m final in Atlanta (9.84 s, +0.7 \(~{}{\rm ms}^{-1}\) performance in Table 1) was reported as 27.8\({}^{\circ}{\rm C}\).
The effects of density altitude are for the most part overshadowed by wind effects and are generally within 0.01 s of each other after rounding to two decimal places. However, larger variations are observable in certain cases. The most notable differences are in the 9.80 s performance (Maurice Greene, USA) at the 1999 World Championships in Seville, ESP, due to the unusually high temperature, as well as the 9.85 s performance in Ostrava, CZE (Asafa Powell, Jamaica). The temperature measurement during the latter was reportedly a chilly 10\({}^{\circ}{\rm C}\) and humidity levels near 100%.
At the time of writing of this manuscript, the world record in the men's 100 meter sprint is 9.77 seconds by Asafa Powell, set on 14 June 2005 in | [] |
|
only when the two beams counter propagate, and the coupling laser has a larger wavenumber \(k_{2}\) than the probe laser \(k_{1}\), AT splitting of the same magnitude can be observed independently of the molecular/atomic weight or the temperature of the system. In this region only a moderate coupling field Rabi frequency is required to split the experimental Optical Optical Double Resonance (OODR) spectrum and to observe the AT effect independent of the Doppler linewidth \(\Delta\nu_{D}\). In order to explain this be
Figure 4: Threshold Rabi frequency \(\Omega_{2}^{T}\) of the coupling field, as a function of the ratio \(k_{1}/k_{2}\). AT splitting can be observed only in region II. The \(\Omega_{2}^{T}\) value is much smaller for \(-1<k_{1}/k_{2}<0\) than for \(-1>k_{1}/k_{2}\) or \(k_{1}/k_{2}>0\) (co-propagating geometry). The two dots indicate the \(k_{1}/k_{2}\) ratio of our experiments. | [
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f water vapor more clear.
The reciprocal of the modified efficiency \(\frac{1}{\varphi_{i}(p)}\) for air in \(\frac{1}{\%}\) are plotted as a function of pressure in Figure 3. The results of LS fitting to eq.(2) are shown by solid (dry), dashed (21%) and dotted (56%) lines.
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Figure 3: The reciprocal of the modified efficiency \(\frac{1}{\varphi_{i}(p)}\) in dry and damp air are plotted as a function of pressure. The LS-fit lines are also shown.
Figure 2: The ratio of \(\epsilon\) at SH=0, 3.0 and 8.0 g/kg to that of \(\epsilon\) of dry air(SH=0) are plotted as a function of SH at three pressures. | [
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**A lattice scheme for**
**stochastic partial differential equations**
**of elliptic type in dimension \(d\geq 4\)**
\begin{tabular}{}
\end{tabular}
**Abstract:** We study a stochastic boundary value problem on \((0,1)^{d}\) of elliptic type in dimension \(d\geq 4\), driven by a coloured noise. An approximation scheme based on a suitable discretization of the Laplacian on a lattice of \((0,1)^{d}\) is presented; we also give the rate of convergence to the original SPDE in \(L^{p}(\Omega;L^{2}(D))\)-norm, for some values of \(p\).
_Keywords:_ stochastic partial differential equations; numerical approximations; coloured noise
_2000 MSC:_ 60H15, 60H35, 35J05
* \({}^{\dagger}\)Partially supported by the grant BFM 2002-04013-C02-02 from the Direccion General de Universidades, Ministerio de Educacion y Ciencia, Spain.
* \({}^{*}\)Supported by the grant BFM 2003-01345 from the Direccion General de Investigacion, Ministerio de Educacion y Ciencia, Spain.
## 1 | [] |
|
independent_ quantum theory of gravity [3], [4]. It is discussed that the spacetime metric is coupled to matter sources through Einstein equations and quantum theory applies to these matter sources. Thus, any attempt to treat gravity classically leads to serious problems and the only way to avoid these difficulties is to treat the spacetime metric in a probabilistic fashion which means quantum gravity.
In what follows, we will try to develop a new idea of quantum gravity which may shed light on the current problems in quantizing gravity, cosmological constant and hierarchy problems. We emphasize that this idea is still far from being complete, but it is hoped that it can stimulate further investigations in this direction.
## 2 Gravitational complementary principle
The so-called complementary principle, introduced by Bohr, is one of the most important basis of quantum mechanics. This fundamental principle in mathematical language is equal to the Heisenberg uncertainty principle which leads to the basic commutation relations on which the standard quantum mechanics stands. In quantum mechanics, position and momentum, just like particle behavior and wave aspects of a system respectively, are complement properties of the system and the theory _does not admit the possibility of an experiment in which both could be established simultaneously_. In this way, Bohr complementary principle BCP resolved some serious difficulties in the advent of quantum mechanics. The two slit experiment was one of confusing behaviors of nature which was satisfactory described by this principle. The prescription was to describe this experiment completely by the wave aspect and avoid any particle behavior. In fact, any attempt to describe it by the particle behavior would destroy | [] |
|
# \(\bm{D_{s}(2317)}\) as a four-quark state in QCD sum rules
Hungchong Kim
[email protected] Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
Yongseok Oh
[email protected] Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, U.S.A.
###### Abstract
We perform a QCD sum rule study of the open-charmed \(D_{s}(2317)\) as a four-quark state. Using the diquark-antidiquark picture for the four-quark state, we consider four possible interpolating fields for \(D_{s}(2317)\), namely, scalar-scalar, pseudoscalar-pseudoscalar, vector-vector, and axial-vector-axial-vector types. We test all four currents by constructing four separate sum rules. The sum rule with the scalar-scalar current gives a stable value for the \(D_{s}\) mass which qualitatively agrees with the experimental value, and the result is not sensitive to the continuum threshold. The vector-vector sum rule also gives a stable result with small sensitivity to the continuum threshold and the extracted mass is somewhat lower than the scalar-scalar current value. On the other hand, the two sum rules in the pseudoscalar and axial-vector channels are found to yield the mass highly sensitive to the continuum threshold, which implies that a four-quark state with the combination of pseudoscalar-pseudoscalar or axial-vector-axial-vector type would be disfavored. These results would indicate that \(D_{s}(2317)\) is a bound state of scalar-diquark and scalar-antidiquark and/or vector-diquark and vector-antidiquark.
pacs: 14.40.Lb, 11.55.Hx, 12.38.Lg +
[FOOTNO | [] |
|
4.3 The 'transit candidates'
The folded light curves of the 18 binaries that passed the basic steps of transit selection are shown in Fig. 4. Except for the following 3 stars, all exhibit obvious signs of stellar binary components (more or less well-defined secondary eclipse, uneven distribution of the eclipses due to eccentric orbit). Closer inspection of the remaining 3 stars shows the following:
* ***052730.57-695:** The secondary eclipse preceding the primary is only marginally visible (it is somewhat more easily identified in the 1P-folded light curve).
* ***051734.54-692:** Observed in fields #7 and 8 (field #8 data are shown). The light curve from field #7 contains more data and yields four times longer period than the one in field #8. The 4P-folded light curve shows two clear eclipses of different depth.
* ***051108.69-691:** Period is almost exactly 8 days. Folding with P/2 definitely shows eccentricity, as the two minima are offset from each other in phase.
5. Conclusions
* *Detailed light curve analysis combined with constraints posed by theoretical transit lengths shows that in a hypothetical blending scenario NONE of the 2495 binaries in the OGLE LMC database could be false positive planetary candidates.
* *Chance of confusion due to blending is especially low among short-period (\(P<1.5\) d) binaries.
* *Since for the same signal the signal-to-noise ratio scales with the square of the standard deviation (\(\sigma\)) of the noise and the present OGLE sample has fairly low \(\sigma\), in current wide field surveys one needs to gather a large number of data points per star in order to reach the above level of confidence in selecting false positives.
* *This test indicates that careful analysis of the light curves (if data quality permits) indeed leads to the elimination of large number of binary blends, thereby narrowing down the list of potential planetary candidates. | [
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(2)
the inhomogeneous Maxwell equations become
C,r=4pr2Jue2b (3)
and
C,u=-4pr2Jre2b, (4)
where the comma subscript represents partial derivative with respect to the indicated coordinate. The function \(C(u,r)\) is naturally interpreted as the charge within the radius \(r\) at time \(u\). Thus, the inhomogeneous field equations entail the conservation of charge inside a sphere comoving with the fluid, expressed by
\[{\bf v}(C)=0.\] (5)
Let \(\omega\) be the velocity of matter as seen by a Minkowskian observer moving at \(-\omega\) with respect to the local comoving frame; the matter velocity in radiation coordinates is then given by
\[\frac{dr}{du}=\frac{V}{r}\frac{\omega}{1-\omega}.\] (6)
Introducing the mass function by
\[V=e^{2\beta}(r-2\tilde{m}(u,r)+C^{2}/r),\] (7)
we can write the Einstein equations as
r+po21-o2+2oq1-o2=e-2b(CC,u/r-m~,u)4pr(r-2m~+C2/r)+m~,r-CC,r/r4pr2, (8)
r-po1+o-(1-o1+o)q=m~,r-CC,r/r4pr2, (9)
(1-o1+o)(r+p)-2(1-o1+o)q=b,r2pr2(r-2m~+C2/r), (10)
p=-14pb,ure-2b+18p(1-2m~/r+C2/r2)(2b,rr+4b,r2-b,r/r)
+18pr[3b,r(1-2m~,r)-m~,rr]+3b,r8pr(2CC,r/r-C2/r2)
+18pr2(C,r2+CC,rr-2CC,r/r). (11)
We describe the exterior space-time by the Reissner-Nordstrom-Vaidya metric
\[ds_{+}^{2}=\left(1-\frac{2m(u)}{r}+\frac{C_{T}^{2}}{r^{2}}\right)du^{2}+2du\, dr-r^{2}\left(d\theta^{2}+\sin{}^{2}\theta\,d\phi^{2}\right),\] (12)
where \(m(u)\) is the total mass and \(C_{T}\) the total charge. It can be shown that the junction conditions to match the metrics (1) and (12), across the moving | [] |
|
GeV. Even though the asymmetry is a lot bigger at \(\sqrt{s}=1000\) GeV than at \(\sqrt{s}=500\) GeV, it is more sensitive to the CP-odd phases for latter case. The change in \(A_{FB}\) can be as much as \(50\%\) and can even change sign (there is an almost symmetric pattern with respect to \(\theta=\pi\) line). However, the increase in \(A_{FB}\) can not exceed \(10\%\) at \(\sqrt{s}=1000\) GeV. For \(\theta=3\pi/2\) and \(\xi=\pi/2\), CP-odd phases effects cancel each other. Maximum constructive effects occur around \(\pi/2\) for both angles. As depicted in Fig. 8, the maximum asymmetry for \(e^{+}e^{-}\to\chi^{+}_{2}\chi^{-}_{2}\) is obtained at around \(\theta,\xi\sim\pi/2\) at both \(\sqrt{s}=500\) and \(1000\) GeV (around \(30\%\) and \(20\%\) increase, respectively). Maximum destructive interference effects are slightly smaller and take place for \(\theta\sim 3\pi/2,\xi\sim\pi/2\). The final set of graphs for \(A_{FB}\) is shown in Fig. 9 for the process \(e^{+}e^{-}\to\chi^{+}_{1}\chi^{-}_{2}\). At \(\sqrt{s}=500\) GeV, CP-odd phases have constructive effects in almost the entire range and the asymmetry can reach a maximum \(23\%\), for which \(A_{FB}\)
Figure 7: The CP phase dependence of the forward-backward asymmetry (\(A_{FB}\)) in the process \(e^{+}e^{-}\to\chi^{+}_{1}\chi^{-}_{1}\). The left (right) figure is at \(\sqrt{s}=500(1000)\) GeV. All the other parameters of the model are chosen as stated in Section III. Also, \(M_{Z_{R}}=500\) GeV, \(\Gamma_{Z_{R}}=20\) GeV, \(M_{\tilde{\nu}_{L}}=150\) GeV and \(M_{\tilde{\nu}_{R}}=600\) GeV. The values for the contours are given in percentage. | [
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In [5], we also noted that there should be another "self-T-duality" of the fibered \(F\)-theory. Consider F-theory on
\[X^{9}\times\begin{array}[]{c}{ 3}\\ \prod\\ { i=1}\end{array}S^{1}_{R}\] (20)
which is a special case of (19). Then this theory should have a \(2\)-brane \(M_{2}\) and a \(3\)-brane \(M_{3}\) where the relationship (9) becomes
\[M_{3}=M_{2}\times S^{1}.\]
In particular, then, \(M_{2}\) can be wrapped on
\[\begin{array}[]{c}{ 3}\\ \prod\\ { i=2}\end{array}S^{1}_{R}\]
and \(M_{3}\) on
\[\begin{array}[]{c}{ 3}\\ \prod\\ { i=1}\end{array}S^{1}_{R}.\]
If we shrink the radius of the first copy of \(S^{1}_{R}\) to \(0\), then, \(M_{3}\) will lose a dimension, but \(M_{2}\) will expand by the new dimension, and we see than that the system \((M_{2},M_{3})\) is self-dual.
Now let us look at this from the point of view of signatures, as considered in Hull [19]. Hull constructs IIA and IIB-like theories as well as M-theory in a variety of signatures. Although these theories pass a number of consistency checks, proposing those theories and then checking their consistency is not the main point of [19]. Rather, the main point is that these theories _must exist_ if we make one simple assumption, namely that in a physical spacetime, the time dimension can be compact (i.e. topologically an \(S^{1}\)). This assumption seems to be widely accepted now, in fact many arguments are only strictly correct if the entire spacetime manifold is compact. Given this assumption, the theories of [19] are simply constructed by applying T-duality in the time-like dimensions.
In particular, following [19], if we take a T-dual of a signature \((9,1)\) type IIB theory in the timelike dimension, we obtain a theory in signature \((9,1)\) denoted \(\text{IIA}^{*}\). It differs from IIA in that in the low energy action, the signs in the RR-sector are reversed. Accordingly, instead of branes which are | [] |
|
world volumes of dimension \((2k-1,1)\) in type IIA, we have branes which are world volumes of dimension \((2k,0)\), i.e. time instantons.
Hull continues to examine the theory \(\text{IIA}^{*}\), in particular its strong coupling limit. He concludes that although the strong coupling limit is \(11\)-dimensional, because of the sign reversal, the additional dimension is in fact time-like, i.e. of signature \((9,2)\). He calls this theory \(\text{M}^{*}\).
Now let us look at this theory from the point of view of [5]. In particular, our \(\text{IIA}^{*}\)-theory is on a spacetime of the form
\[X^{9}\times S^{1}_{NS,t}\] (21)
where \(X^{9}\) is space-like, and the subscripts \(NS\), \(t\) stand for 'Neveu-Schwarz' and 'timelike', respectively. Therefore, \(\text{M}^{*}\) is on
\[X^{9}\times S^{1}_{NS,t}\times S^{1}_{R,t}.\] (22)
Now F-theory is on a spin cobordism of the manifold (22) to \(0\). This manifold is of the form
\[X^{9}\times E^{\prime}_{t}\times S^{1}_{R,t}\] (23)
where \(E\) is a \(2\)-dimensional timelike (signature \((0,2)\)-) cobordism of \(S^{1}_{NS,t}\) with \(0\).
Now let us apply the technique of [5] of shrinking the boundary of \(E^{\prime}_{t}\) to a point (while preserving the bulk). In this limit, we obtain a theory on
\[X^{9}\times E_{t}\times S^{1}_{R,t}\] (24)
where \(E_{t}\) is \(E^{\prime}_{t}\) with a disk attached. It is possible to choose \(E_{t}\) to be any Riemann surface, in particular
\[E_{t}=S^{1}_{R,t}\times S^{1}_{R,t},\] (25)
in which case (24) becomes
\[X^{9}\times\begin{array}[]{c}{ 3}\\ \prod\\ { i=1}\end{array}S^{1}_{R,t}.\] (26)
The spacetime \(X^{9}\times S^{1}_{R,t}\) where \(S^{1}_{R,t}\) is the first factor (26) is now T-dual to the original spacetime (21), and is therefore of type IIB. Therefore, we have | [] |
|
# PP-Wave Light-Cone Free String Field Theory at Finite Temperature
M. C. B Abdalla1, A. L. Gadelha2 and Daniel L. Nedel3
Footnote 1: [email protected]
Footnote 2: [email protected]
Footnote 3: [email protected]
Instituto de Fisica Teorica, Unesp, Pamplona 145, Sao Paulo, SP, 01405-900, Brazil
###### Abstract
In this paper, a real-time formulation of light-cone pp-wave string field theory at finite temperature is presented. This is achieved by developing the thermo field dynamics (TFD) formalism in a second quantized string scenario. The equilibrirum thermodynamic quantities for a pp-wave ideal string gas are derived directly from expectation values on the second quantized string thermal vacuum. Also, we derive the real-time thermal pp-wave closed string propagator. In the flat space limit it is shown that this propagator can be written in terms of Theta functions, exactly as the zero temperature one. At the end, we show how supestrings interactions can be introduced, making this approach suitable to study the BMN dictionary at finite temperature.
## I | [] |
|
* [17] V.V. Petrov, Limit Theorems of Probability Theory, Clarendon Press, Oxford, 1995.
* [18] N.E. Ratanov, Stabilization of statistical solutions of second order hyperbolic equations, _Russian Mathematical Surveys_**39** (1984), no.1, 179-180.
* [19] N.E. Ratanov, Asymptotic normality of statistical solutions of the wave equation, _Moscow University Mathematics Bulletin_**40** (1985), no.4, 77-79.
* [20] Z. Rieder, J.L. Lebowitz, E. Lieb, Properties of a harmonic crystal in a stationary nonequilibrium state, _J. Math. Phys._**8** (1967), no.5, 1073.
* [21] H. Spohn, J. Lebowitz, Stationary non-equilibrium states of infinite harmonic systems. _Comm. Math. Phys._**54** (1977), no. 2, 97-120.
* [22] B.R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989.
* [23] M.I. Vishik, A.V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, Dordrecht, 1988. | [] |
|
C. G. . Callan, C. Lovelace, C. R. Nappi and S. A. Yost, "String Loop Corrections To Beta Functions," Nucl. Phys. B **288** (1987) 525.
* [25] J. Polchinski, "String theory. Vol. 1: An introduction to the bosonic string," Cambridge University Press, Cambridge, (1998)
* [26] S. Kachru and E. Silverstein, "4d conformal theories and strings on orbifolds," Phys. Rev. Lett. **80** (1998) 4855 [arXiv:hep-th/9802183].
* [27] A. Hashimoto and N. Itzhaki, "Non-commutative Yang-Mills and the AdS/CFT correspondence," Phys. Lett. B **465** (1999) 142 [arXiv:hep-th/9907166]. J. M. Maldacena and J. G. Russo, "Large N limit of non-commutative gauge theories," JHEP **9909** (1999) 025 [arXiv:hep-th/9908134]. | [] |
|
ing coefficient. In the same way, the own structure of the sentences and the frequency of the words along the whole text affects C and other indices as well. In order to evaluate the role of these main agents in the determination of the indices, we have re-analyzed the filtered texts after submitting them to four shuffling processes, which are so characterized: \((SA)\) In this process, the beginning and the end of sentences are kept fixed, however the position of the words along the whole text is randomly changed. This procedure breaks completely the structure of concepts in the sentences, but their lengths and the frequency of the words in the whole text are kept unchanged. \((SB)\) Here, the original sequence of the words along the text remains unchanged, but all sentences are forced to have the same average length, as obtained from the analysis of the unperturbed text. \((SC)\) As in \(SA\), the beginning and the end of sentences are kept fixed, and words are randomly chosen from the same vocabulary of the text. However, the words frequency is changed to a white distribution, so that all the words have the same choice probability. \((SD)\) Here, we randomly distribute the same number of links as in the network of the filtered text among the nodes, i.e., obtaining an Erdos-Renyi network.
A summary of results is also included in the Table I. They show that the networks are affected in different ways but, as expected, for all but the \(SD\) procedure, they are very far from random networks. The indices for \(SA\) and \(SB\) are not significantly altered. In the first case it show
Figure 4: Clustering coefficient and average shortest path length evolution, according to the number of sentences in James Joyce’s Ulysses, in the original English version and in a Portuguese translation.
Figure 3: (a) Behavior of C as function of the number of vertices compared with two texts. (b) The same index as function of the number of sentences, for 15 texts. | [
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#### Data arrays
If the triple product amplitudes are meaningless, as is the case for COAST data, NULL values for T3AMP may be used. The closure phases should still be treated as valid.
NWAVE is the number of distinct spectral channels recorded by the single (possibly "virtual") detector, as given by the NAXIS2 keyword of the relevant OI_WAVELENGTH table.
#### Complex visibility and visibility-squared UV coordinates
UCOORD, VCOORD give the coordinates in meters of the point in the UV plane associated with the vector of visibilities. The data points may be averages over some region of the UV plane, but the current version of the standard says nothing about the averaging process. This may change in future versions of the standard.
#### Triple product UV coordinates
The U1COORD, V1COORD, U2COORD, and V2COORD columns contain the coordinates of the bispectrum point - see Sec. 4 for details. Note that U3COORD and V3COORD are implicit.
The corresponding data points may be averages in (bi-) spatial frequency space, but this version of the standard does not attempt to describe the averaging process. | [] |
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